HK1077902B - Method and apparatus for generating m-degree forms in a n-dimension space - Google Patents

Description

Method and apparatus for generating M-degree graphics in N-dimensional space

Technical Field

The present invention relates to the field of computer graphics fabrication, and more particularly, to the field of molded graphics.

Background

The rendering of objects in space is not a new technique. For many centuries, people have attempted to depict graphics or shapes in space for various purposes, ranging from architectural to artistic. Most modern computer graphics fabrication systems construct complex three-dimensional graphics using very simple rendering graphics, i.e., a large number of combined polygons. Polygons are mathematically simple to express and are applied in the generation of display output.

A graphic can be depicted by a surface in two different ways.

The first way to depict graphics is by applying a parametric surface. Parametric surfaces include, for example, Bezier curves, sparses, non-uniform rational B-sparses (nurbs), and the like. This first way of rendering graphics is widely used by computers, in fact, parametric surfaces are tied to the underlying polygon structure of the graphics. However, it will be appreciated that these surfaces are approximate. To overcome such limitations, a large number of parametric surfaces may be incorporated to depict more complex patterns.

Those skilled in the art will appreciate the fact that this approach is resource intensive. The use of such simplified "raw graphics" to construct a depiction of a complex object requires a large number of specifications for the object, as well as extensive data processing to generate and display the graphics. It will also be appreciated that such methods often provide only approximate graphics. Especially when a complex pattern has to be generated. When such figures are enlarged, the border lines between the original figures tend to be noticeable.

A second way of depicting the graphics is by applying an implicit surface. The Implicit surface may be algebraic or geometric, as explained in the chandra jit Bajaj et al "Implicit surface guide to Implicit Surfaces" article published by MorganKaufmann Publishers (Publishers, Inc), the contents of which are incorporated herein by reference.

For example, a sphere centered at the origin may be given its algebraic equation x²+y²+z²＝r²Defined as an algebraic form, the size of the ball is determined by setting its radius 'r'. The algebraic equation for a sphere with its central coordinates (a, b, c) is (x-a)²+(y-b)²+(z-c)²＝r². The ball can be defined as a geometric form by geometric characteristics. In this particular case, the radius and center are used to uniquely define the surface. When a basic portion of a figure is rendered by an algebraic expression, the figure is displayed at a high magnification, and its details are not lost. In the case of a ball, it will also be appreciated that the figure is depicted with four values, namely a, b, c, r, whereas a parametric representation based on Bezier's original figure requires a much larger number of elements to define the figure.

Those skilled in the art will note that it is difficult to provide an equation for a high-order surface that is used to generally depict a complex figure in practice. While a common equation can be used to describe a sphere, it would still be a challenge to provide a common system that can provide an algebraic representation of a graph as defined by its geometric properties, such as center and radius. In most cases, the derivation of algebraic equations corresponding to a graph defined by geometric properties is not only computationally complex and time consuming, but also unpredictable and unstable, i.e. various algorithms may lead to unsolvable equations. While a combination of simpler algebraic forms can be used to produce shapes that depict graphics, it will be appreciated that such an approach is inefficient because it is also highly resource intensive. As a result, a large bandwidth is required for data conversion regarding the shape of a drawing. It will also be appreciated that when using algebraic form, it is generally not possible to imagine the shape.

The applicant Jean Francois Rotge in its paper "L' Arithmetique des Forms: the fifth chapter of Une introduction a la Logique de l 'Espace' indicates how to define a quadratic three-dimensional surface by its geometrical properties. The method proposed in this paper is basically unknown and is not applied to computer graphics and is based on a mathematical operation that cannot be generalized to higher order graphics. Quadratic patterns, such as spheres, ellipsoids, hyperboloids, etc., are also relatively simple compared to higher order patterns. By combining such secondary surfaces, a shape can be created that depicts a complex pattern. Unfortunately, because this method is limited to secondary patterning, the value of molding complex objects is limited.

In view of the above, there is a need for a method and apparatus that overcomes the above-mentioned deficiencies.

Disclosure of Invention

It is an object of the present invention to provide a method and apparatus for determining algebraic coefficients for a fixed m-degree graph in an n-dimensional space for improving the display of a computer graphics system. Preferably m is greater than 2 and n is typically 3. In the case of a four-dimensional graphic, the fourth dimension is preferably time, so that the displayed graphic is dynamic.

It is another object of the present invention to provide a method and apparatus for determining algebraic coefficients for a plurality of m-degree graphs in an n-dimensional space to produce a complex graph for improving the display of the complex graph in a computer graphics system.

It is another object of the present invention to provide a method and apparatus for determining algebraic coefficients for a fixed m-degree graph in an n-dimensional space for improving the display of a computer graphics system, which method and apparatus will operate with integer arithmetic.

It is a further object of the present invention to provide a framework for generating an m-degree graph.

It is a further object of the present invention to provide a method and apparatus that enables an operator to modify a shape in an interactive manner in a controlled manner, and to provide data identifying the modified shape.

It is a further object of the present invention to provide a method and apparatus that enables a user to modify an m-degree pattern in an n-dimensional space in a flexible manner.

According to a broad embodiment of the present invention, there is provided a computer graphics design system having an interface for defining a desired 3-dimensional figure by geometric characteristics, means for converting the geometric characteristics into an algebraic expression having a degree of rendering of the desired figure greater than 2, and means for providing a graphical output of the figure using the algebraic expression, wherein the means for converting is stable for all possible geometric characteristics. According to another broad embodiment of the present invention, there is provided a computer graphics design system having a user interface for defining a desired 3-dimensional graphic by geometric properties, the user interface imposing a constraint on a value defining the geometric properties using a reference cell, the system further comprising means for converting the geometric properties into an algebraic expression having a degree greater than 2 for rendering the desired graphic, means for providing a graphic output of the graphic using the algebraic expression, means for providing the graphic output using the algebraic expression, and means for adjusting the reference cell relative to the desired graphic to further facilitate defining the graphic using the adjusted reference cell.

According to an embodiment of the present invention there is provided a method of generating an m-degree form using algebraic coefficients of equations defining the m-degree form in an n-dimensional space employing a plurality of points defining the form, the method comprising the steps of providing a skeleton comprising more than one geometric surface, each of the more than one geometric surface comprising at least one line, each of the at least one line comprising at least one point of the plurality of points; the method further includes the step of providing a solution equation with equations defining the m-degree graph, the solution equation depending on at least three points, the method including the step of causing each of the plurality of points to generate a corresponding equation by applying the solution equation to each of the plurality of points and to two points located on one of more than one geometric surface, the two selected points defining a line of the skeleton, each of the plurality of points being on the line; comprising the steps of solving the generated corresponding equations in an interactive manner among the generated equations processed in an order with each of the plurality of points, the order existing among the selected generated corresponding equations to a point located on a line having the highest number of points thereon located on at least one geometric surface of the framework on which the highest number of lines are located, to provide algebraic coefficients of the m-degree form, and the method comprises the step of generating the m-degree form with the provided algebraic coefficients of the m-degree form.

According to another embodiment of the present invention, there is provided an apparatus for generating an m-degree figure using algebraic coefficients of equations defining the m-degree figure in an n-dimensional space using a plurality of points defining the figure, the apparatus including an object memory, the object including more than one geometric surface, each of the more than one geometric surface including at least one line, each of the at least one line including at least one point of the plurality of points, the apparatus including a control point to algebraic coefficient converter unit receiving each of the plurality of points and a solution equation defined by the equations defining the m-degree figure and dependent on at least three points, the control point to algebraic coefficient converter unit generating a control point by applying the solution equation to each of the plurality of points and to two points located above one of the more than one geometric surface Corresponding equations, the two selected points defining a line of the skeleton, each of the plurality of points being on the line, and solving the resulting corresponding equations in an interactive manner to provide algebraic coefficients of the m-degree graph; the manner of interaction consists in processing the generated equations with each of the plurality of points in an order consisting in selecting the corresponding equation generated, the corresponding equation corresponding to a point located on a line having the highest number of points thereon, the line being located on at least one geometric surface of the frame having the highest number of lines thereon, the apparatus further comprising an output unit receiving algebraic coefficients of the provided m-degree figures and forming the m-degree figures.

According to another embodiment of the invention there is provided a method of monitoring deformation of a graph m times using algebraic coefficients of equations defining the graph m times in an n-dimensional space employing a plurality of points defining the graph, the method comprising the step of providing a framework in accordance with a planned modification of the graph, the framework comprising more than one geometric surface, each of the more than one geometric surfaces comprising at least one line, each of the at least one line comprising at least one point of the plurality of points, the method comprising the step of modifying at least one portion of the framework provided in accordance with the planned modification of the graph, the method comprising the step of providing a solution equation with equations defining the graph m times, the solution equation depending on at least three points, the method comprising for each of the plurality of points, applying the solution equation to the each point and to two points located above one of the more than one geometric surfaces A step of generating a corresponding equation, the two selected points defining a line of the skeleton, each of a plurality of points being on the line, the method comprising the steps of solving the generated corresponding equation in an interactive manner to provide algebraic coefficients of the m-degree graph; the manner of interaction consists in processing the generated equations with each of the plurality of points in an order consisting in selecting the corresponding equation generated, the corresponding equation corresponding to a point located on a line having the highest number of points thereon, the line being located on at least one geometric surface of the skeleton having the highest number of lines thereon, and the method comprising the step of outputting algebraic coefficients of the modified graph.

Drawings

Further features and advantages of the present invention will be readily apparent to those of ordinary skill in the art from the following detailed description of the invention taken in conjunction with the accompanying drawings.

FIG. 1a is a diagram showing one point in Cartesian geometry;

FIG. 1b is a diagram showing a point with a reference cell in the projection geometry;

FIG. 2 is a diagram showing a quadratic curve in two dimensions with a reference cell; four intersection points are necessary to completely define the quadratic curve;

FIG. 3 shows how a pole can be generated with an associated non-pole in one embodiment;

FIG. 4a shows the number of geometric conditions necessary for a quadric surface in a 3-dimensional space; in this embodiment of the invention three sections are defined, introducing a fixed number of lines and points for each defined section;

FIG. 4b shows the number of geometric conditions necessary for a cubic surface in a 3-dimensional space; in this embodiment of the invention four cross sections are defined, introducing a fixed number of lines and points for each defined cross section;

FIG. 4c shows the number of geometric conditions necessary for a quadric surface in a 3-dimensional space; in this embodiment of the invention five sections are defined, introducing a fixed number of lines and points for each defined section;

FIG. 5a shows a first embodiment of a cross-section of a quadric surface defining a quadric dome surface; in this embodiment, all the cross sections defining the quartic conic surface intersect on a line perpendicular to the horizontal plane of the quartic conic surface;

FIG. 5b shows a second embodiment of a cross-section of a quadric surface defining a quadric dome surface; in this embodiment, all the cross sections defining the quartic dome intersect on a parallel line included in the horizontal plane of the quartic dome;

FIG. 6 is a block diagram showing each element of the preferred embodiment of the present invention;

FIG. 7 is a flow chart showing various steps performed in a preferred embodiment of the present invention; according to a first step, providing data relating to a graphic; then according to the second step, modifying the control point of the graph; then according to the third step, calculating the algebraic coefficient of the modified graph;

FIG. 8 is a flow chart showing another embodiment of how data relating to a graphic is provided; in this embodiment, data is provided using existing control points;

FIG. 9 is a flow diagram showing another alternative embodiment of how data relating to a graphic is provided; in this alternative embodiment, the data is provided using existing algebraic equations which are then converted to control points;

FIG. 10 is a flow diagram showing another alternative embodiment of how data relating to a graphic is provided; providing data with an existing graphic in the alternative embodiment;

FIG. 11 is a flow diagram showing another alternative embodiment of how data relating to a graphic is provided; in this further alternative embodiment, the data is provided in a mixture of both algebraic and geometric information;

FIG. 12 is a flow chart showing another embodiment of how data relating to a graphic is provided; the control points entered in the preferred embodiment of the invention using the various cross sections in this embodiment provide data;

FIG. 13 is a flow chart showing how algebraic coefficients defining a graph are computed using control points of the graph;

FIG. 14 is a flow chart showing how poles are calculated with at least non-poles;

FIG. 15 is a flow chart showing how control points move in a line;

FIG. 16 is a flow chart showing how lines move;

FIG. 17 is a screen shot of a first quadric, which is an ovoid;

FIG. 18 is a screen shot showing three sections defining a first quadric, control points and a reference cell;

FIG. 19 is a screen shot showing a quadric surface and its three defined sections, control points and reference cells;

FIG. 20 is a screen shot of a second quadric, which is an ovoid;

FIG. 21 is a screen shot showing three sections of a second quadric, control points and reference cells;

FIG. 22 is a screen shot showing a second quadric surface, three sections of the second quadric surface, control points and a reference cell;

FIG. 23 is a screen shot of a third quadric, which is an ovoid;

FIG. 24 is a screen shot showing three sections of a third quadric, control points and reference cells;

FIG. 25 is a screen shot showing a third quadric, three sections of the third quadric, control points and a reference cell;

FIG. 26 is a screen shot of a fourth quadric, which is a sphere;

FIG. 27 is a screen shot showing three sections of the fourth quadric, control points and reference cells;

FIG. 28 is a screen shot showing a fourth quadric, three sections of the fourth quadric, control points and a reference cell;

FIG. 29 is a flowchart showing the steps performed in sequence to generate algebraic coefficients when the expression for the coefficients is known;

FIG. 30 is a graph showing the origin with λ, μ and ρ, σ;

FIG. 31 is a diagram showing different cross-sections for a quadric surface and how the cross-section used is positioned relative to a reference cell in a preferred embodiment of the invention;

FIG. 32 is a flow chart of how the equations are derived;

FIG. 33 is a flowchart of the steps to derive coefficient expressions;

FIG. 34a is a table of algebraic equations providing a quadric;

FIG. 34b is a table of algebraic equations providing a quadric;

FIG. 35 is a screen shot of 5 sections showing a quadric surface, all five sections intersecting at one edge of a reference cell;

FIG. 36 is a graph showing a quadric surface with 15 lines; 5 of the 15 lines define a first cross-section, 4 of the 15 lines define a second cross-section, 3 of the 15 lines define a third cross-section, 2 of the 15 lines define a fourth cross-section, and 1 of the 15 lines define a fifth cross-section;

FIG. 37 is a screen shot of a first cross-section showing a quadric surface, the first cross-section including 5 lines, all of the lines of the first cross-section intersecting at xI;

FIG. 38 is a screen shot of a second cross-section showing a quadric surface, the second cross-section including 4 lines, all of the lines of the second cross-section intersecting at xII;

FIG. 39 is a screen shot of a third section showing a quadric surface, the third section including 3 lines, all of the lines of the third section intersecting at xIII;

FIG. 40 is a screen shot of a fourth cross section showing a quadric surface, the fourth cross section including 2 lines, all of the lines of the fourth cross section intersecting at xIV;

FIG. 41 is a screen shot showing a fifth cross-section of a quadric surface, the fifth cross-section including 1 line intersecting the reference cell at xV;

FIG. 42 shows the system of equations Eq.2;

FIG. 43 shows the system of equations Eq.3;

FIG. 44 shows the system of equations Eq.5;

FIG. 45 shows equation system Eq.7;

FIG. 46 shows the system of equations Eq.8;

FIG. 47 shows the system of equations Eq.10;

FIG. 48 shows the system of equations Eq.11;

FIG. 49 shows the system of equations Eq.14;

FIG. 50 shows the system of equations Eq.17;

FIG. 51 shows the system of equations Eq.19;

FIG. 52 shows the system of equations Eq.21;

FIG. 53 shows the system of equations Eq.23;

FIG. 54 shows the system of equations Eq.24;

FIG. 55 shows the system of equations Eq.27;

FIG. 56A shows the system of equations Eq.29;

FIG. 56B shows the system of equations Eq.31;

FIG. 56C shows the system of equations Eq.33;

FIG. 56D shows the system of equations Eq.35;

FIG. 57 shows the system of equations Eq.36;

FIG. 58 shows the system of equations Eq.38;

FIG. 59 shows the system of equations Eq.40;

FIG. 60 shows the system of equations Eq.42;

FIG. 61A shows the system of equations Eq.44;

FIG. 61B shows the system of equations Eq.46;

FIG. 61C shows the system of equations Eq.48;

FIG. 62 shows the system of equations Eq.49;

FIG. 63 shows the system of equations Eq.53;

FIG. 64A shows the system of equations Eq.55;

FIG. 64B shows the system of equations Eq.57;

FIG. 65 shows the system of equations Eq.60;

FIG. 66 is a table showing various parameters showing the characteristics of lines of 1, 2 and 3 degrees polarity for 2, 3 and 4 degree algebraic surfaces;

FIG. 67 is a photograph of a screen showing a quadric surface and a pole and a non-pole located on a 1 st-order pole surface;

FIG. 68 is a screen shot showing a quadric surface and two poles and one non-pole located on a 2 nd pole surface;

FIG. 69 is a photograph of a screen showing a quadric surface and three poles and one non-pole located on a 3-pole surface;

FIG. 70 shows a decomposition of one determinant into a sum of smaller determinants;

FIG. 71 shows a decomposition of a determinant into a sum of smaller determinants in the case of a quadric surface;

FIG. 72 shows 10 small 2 x 2 determinant;

fig. 73 shows 10 3 × 3 small determinant; and

fig. 74 shows 54 × 4 small determinant.

Detailed Description

Brief introduction to projection geometry

The projection geometry originated from the work of Pappus and Gerard Desargue, the guidelines of the theory were created by the work of j.v. poncelet (1822), and the axiom basis thereof was established by k.g.c. von Staudt (1847). Mathematicians have adopted this clean or deductive approach, in which algebraic and analytic approaches are avoided and the processing method is purely geometric.

As is well known to those skilled in the art, m-th surfaces in N-dimensional space have an algebraic coefficient N_CIs equal to:

number N defining the geometric condition of the surface_GEqual to:

number N with algebraic coefficients on m-th surfaces in 3-dimensional space_CThe coefficient is therefore equal to:

for a quadric surface, m is 2, N_C10, and in the case of quadric, m is 4, N_C35. Number N of required geometric conditions_GEqual to:

thus, 9 geometries are required for quadric surfaces in 3-dimensional space and 34 geometries are required for quadric surfaces in 3-dimensional space. The required geometry will be entered into the preferred embodiment of the invention in a certain way, as will be described below.

Referring now to fig. 1a, a reference point M in euclidean geometry is shown. Point M is characterized by three coordinates, e.g., (2, 1, 1), which are the x, y, and z coordinates of the reference point, respectively. Referring now to FIG. 1b, point M is shown with its reference cell. In the projection geometry, the 3-dimensional reference cell consists of 4 points. Point M has 4 coordinates in such a reference cell.

The coordinates of a point in projection space from euclidean space can be expressed by a transfer matrix. The coordinates of points in euclidean space from the projection space can be expressed by an inverse of the transfer matrix.

In 2-dimensional euclidean space, if a point has coordinates x and y (M (x, y)) in euclidean space, M will have a third coordinate in the projection plane. More specifically, a point in n-dimensional Euclidean space is depicted as a point in (n +1) -dimensional projection space. M (x, y, l) is the representation of point M in projection space. In projection space, all scales are unimportant, so (x, y, l) ═ a · x, a · y, a · l. But to avoid a being 0. Since the scale is not important, (x, y, z) is referred to as the homogeneous coordinate of the point. To convert the representation of the points expressed in projection space back into euclidean space, the coordinates are divided by a scaling factor. In this regard, it is important to note that the projection plane includes more points than the Euclidean space. For example a point whose final coordinate in projection space is zero. These points are referred to as ideal points or infinite points. All the ideal points lie on a line called the ideal line or infinite line. In two dimensions, the line is represented by (0, 0, 1).

Referring now to FIG. 2, a cone in 2-dimensional space is shown. The cone is defined by 5 geometric conditions;

a reference monomer is first selected. The geometry is then entered with an entity defined as a control point. As will be appreciated by those skilled in the art, the geometry may be entered with other types of information. A point is usually defined as the intersection of a curve and another line, in this case a wire bundle.

To extend the concept of points, in a preferred embodiment of the invention, the control points can be divided into non-poles and poles. It will be appreciated that curves such as Lama curves cannot be defined with conventional intersection points.

Referring now to fig. 3, a pole and an associated non-pole of a cone in 2-dimensional space are shown.

According to a first step, a non-pole is selected on the wire bundle. According to a second step, the non-polar point becomes the intersection of the two tangent lines to the surface. According to a third step, the intersection of the two contacts generated by the two tangent lines becomes the pole. The pole is defined as the intersection of the polar line and the beam. Those skilled in the art will appreciate that when the beam is tangent to the shape, the poles and non-poles will merge into one. Thus, the full concept of the intersection point will be outlined in the use of poles and non-poles. Imaginary points are therefore not used in the preferred embodiment of the invention.

One pole and one associated non-pole define a geometric condition. In the case of a quadratic surface in 2-dimensional space, 5 geometric conditions are necessary. 10 control points can be used to provide the 5 geometries. The 10 control points include 5 non-poles and 5 associated poles. Alternatively, 5 intersections may be provided in order to meet the requirements of 5 geometric conditions.

In the case of a 3-dimensional quadric, the geometric condition number is equal to:

in a preferred embodiment of the invention 18 control points are provided, of which 9 are non-poles and the other 9 are the relevant poles of the 9 first non-poles. It will be appreciated that the non-poles and poles allow the user to set the beam at the location of the study. This advantage will be utilized in the techniques described above.

In the case of quadric surfaces in 3-dimensional space, the geometric condition number is equal to:

in a preferred embodiment of the invention 68 control points are provided, of which 34 are non-poles and the other 34 are the relevant poles of the 34 first non-poles.

One skilled in the art will appreciate that if the shape is n-times, the surface defining the polar plane is n-1 times the shape.

In order to be able to identify the algebraic coefficients of a shape using the information provided by the control points, it is important to combine the information provided by the control points.

In a preferred embodiment of the invention, in order to achieve a solution process with the interaction of the provided geometrical conditions, cross sections are defined to combine the control points.

In this regard, it is important to understand how the number of sections is determined in the preferred embodiment of the present invention. The number of geometrical conditions determines the number of sections.

Referring now to the Pascal triangle, it is known that:

where m is the degree of the surface and n is the dimension of space.

Preferably, the number of sections is defined by the formula:

referring back now to the following equation:

m +1 cross sections can be identified and broken down into other elements. In a preferred embodiment of the invention, each section is further broken down into lines.

The last equation shows that each section can be decomposed into lines. Each line includes a certain number of control points. For example, the first section S₁Is defined by the formula:

the cross-section comprises 1 line with 2 control points, one of which is a non-pole and the other of which is the associated pole of the non-pole.

Second cross section S₂Is defined by the formula:

the cross section comprises 2 lines. The first line will provide

A geometric condition that it will includeA control point; half of which are non-poles and the other half are the associated poles of the first half.

The second line will provide

A geometric condition that it will includeA control point; half of which are non-poles and the other half are the associated poles of the first half.

S_m+1The cross-section is defined by the formula:

the cross-section includes m +1 lines. Section S_m+1Will provideA geometric condition that it will includeA control point; half of which are non-poles and the other half are the associated poles of the first half.

Section S_m+1Will provide a second lineA geometric condition that it will includeA control point; half of which are non-poles and the other half are the associated poles of the first half.

Section S_m+1Will provide the m +1 th lineAnd (4) a geometric condition. It will be appreciated that 1 geometry is subtracted from the total. Which will provide A control point; half of which are non-poles and the other half are the associated poles of the first half.

In a preferred embodiment of the invention, control points are provided to each section in a certain way. More specifically, the control point is first provided to the section S_m+1Then supplied to S_mUp to S₁And finally to S₀。

Referring now to FIG. 4a, a decomposition of each section into lines for a quadric of a 3-dimensional space is shown. For example, section S₃Comprising 3 lines, the first line provides 2 geometries with 4 control points. Section S₃The second line of (2) provides 2 geometries with 4 control points. Section S₃The third line of (2) provides 1 geometry with 2 control points.

Referring now to FIG. 4b, a decomposition of each section into lines for a cubic surface of 3-dimensional space is shown. For example, section S₄Comprising 4 lines, the first line provides 3 geometries with 6 control points. Section S₄The second line of (2) provides 3 geometries with 6 control points. Section S₄The third line of (2) provides 2 geometries with 4 control points. Section S₄The fourth line of (a) provides 1 geometry with 2 control points.

Referring now to FIG. 4c, a decomposition of each cross section into lines for a quadric surface of a 3-dimensional space is shown.

In a preferred embodiment, the user may provide control points using a charting user interface. The charting user interface is located on a computer.

The user selects a pixel by using a mouse or any device that allows the selection of pixels on the screen. The selected point is characterized by the coordinates of the selected pixel, which in the preferred embodiment includes two integer elements. The coordinates of the pixels are then converted in the preferred embodiment of the invention to world coordinates known to those skilled in the art.

The conversion of the coordinates of the pixels into world coordinates is performed by matrix multiplication. In a preferred embodiment, the world coordinates include three "floating" values. Also in the preferred embodiment, and then converted to projected homogeneous coordinates. The projected homogeneous coordinates are obtained by adding a new coordinate of 1. In the case where the new coordinate is desired to be expressed as "infinity", the new coordinate may be 0.

The polar plane can be expressed by the derivation of the equation for the surface with respect to each coordinate of the reference cell.

Referring now to FIG. 5a, a first embodiment of defining a cross-section defining a quadric with a quadric surface is shown.

More specifically, 5 cross sections are defined as explained in fig. 4 c.

It will be appreciated that 4 sections are necessary to define a cubic surface, as explained in figure 4 b.

Referring now to fig. 6, a preferred embodiment of the present invention is shown. The preferred embodiment of the present invention includes a database of equations 12, a graph database 14, a base graph selection unit 10, a graph to control points converter 16, an algebraic coefficients to control points converter 18, a section management unit 20, a projection space object store 22, a control points editor 24, a control points to algebraic coefficients converter 26, an output interface 28, a ray tracing unit 30 and a graph display 32.

The equation database 12 includes equations for various patterns. The equation database includes algebraic coefficients of algebraic equations of the graph and an identifier of the graph. The pattern is selected using an identifier. It will be appreciated that the number of degrees of the algebraic equation defining the graph is not limited to a certain value as explained below, and thus any type of graph generation may be achieved.

The graphic database 14 includes, for each graphic, an identifier of the graphic and information about the graphic. The information about a graphic includes control points defining the graphic. The information about the pattern further includes a reference cell. In another embodiment, the information may be geometric information about the figure, such as radius and center in the case of a spherical figure.

The base graphic selection unit 10 enables a user to select data from at least the equations database 12 and the graphics database 14 and provide the selected data to the projection space object memory 22 if the selected data includes control points and reference cells, to the graphic to control points converter 16 if the selected data includes graphic definitions of objects, and to the algebraic coefficients to control points converter 18 if the selected data includes algebraic coefficients.

The graphic to control point converter 16 performs a conversion of the defined graphic to control point. The converter is dedicated to converting each type of graphics defined to a control point. For example, for a sphere, the graph is defined by the coordinates of the radius and the center. The graphics-to-control point converter 16 converts this information into a set of control points.

In a preferred embodiment, the user provides a reference cell in order to generate the control points. In another embodiment, the reference cells are automatically generated by the pattern-to-control point converter 16.

The algebraic coefficient to control point converter 18 performs a conversion of an algebraic coefficient defining a graphic to a set of control points that will define the graphic in projection space. In a preferred embodiment, the user may provide a reference cell. A reference monomer will be generated by reference to that monomer. In another embodiment of the present invention, the reference cell is automatically generated by the algebraic factor to control point converter 18.

The cross-section management unit 20 allows a user to create, edit or delete a cross-section of the projection space object store 22. The section management unit 20 may also use algebraic coefficients when the user wishes to redefine a new section. The projection space object memory 22 stores data of the graphics in the projection space.

In a preferred embodiment of the invention, the data of the pattern in the projection space object memory 22 comprises control points of the pattern and reference cells. The control points include poles and associated non-poles. The control points are located on lines of a wire bundle, a subset of at least one line defining a cross-section as previously described.

The control point editor 24 enables a user to create, edit or delete control points. In a preferred embodiment of the invention, the control points comprise poles and associated non-poles, as described above.

The control point to algebraic coefficient converter 26 performs a conversion of the control points defining a graph to algebraic coefficients defining the graph. Since the control points are related to the reference cell, the reference cell is preferably also used for making the transition.

The output interface receives the algebraic coefficients and provides an interface for outputting the algebraic coefficients defining the graph.

Ray tracing unit 30 receives the algebraic coefficients and provides the data to a graphics display 32.

Ray tracing unit 30 operates under Principles described on pages 710 and 702 of the paper Computer Graphics Principles and Practices by James d.

Chart display 32 enables a user to view the geometry using a chart interface. Preferably, the ray tracing unit 30 further receives a view signal point. In one embodiment of the invention, the graphics display operates under the i386 operating system, and in another embodiment, the graphics display operates under the MAC operating system.

Referring now to fig. 7, a preferred embodiment of the present invention is shown.

According to step 36 of the present invention, data is provided around a graph. In a preferred embodiment, the data provided comprises a reference cell and control points. The pattern is defined in projection space with reference cells and control points. Referring again to FIG. 7 of the present application and in accordance with step 38 of the present invention, the control points defining the graph are modified. It will be appreciated that the user may modify the control points by modifying only the reference cell.

Algebraic coefficients of the graph defined by the modified control points are calculated using the modified control points and the reference cell, according to step 40. Algebraic coefficients of the graph can be used to provide a euclidean view of the object.

Referring now to FIG. 8, there is shown one embodiment of how data is provided in accordance with the 36-step method of the present invention. In this embodiment and according to step 42, a graphic is selected in the graphic database 14. The pattern is selected by the base pattern selection unit 10 with an identifier. As described above, the information attached to the identifier includes the control point and the reference cell. Step 44 according to the invention provides the control point and the reference monomer.

Referring now to fig. 9, there is shown another embodiment of how data is provided in accordance with the 36-step method of the present invention. In this embodiment and based on 46 steps, an algebraic equation is selected. The algebraic equations depend on the required pattern selection. Algebraic equations are selected in the equation database 12 by the basic graph selection unit 10. The algebraic equations are converted into control points according to the 48 steps. The algebraic equations are converted by algebraic coefficient to control point converter 14. In a preferred embodiment, the reference cell is provided by the user. In another embodiment, the reference cell is automatically generated.

Referring now to fig. 10, there is shown another embodiment of how data is provided in accordance with the 36-step method of the present invention. One geometry is selected according to the 50 steps of this embodiment. The geometry is selected using the basic graphic selection unit 10 and the graphic database 14. The selected geometry is converted to control points by the pattern-to-control point converter 16, according to step 52 of fig. 10. In a preferred embodiment, the reference cell is provided by the user. In another embodiment, the reference cell is automatically generated. The control points generated will be generated relative to the reference monomer.

Referring now to fig. 11, there is shown another embodiment of how data is provided in accordance with the 36-step method of the present invention.

Information about the first geometrical cross section 54 is provided according to step 54. This information is provided by the basic graphics selection unit 10. This information may be the algebraic equation of the first geometric section, in which case the algebraic equation is stored in the equation database 12.

Additional geometric cross-sections and their control points are generated according to step 56. In the preferred embodiment, control points are generated using a control point editor 24. Also in the preferred embodiment, other geometric cross-sections are created with the cross-section management unit 20. In a preferred embodiment, the reference cell is provided by the user. In another embodiment, the reference cell is automatically generated.

According to step 58, the control points of each section are provided with reference cells.

Referring now to fig. 12, there is shown another embodiment of how data is provided in accordance with the 36-step method of the present invention. In this embodiment the data is generated by the user. A geometric cross-section was produced according to step 60. In a preferred embodiment, the reference cell is provided by the user. In another embodiment, the reference cell is automatically generated. The geometric cross-section is created with a cross-section management unit 20. According to step 62, the control point editor 24 is used to place the non-poles associated with the cross section. According to step 64, the control point editor 24 is used to place the poles associated with the section. Those skilled in the art will appreciate that the poles may be placed in front of the non-poles in the resulting cross-section.

Depending on the number of times the pattern is drawn, a certain number of control points are necessary for each section in order to draw the pattern as described above. The 66-step check according to the invention is carried out once, with the aim of checking whether all the necessary cross sections have been produced. As mentioned above, a certain number of cross-sections are necessary for a complete definition of a pattern. Steps 6062 and 64 are reached at additional times if the necessary cross-sections have not been fully produced.

If all necessary sections have been created, control points are provided according to step 68. The control points include the poles and non-poles of each cross section. A reference monomer is also provided as the control point is defined with the reference monomer.

Referring now to FIG. 13, there is shown how the control points defining the graph are converted into algebraic coefficients.

Control points and reference monomers are provided according to step 78. A check is made according to 80 steps, the purpose of which is to check whether all control points are already fully available. If all control points are available, algebraic coefficients are calculated according to step 92 of FIG. 13. Algebraic coefficients are provided according to step 94 of fig. 13.

The coordinates of the control point selected according to step 84 are converted into one-dimensional with the extreme value of the segment in which the control point is located. This step is described in more detail below.

According to step 86, the one-dimensional representation of the selected control point is stored.

Steps 82, 84 and 86 are performed for each available control point. Referring now to fig. 14 and in accordance with step 96, a reference cell is provided by the user. In another embodiment, the reference cell is automatically generated. The non-poles are provided by the user according to step 98. In the preferred embodiment, the non-poles are provided by the control point editor 24.

The expression of the polar plane is calculated from 100 steps, which is calculated using the algebraic equation of the interpreted graph.

An experiment was performed on 102 steps in order to find out if at least one of the more than one non-poles is available. If at least one of the more than one non-poles is available, the available non-poles are selected according to step 104.

According to step 106, the pole of the selected non-pole is calculated using at least the expression of the pole plane.

If no non-poles are available and according to step 108, the control point and the reference cell are provided.

Referring now to fig. 15, there is shown how the control point editor works. One control point is selected according to step 140. One pole and one non-pole may be selected.

In a preferred embodiment of the invention a control point is selected on the user interface with a mouse.

In accordance with step 142 of the present invention, the selected control point is moved on the template line with the cursor of the mouse in the preferred embodiment of the present invention.

A new control point is provided according to step 146. The new control point may be a pole as well as a non-pole.

According to fig. 16, it is shown how the section managing unit works. In a preferred embodiment of the present invention, one line is selected according to step 148 of the present invention. The line is selected with a mouse and click keys. The selected line is moved with the mouse and preferably predetermined keys according to step 150. According to step 152, the control points located on the line and any other subordinate control points, if any, are updated.

A new control point is provided according to step 154.

Referring now to FIG. 17, two such surfaces in 3-dimensional space are shown.

Referring now to FIG. 18, there is shown how a quadric surface is produced using a preferred embodiment of the present invention. The quadric surface is created with a single reference body, three defined cross sections and control points on each line of a bundle. Three cross sections are defined as described above. The first cross-section 156 is defined by lines 159, 160, and 161. The second cross-section 157 is defined by lines 162 and 163. The third cross-section 158 is defined by a line 164. Control points are inserted on each of the three cross sections. As mentioned above, and in a preferred embodiment of the present invention, the first cross-section 156 comprises three lines, the first line 159 comprises 4 control points (since 2 geometries are necessary), the second line 161 of the first cross-section 156 comprises 4 control points (since 2 geometries are necessary), and the third line 160 of the first cross-section 156 comprises 2 control points (since 1 geometry is necessary).

The first line 162 of the second cross section 157 comprises 4 control points (since 2 geometries are necessary) and the second line 163 of the second cross section 157 comprises 2 control points (since 1 geometry is necessary).

The first line 164 of the third cross-section 158 comprises 2 control points (since 1 geometry is necessary).

Figure 19 shows a quadric surface as shown in figure 18 nested with reference cells and control points.

Fig. 20 shows a second quadric surface.

Referring now to FIG. 21, a reference cell and control points defining the second quadric surface are shown. Three cross sections are also defined in this embodiment. The reader will understand that in this embodiment the 7 non-pole control points are combined with their respective pole control points.

Referring now to FIG. 22, the user will understand that when a polar control point is combined with its associated non-polar control point, the resulting point is located on the surface of the quadric.

Referring now to FIG. 23, a third quadric surface is shown. In the third example of the quadric surface, one point defining the reference cell is sent to "infinity", as shown in fig. 24. The same number of sections, lines and control points are necessary. Sending a point to infinity allows the user the possibility of approaching a new type of quadric. FIG. 25 shows a third quadric surface with control points and reference cells.

Fig. 26 shows a fourth quadric surface. In the fourth example of the quadric surface, 2 points defining a reference cell are sent to "infinity", as shown in fig. 27. This embodiment allows for example to create a ball.

FIG. 28 shows a fourth quadric surface with control points and reference cells.

Control point to algebraic coefficient converter

Referring now to FIG. 29, there is shown how control point to algebraic coefficient converter 26 operates. More specifically, the figure shows how the control points entered by the user are used to determine the coefficients from the algebraic equations. As mentioned above, in a preferred embodiment of the invention, the control points comprise poles and non-poles.

According to step 200 of fig. 29, the cells and the control points are provided to a control point to algebraic coefficient converter 26. As described above, the monomers and control points are generated from the projection space object store 22.

According to step 202 of fig. 29 and in a preferred embodiment of the invention, λ, μ and ρ, σ are calculated for the control points. λ, μ is used to designate one non-pole, while ρ, σ is used to designate one pole. Referring now to fig. 30, the results using λ, μ and ρ, σ are shown. To this extent, each control point can be referenced to four coordinates (i.e., x) on the cell using its own reference point₀，x₁，x₂，x₃) And (4) introducing. With the mechanism explained below, each point will be introduced with two coordinates, λ, μ or ρ, σ, depending on whether the control point is a pole or a non-pole as described above. By making the following variable changes:

x₀＝μ·π₀

x₁＝μ·π₁

the purpose can be achieved.

x₂＝μ·π₂

x₃＝μ·π₃

To achieve this, two points must be provided which will define a line, and therefore pi₀，π₁，π₂，π₃. Thus, when two points forming a line on which the control points are located are defined, each control point can be expressed in one dimension with two coordinates. One skilled in the art will note that such a scheme would provide a convenient way to simplify the mechanism, since a point on any dimension is always on a line.

Referring now to FIG. 31, there is shown a cross-section (i.e., S) in the first and second sections in a preferred embodiment of the present invention₅And S₄) The line used in the case of (1). The skilled person may define the support lines of the control points in other ways, but it will be noted that by introducing an advantageous simplification, the scheme depicted in fig. 31 allows a convenient calculation of the algebraic coefficients. For example, for all the positions in the section S₅Point of (d), coordinate pi₃Will be equal to zero and will be for all positions in section S₄Point of (d), coordinate pi₂Will be equal to zero.

Now, referring back to FIG. 29 and selecting a section based on step 204. The selected cross-section is the remaining cross-section where there is the greatest number of control points in the preferred embodiment of the invention. For example, in the case of a quadric surface, the first cross-section selected is the cross-section S₅The cross section comprises 14 control points, as shown in fig. 4 b.

A line and the corresponding control point on the selected line are selected according to step 206. In a preferred embodiment of the invention, the selected line is the remaining line where there is the greatest number of control points. For example, the line 1 is in the section S₅Is selected first.

A check is made on the basis of step 208 to find out if the position of the singularity is together with a point on the selected line. A singularity is defined as more than one control point having the same spatial position. If such singularities are detected, a corresponding equation is used according to step 210; if no singularities are detected, the standard equation is used according to step 212.

According to step 214, a check is then made to find out whether all lines on which the control points are located have been processed. If this is the case, and according to step 216, the calculated coefficients and/or information are provided. The provided information may in some cases be an equation.

A check is made according to 218 steps with the aim of finding out whether all cross sections have been processed. If this is the case and according to step 220, the calculated coefficients will be provided. If a section has not been processed, a new section is selected according to step 204.

For example, in the case of a quadric surface, the first cross-section processed is named S₅Is processed with a second section named S₄Is a processed third cross section is a cross section S₃The fourth cross section is S₂The last section processed is named S₁Cross-section of (a). When processing the first section S₅Then, the first processed line is line 1, the second processed line is line 2, the third processed line is line 3, the fourth processed line is line 4, and the fifth processed line is line 5. When processing the second section S₄Then, the first processed line is line 1, the second processed line is line 2, the third processed line is line 3, and the fourth processed line is line 4. When processing the third section S₃Then, the first processed line is line 1, the second processed line is line 2, and the third processed line is line 3. When processing the fourth section S₂The first processed line is line 1 and the second processed line is line 2. Finally, when processing the fifth section S₁The first line processed is line 1.

Also in the case of a quadric, the first section would provide 16 coefficients; the first section will provide 16 coefficients; the second section will provide a factor of 10; the third section will provide 3 coefficients and three equations to be used later; the fourth section will provide 4 coefficients and one equation and the fifth section will provide the coefficients of the last 3 equations.

The skilled person will note that the developed method described above allows working with any type of dimensionality, since the information comprised in each control point is processed at the level of the line. It is now important to understand how the standard equations and corresponding equations used to find the coefficients in steps 210 and 212 are generated. One skilled in the art would then be able to find the standard equations and corresponding equations used in steps 210 and 212 and thus be able to determine algebraic coefficients in any dimension.

General principles regarding methods for finding standard equations and corresponding equations

The user may provide control points on a line as described above via the user interface, and each control point includes a pole and a non-pole as described above.

An iterative method will be used in order to find the standard equations and the corresponding equations. It will be appreciated that iterative methods enable one skilled in the art to find algebraic coefficients in an n-dimensional space. It will also be appreciated that the methods described below are preferred methods; those skilled in the art will be able to find other variables. Referring now to FIG. 32, there is shown how the general equations are generated.

Algebraic equations are provided in terms of 230 steps. The algebraic equation depends on x₀，x₁，x₂，x₃. Referring now to FIGS. 34a and 34b, algebraic equations for a quadric and a quadric are shown. Each element of the polar plane is calculated according to 232 steps. The elements of the polar plane are calculated by taking the derivatives of the algebraic equations with respect to the variables of the elements. E.g. x of polar plane₀Element equal to related to x₀The derivative of the algebraic equation of (c).

The condition for polarity is expressed according to step 234. More specifically, the condition is expressed as:

F_x0·z₀+F_x1·z₁+F_x2·z₂＝0(Eq.0)

in the formula F_x0，F_x1，F_x2，F_x3Is polar plane relative to x₀，x₁，x₂，x₃Element of (a), z₀，z₁，z₂，z₃The coordinates of the poles.

The change of the variable is performed according to the 236 steps. The variables are changed with λ, μ and ρ, σ.

More specifically, the variables that were made were changed to:

x₀＝μ·π₀

x₁＝μ·π₁

x₂＝μ·π₂

x₃＝μ·π₃

z₀＝ρ·π₀

z₁＝ρ·π₁

z₂＝σ·π₂

z₃＝σ·π₃

eq.0 regarding the change of the variable made in step 236 now depends on λ, μ, ρ, σ and x₀，x₁，x₂，x₃. The new approach can be used with any control point located on any line of any cross section. The new equations provide a general tool to provide the correspondence between algebraic coefficients and geometrical information provided by the control points.

The obtained general equation is provided according to step 238.

However, as described below, a particular solution equation will be used to determine the coefficients using the general equation applied to each control point.

One skilled in the art can propose a simple solution to the resulting system of equations by applying the general equations to each control point and inverting the resulting system of equations. However, this inversion operation is too time consuming and such an operation is not possible for high order systems, and finally an odd point would make the operation impossible. There is therefore a need for a derivation method to solve the system of equations.

To reveal the obligation of the preferred embodiment, FIG. 33 provides a mechanism for overriding the equations provided at step 238.

The general equation is provided in accordance with step 240.

One section is selected according to step 242 and in a preferred embodiment the selection of the section is made according to the scheme described above. By selecting a section to set pi₂And pi₃The value of (c).

Selecting a line according to step 244; in a preferred embodiment, the selection of the line is made in a selected cross section according to the scheme described above. Setting pi by selecting a line₀And pi₁The value of (c).

According to step 246, a non-pole and its associated pole are selected on the selected line of the selected cross-section. According to step 248, the selected non-poles and their associated poles are input into the provided general equation. The values of λ, μ and ρ, σ are set by selecting one non-pole and its associated pole.

According to step 248, the information provided by the selected pair of points is used to calculate the coefficients.

More specifically, the general equation is used with values of λ, μ and ρ, σ, and a system of equations is generated on the line level. It will be appreciated that for each section, pi₂And pi₃Is stationary. Thus, passing through π is performed in each equation of a system of equations generated in the same cross-section₂And pi₃Or pi₂And pi₃Any linear combination of (a) and (b). Factorization by algebraic coefficients and their linear combinations is also performed. This will result in a second system of equationsIn this second system of equations, the unknown variable is now a polynomial that depends on the combination of coefficients, rather than on the original system of equations that relied on the coefficients. These second system of equations are then solved using the Kramer solution equation scheme, allowing, in the preferred embodiment, the original system of equations to be solved.

It will be appreciated that each time an initial or second type of system of equations is solved, a new unknown variable for the homogeneous factor is used to provide the result of solving the equations. The homogeneous factors will be solved at the end of the solution process, and this will allow to frequently find each homogeneous factor introduced in the solution process.

A check is made when generating the system of equations of the initial type and of the second type, with the aim of finding whether factorization by a known polynomial dependent on λ, μ and ρ, σ is possible. The known polynomial is a polynomial that describes a particular geometric configuration; such a specific configuration may be that the two control points are located at the same physical location (meaning λ)₁＝λ₂，μ₁＝μ₂) In the configuration of (a). In the last-mentioned case, the polynomial will be λ₁·μ₂-λ₂·μ₁. In the case of a quartic surface, the specific geometry searched for is the case of two-dots, and the case of three-dots. If such polynomials are detected, a division is performed by the polynomials in the determinant in order to avoid these specific cases that would fail the solution of the equation, since the determinant would be equal to zero. It is therefore important to be careful at this level. In case of detection of a singularity, a new equation for the coefficient will be provided.

More specifically, fig. 71 shows a determinant for which a solution equation scheme is performed with at least two points located at the same place.

As shown in fig. 70, a determinant may be expanded by any row or column. Such an expansion is explained in Erwin Kreuszig's paper, see section Edition of Advanced Engineering materials, page 373, which is incorporated herein by reference.

Referring now to FIG. 71, an expanded solution of a determinant expanding into a smaller determinant sum is shown.

In the degenerate case, exactly the same rows and thus the complementary small determinant will disappear. Preferably, a one-time determinant decomposition is performed to provide a small determinant in case of multiplicity m.

In the case of, for example, a multiplicity of 2, where λ₃＝k·λ₂，μ₃＝k·μ₂Fig. 72 shows the decomposition of the determinant shown in fig. 71 into a smaller determinant with size 2. If in the small determinant shown in FIG. 72, λ is₃＝k·λ₂，μ₃＝k·μ₂By substitution, one can understand₃·ρ₂-σ₂·ρ₃Will be a co-vanishing factor of these small determinants. This determinant will then be calculated with the other smaller determinants.

In the case of two other double points, the method disclosed above will be applied to two other identical lines in order to extract a co-vanishing factor.

Referring now to fig. 73, there is shown a decomposition of the determinant shown in fig. 71 into a smaller determinant having size 3. Such decomposition is performed with the three points being placed at the same location. For example, if λ₃＝k·λ₂，μ₃＝k·μ₂，λ₃＝l·λ₂，μ₃＝l·μ₂The small determinant shown in FIG. 73 will be (λ)₂·μ₁-λ₁·μ₂)，(σ₂·ρ₁-σ₁·ρ₂) As a co-elimination factor. By eliminating such a vanishing factor from the small determinant shown in FIG. 73, the sum x can be calculated₁，x₂，x₃，x₄Each coefficient being related.

Referring now to fig. 74, the decomposition of the determinant shown in fig. 71 into a smaller determinant having a dimension of 4 is shown. Such a decomposition is performed in the case of four points, i.e. four points are placed at the same position. By eliminating the common vanishing factor from the small determinant shown in FIG. 74, the sum x can be calculated₁，x₂，x₃，x₄Each coefficient being related.

A check is made on a 250 step basis in order to find out whether other control points are available on the selected line of the selected cross section.

A check is made on step 252 to find out if additional lines are available on the selected cross-section.

A check is made on step 254 to find out if other cross sections are available. If no other cross-section is available, the calculated coefficients will be provided.

Determining the coefficients of a quadric surface in accordance with a preferred embodiment of the present invention

It will be appreciated by those skilled in the art that this embodiment shows an example of determining the coefficients of a quadric surface in accordance with a preferred embodiment of the present invention.

As mentioned above and in the preferred embodiment of the invention, a quadric surface is defined by 5 sections. The 5 sections defining the quadric surface are shown in figure 35.

FIG. 36 shows 15 lines l_a，l_b，l_c，l_d，l_e，l_f，l_g，l_h，l_i，l_j，l_k，l_l，l_m，l_n，l_o。

The first cross-section comprises 5 lines/selected in the first cross-section_a，l_b，l_c，l_d，l_eAs shown in fig. 37. The first line of the first cross section comprises 4 poles and their associated 4 non-poles. The second line of the first cross section comprises4 poles and their associated 4 non-poles. The third line of the first cross-section comprises 3 poles and their associated 3 non-poles. The fourth line of the first cross section includes 2 poles and their associated 2 non-poles. The fifth line of the first cross-section comprises 1 pole and its associated 1 non-pole.

The second cross-section comprises 4 lines l selected in the second cross-section_f，l_g，l_h，l_iAs shown in fig. 38. The first line of the second cross section comprises 4 poles and its associated 4 non-poles. The second line of the second cross section comprises 3 poles and its associated 3 non-poles. The third line of the second cross section comprises 2 poles and its associated 2 non-poles. The fourth line of the second cross section includes 1 pole and its associated 1 non-pole.

The third cross-section comprises 3 lines l selected in the third cross-section_j，l_k，l_lAs shown in fig. 39. The first line of the third cross-section comprises 3 poles and their associated 3 non-poles. The second line of the third cross-section comprises 2 poles and its associated 2 non-poles. The third line of the third cross-section comprises 1 pole and its associated 1 non-pole.

The fourth cross-section comprises 2 lines l selected in the fourth cross-section_m，l_nAs shown in fig. 40. The first line of the fourth cross-section comprises 2 poles and its associated 2 non-poles. The second line of the fourth cross-section comprises 1 pole and its associated 1 non-pole.

The fifth cross-section comprises 1 line l selected in the fifth cross-section_oAs shown in fig. 41. The line of the fifth cross-section comprises 1 pole and its associated 1 non-pole.

The general equation for quadric surfaces is provided according to step 230 of FIG. 32.

Z4＝z434x2x3³+z43x0²x1²+z49x1³x2+z414x1x2³+z416x0³x3+z431x2²x3²+z419x1³x3+z413x0x2³+z41²x1²x2²+z410x0²x2²+z46x0³x2+z428x1²x3²+z432x0x3³+z44x0x1³+z415x2⁴+z433x1x3³+z42x0³x1+z426x0²x3²+z425x+z41x0⁴+z435x3⁴+z47x0²x1x2+z45x1⁴+z48x0x1²x2+z411x0x1x2²+z417x0²x1x3+z418x0x1²+z420x0²x2x3+z422x1²x2x3+z423x0x2²x3+z424x1x2²x3+z427x0x1x3²+z429x0x2x3²+z430x1x2x3²+z421x0x1x2x3

The expression for the polar plane is calculated according to 232 steps. The expression for the polar plane is calculated using the general equation for quadric surfaces as described above.

The polarity condition is expressed according to step 234. The polarity expression is expressed as described above.

The variables are changed with λ, μ and ρ, σ according to step 236.

The expression of the pole plane of the quadric surface is calculated according to step 238 of fig. 32. The resulting expression for the polar plane is:

4αt₀+βt₁+2γt₂+δt₃+4εt₄＝0

in the formula:

α＝ρλ³；

β＝λ²(λσ+3μρ)；

γ＝λμ(λσ+μρ)；

δ＝μ²(3λσ+μρ)；

ε＝σμ³；

t₀＝z₄₃₅π₃ ⁴+z₄₃₄π₃ ³π₂+z₄₃₁π₃ ²π₂ ²+z₄₂₅π₃π₂ ³+z₄₁₅π₂ ⁴；

t₁＝(z₄₃₃π₃ ³+z₄₃₀π₃ ²π₂+z₄₃₁π₃ ²π₂ ²+z4₁₄π₂ ³)π₁+(z₄₃₂π₃ ³+z₄₂₉π₃ ²π₂+z₄₂₃π₃π₂ ²+z₄₁₃π₂ ³)π₀

t₂＝(z₄₂₈π₃ ²+z₄₂₂π₃π₂+z₄₁₂π₂ ²)π₁ ²+(z₄₂₇π₃ ²+z₄₂₁π₃π₂+z₄₁₁π₂ ²)π₀π₁+(z₄₂₆π₃ ²+z₄₂₀π₃π₂+z₄₁₀π₂ ²)π₀ ²

t₃＝(z₄₁₉π₃+z₄₉π₂)π₁ ³+(z₄₁₈π₃+z₄₈π₂)π₁ ²π0+(z₄₁₇π₃+z₄₇π₂)π₁π₀ ²+(z₄₁₆π₃+z₄₆π₂)π₀ ³(ii) a And

t₄＝z₄₅π₁ ⁴+z₄₄π₁ ³π₀+z₄₃π₁ ²π₀ ²+z₄₂π₁π₀ ³+z₄₁π₀ ⁴

in a preferred embodiment of the invention, Ψ_iI-0.. 14 is defined as:

Ψ₀＝z₄₃₅π₃ ⁴+z₄₃₄π₃ ³π₂+z₄₃₁π₃ ²π₂ ²+z₄₂₅π₃π₂ ³+z₄₁₅π₂ ⁴；

Ψ₁＝z₄₃₃π₃ ³+z₄₃₀π₃ ²π2+z₄₂₄π₃π₂ ²+z₄₁₄π₂ ³；

Ψ₂＝z₄₃₂π₃ ³+z₄₂₉π₃ ²π₂+z₄₂₃π₃π₂ ²+z₄₁₃π₂ ³；

Ψ₃＝z₄₂₈π₃ ²+z₄₂₂π₃π₂+z₄₁₂π₂ ²；

Ψ₄＝z₄₂₇π₃ ²+z₄₂₁π₃π₂+z₄₁₁π₂ ²；

Ψ₅＝z₄₂₆π₃ ²+z₄₂₀π₃π₂+z₄₁₀π₂ ²；

Ψ₆＝z₄₁₉π₃+z₄₉π₂；

Ψ₇＝z₄₁₈π₃+z₄₈π₂；

Ψ₈＝z₄₁₇π₃+z₄₇π₂；

Ψ₉＝z₄₁₆π₃+z₄₆π₂；

Ψ₁₀＝z₄₅；

Ψ₁₁＝z₄₄；

Ψ₁₂＝z₄₃；

Ψ₁₃＝z₄₂(ii) a And

Ψ₁₄＝z₄₁。

one skilled in the art will understand that Ψ_iI 0.. 14 characterizes a selected section, since they depend only on pi₂And pi₃(ii) a They are fixed for the selected cross-section, and t_iI 0.. 14 characterize lines of a particular cross-section, since they depend only on pi₀And pi₁。

Ψ_iI 0.. 14 is related to i 0.. 14 as follows:

t₀＝Ψ₀；

t₁＝Ψ₁π₁+Ψ₂π₀＝Ψ₃π₁ ²+Ψ₄π₀π₁+Ψ₅π₀ ²；

t₃＝Ψ₆π₁ ³+Ψ₇π₁ ²π₀+Ψ₈π₁π₀ ²+Ψ₉π₀ ³；

t₄＝Ψ₁₀π₁ ⁴+Ψ₁₁π₁ ³π₀+Ψ₁₂π₁ ²π₀ ²+Ψ₁₃π₁π₀ ³+Ψ₁₄π₀ ⁴。

hereinafter, Ψ_i ^JWill refer to Ψ of section J_i，t_iaWill indicate t of line a_i。

One section is selected according to step 242 of fig. 33. In a preferred embodiment of the invention, the first cross section is selected.

First line of first cross section (line a)

A line is selected according to step 244 of figure 33. In a preferred embodiment of the invention the first cross-section ([ a ] is selected₀:a₁:0:0]) The first line of (a).

A pair of points is selected according to step 246. By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the first line of the first section yields an equation system, eq.1, with t_iI is 0.. 4 as an unknown.

Cofactor x associated respectively with each element of the added row in the system of equations Eq.1₀，x₁，x₂，x₃，x₄Is added to the common equation system eq.2.

The system of equations eq.2 is shown in fig. 42.

From the system of equations Eq.2, t_iaI-0.. 4 is expressed as Ψ_i ^II is a function of 0.. 14.

Thus can provide psi_i ^IΨ satisfied by i 1.. 14₀ ^IAnd an equation.

Second line (line b) of the first cross section

The second line of the first cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246. By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the second line of the first section yields an equation system, eq.3, with t_ibI is 0.. 4 as an unknown.

Respectively associated with each element of the added row in the system of equations Eq.3Cofactor x₅，x₆，x₇，x₈，x₉Is added to the system of equations.

The system of equations eq.3 is shown in fig. 43.

From the system of equations Eq.3, t_ibI-0.. 4 is expressed as Ψ_i ^II is a function of 0.. 14. By t_1aAnd t_1bThe expression of (c) yields the equation system eq.4.

Adding a cofactor x associated respectively with each element of the added row in the system of equations Eq.4₁₀，x₁₁，x₁₂To yield the system of equations eq.5. The system of equations eq.5 is shown in fig. 44.

Psi can be provided using equation system Eq.5₁ ^IAnd Ψ₂ ^I。

Third line (line c) of the first cross-section

The third line of the first cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246. By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the third line of the first section yields an equation system eq.6 with t_icI is 0.. 4 as an unknown.

By passing through t_icPsi dependence of i ═ 0, 1_i ^IThe calculated value of i ═ 0.. 2 replaces t_icI-0, 1, and by adding a cofactor x respectively associated with each element of the added row in the system of equations₁₃，x₁₄，x₁₅，x₁₆The line of (c) yields the equation system eq.7. The system of equations eq.7 is shown in fig. 45. Then t may be provided_icAn expression of i ═ 2.. 4.

Using t generated by first, second and third lines of a first cross-section_2iC, yielding an expression dependent on Ψ_i ^IEquation system eq.8 of 3.. 5. In a preferred embodiment of the invention, the cofactor x₁₇，x₁₈，x₁₉，x₂₀Is added to the system of equations eq.8, each cofactor is associated with each element of the added row in the system of equations eq.8, respectively.

The system of equations eq.8 is shown in fig. 46.

Providing Ψ by solving equation System Eq.8_i ^I，i＝3...5。

Fourth line of the first section (line d)

The fourth line of the first cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246. By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the fourth line of the first section yields an equation system eq.9 with t_icI is 0.. 4 as an unknown.

By passing through t_idPsi dependence of i-0_i ^IReplacing t with the calculated value of i ═ 0.. 5_idI-0.. 2, and by adding a cofactor x that is respectively associated with each element of the added row in the system of equations eq.9₂₁，x₂₂，x₂₃The line of (c) yields the system of equations eq.10. The system of equations eq.10 is shown in fig. 47. T can then be provided by solving the system of equations Eq.10_idAnd i is an expression of 3, 4.

Using t generated from first, second, third and fourth lines of the first cross-section_3iAn expression for i.. d, which yields the expression associated with Ψ_i ^IEquation system eq.11 for i 6.. 9

In a preferred embodiment of the invention, the cofactor x₂₄，x₂₅，x₂₆，x₂₇，x₂₈Is added to the system of equations eq.11, each cofactor being associated with a respective row added to the system of equations eq.11Each element. The resulting system of equations eq.11 is shown in fig. 48.

Ψ can be calculated by solving the equation System Eq.11_i ^I，i＝6...9。

Fifth line of the first section (line e)

The fifth line of the first cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246. By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to the poles and non-poles of the fifth line of the first section yields the equation system eq.12.

Using psi generated from first, second, third and fourth lines of a first cross-section_i ^I9 may introduce y_iI ═ 3.. 5. In equation 4 α t applied to the fifth line₀+βt₁+2γt₂+δt₃+4εt₄Substitution of y in ═ 0_iI-3.. 5 and applications t_4iThe expression i ═ a.. d, can be used in Ψ_i ^IThe equation system eq.13 is generated at 0.. 9.

Cofactor x associated with each element of the added row, respectively₂₉，x₃₀，x₃₁，x₃₂，x₃₃，x₃₄Is added to the system of equations eq.13. The resulting system of equations eq.14 is shown in fig. 49.

Solving the equation System Eq.14 to provide Ψ_i ^I，i＝10...14。

It can thus be understood that Ψ now_i ^II is defined in its entirety as 0.. 14. The skilled person will then understand that the factor z is provided in the first cross section₄₅，z₄₆，z₄₇，z₄₈。

A new section is selected according to step 254 of fig. 33 because no other lines are available in the current section. In a preferred embodiment and as described above, the second cross-section is selected after processing the first cross-section.

First line (line f) of the second section

The first line of the second cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246. By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the first line of the second section yields an equation system eq.15 with t_ifI is 0.. 4 as an unknown. T is_4fCalculated using the results from the first section, as will be noted by those skilled in the art.

Then define y₆。

A new system of equations eq.16 is thus generated using system of equations eq.15.

Cofactor x associated separately with each element of the added row₃₅，x₃₆，x₃₇，x₃₈，x₃₉Is added to the system of equations eq.16. The resulting system of equations eq.17 is shown in fig. 50.

Ψ can be calculated by solving the system of equations Eq.17 shown in FIG. 50₀ ^II。t_ifThe expression of i ═ 1.. 3 is also available.

Second line (line g) of second cross section

The second line of the second cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the second line of the second section yields an equation system eq.18 with t_ifI is 0.. 4 as an unknown. Introduction of y₇To provide t_4g。

Then t is replaced in the equation system Eq.18_0gAnd t_4g. And then a cofactor x associated with each element of the added row, respectively₄₀，x₄₁，x₄₂，x₄₃Is added to the system of equations eq.18. The resulting system of equations eq.19 is shown in fig. 51.

T may be provided by solving the system of equations Eq.19 shown in FIG. 51_igAn expression of 1.. 3.

t_1gAnd t_ifIs used to generate a new system of equations eq.20. And then a cofactor x associated with each element of the added row, respectively₄₄，x₄₅，x₄₆Is added to the system of equations eq.20. The resulting system of equations eq.21 is shown in fig. 52.

Ψ can be provided by solving the equation system Eq.21_i ^II，i＝1...2。

Third line (line h) of second cross-section

The third line of the second cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to all poles and non-poles of the third line of the second section yields an equation system eq.22 with t_ihI is 0.. 4 as an unknown.

t_ihI-0, 1 using the previously calculated Ψ_i ^III is calculated as 1, 2. t is t_4hCalculations were performed using previous cross-sectional calculations. By t_4hIntroduction of y₈. And also introduce y₉。

Then y is replaced in the equation system Eq.22₈And y₉. And then a cofactor x associated with each element of the added row, respectively₄₇，x₄₈，x₄₉Is added to the system of equations eq.22. Equation of the resultThe system eq.23 is shown in fig. 53.

Then obtain t_ihAnd i is an expression of 2 and 3.

Then using t_2iI ═ f, g, h yields the system of equations eq.23. And then a cofactor x associated with each element of the added row, respectively₅₀，x₅₁，x₅₂，x₅₃Is added to the system of equations eq.23. The resulting system of equations eq.24 is shown in fig. 54. The system of equations Eq.24 solving the results provides Ψ_i ^II，i＝3...5。

Fourth line of the second section (line i)

A fourth line of the second cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

By applying equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Applying 0 to the poles and non-poles of the fourth line of the second section yields an equation system eq.25.

Introduction of y_i10.. 12. Then Ψ_i ^III 0.. 2 is injected into equation eq.25.

Then t_3iI ═ f, g, h, i are used to generate the system of equations eq.26.

And then a cofactor x associated with each element of the added row, respectively₅₄，x₅₅，x₅₆，x₅₇，x₅₈Is added to the system of equations eq.26. The resulting system of equations eq.27 is shown in fig. 55.

Solving the equation System Eq.27 to provide Ψ_i ^II，i＝6...9。

Then Ψ₆ ⁱII is used to generate an equation system eq.28 having z₄₁₉，z₄₉As an unknown. And then a cofactor x associated with each element of the added row, respectively₅₉，x₆₀，x₆₁Is added to the system of equations eq.28. The resulting system of equations eq.29 is shown in fig. 56A. Solving the equation System Eq.29 provides z₄₁₉，z₄₉。

Then Ψ₇ ⁱII is used to generate an equation system eq.30 having z₄₁₈，z₄₈As an unknown. And then a cofactor x associated with each element of the added row, respectively₆₂，x₆₃，x₆₁Is added to the system of equations eq.30. The resulting system of equations eq.31 is shown in fig. 56B. Solving the equation System Eq.31 provides z₄₁₈，z₄₈。

Then Ψ₈ ⁱII is used to generate an equation system eq.32 having z₄₁₇，z₄₇As an unknown. And then a cofactor x associated with each element of the added row, respectively₆₄，x₆₅，x₆₁Is added to the system of equations eq.32. The resulting system of equations eq.33 is shown in fig. 56C. Solving the equation System Eq.33 provides z₄₁₇，z₄₇。

Then Ψ₉ ⁱII is used to generate an equation system eq.34 having z₄₁₆，z₄₆As an unknown. And then a cofactor x associated with each element of the added row, respectively₆₆，x₆₇，x₆₁Is added to the system of equations eq.34. The resulting system of equations eq.35 is shown in fig. 56D. Solving the equation System Eq.35 provides z₄₁₆，z₄₆。

Psi can be provided from the foregoing equation for the second section_i ^II，i＝1...14。

A new section is selected according to step 254 of fig. 33 because no other lines are available in the current section. A third cross-section is selected after solving the second cross-section.

First line of third section (line j)

Those skilled in the artThe practitioner will understand that Ψ is known_i ^III 6.. 14 because they depend on previously calculated coefficients.

The first line of the third cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

Then calculate t_3j，t_4jAnd in equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Replace it in 0.

Then introduce y_i，i＝13...16。

The system of equations eq.36 is then generated with the poles and non-poles of the first line of the third cross section.

And then a cofactor x associated with each element of the added row, respectively₆₈，x₆₉，x₇₀，x₇₁Is added to the system of equations eq.36. The resulting system of equations is shown in fig. 57.

The resulting system of equations then provides t_ijAn expression of i ═ 0.. 2.

Second line of third section (line k)

The second line of the third cross-section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

Then sum equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄Together with 0 the system of equations eq.37 is generated using the poles and non-poles of the second line of the third section. Then using t_ikI is 0, 3, 4 introduces y_i20, i ═ 17. And then a cofactor x associated with each element of the added row, respectively₇₂，x₇₃，x₇₄Is added to the system of equations eq.37. The resulting system of equations eq.38 is shown in fig. 58.

T is then provided using the system of equations Eq.38_ik1.. 2 ofAnd (5) expressing.

By t_IiJ, k may yield an equation system eq.39 having Ψ_i ^IIIAnd i is 1, 2 as an unknown number. And then a cofactor x associated with each element of the added row, respectively₇₅，x₇₆，x₇₇Is added to the system of equations eq.39. The resulting system of equations eq.40 is shown in fig. 59. Solving equation System Eq.40 may provide Ψ_i ^III，i＝1，2。

Third line of third section (line l)

A third line of the third cross section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

Then sum equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄The poles and non-poles of the third line of the third section are applied together 0.

Then t is_ilI-0.. 4 injects the equation.

By t_2iJ, k and this equation yield the system of equations eq.41.

And then a cofactor x associated with each element of the added row, respectively₇₈，x₇₉，x₈₀，x₈₁Is added to the system of equations eq.41. The resulting system of equations eq.42 is shown in fig. 60.

Psi can be generated using equation system Eq.42_i ^III，i＝3...5。

By means of psi₃ ⁱI.. III may yield an equation system eq.43 to be used for calculating z₄₂₂，z₄₂₈，z₄₁₂. And then a cofactor x associated with each element of the added row, respectively₈₂，x₈₃，x₈₄，x₈₅Is added to the system of equations eq.43. The resulting system of equations eq.44 is shown in fig. 61A. Equation solving system Eq.44Supply z₄₂₂，z₄₂₈，z₄₁₂Is described in (1).

By means of psi₄ ⁱI.. III may yield an equation system eq.45 that will be used to calculate z₄₂₇，z₄₂₁，z₄₁₁. And then a cofactor x associated with each element of the added row, respectively₈₅，x₈₇，x₈₈，x₈₅Is added to the system of equations eq.45. The resulting system of equations Eq.46 is shown in FIG. 61B. Solving the equation System Eq.46 provides z₄₂₇，z₄₂₁，z₄₁₁Is described in (1).

By means of psi₅ ⁱI.. III may yield an equation system eq.47 to be used for calculating z₄₂₆，z₄₂₀，z₄₁₀. And then a cofactor x associated with each element of the added row, respectively₈₉，x₉₀，x₉₁，x₈₅Is added to the system of equations eq.47. The resulting system of equations eq.48 is shown in fig. 61C. Solving the equation System Eq.48 provides z₄₂₆，z₄₂₀，z₄₁₀Is described in (1).

A new section is selected according to step 254 of fig. 33 because no other lines are available in the current section. In a preferred embodiment and as described above, the fourth cross-section is selected after the third cross-section is solved.

First line of fourth section (line m)

Those skilled in the art will appreciate that Ψ is known from previous calculations_i ^IV，i＝3...14。

The first line of the fourth cross-section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

The poles and non-poles are then applied to equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄The equation system eq.49, which has t, is generated when the equation system is 0_imI is 0, 1 is asAnd (6) knowing the number.

Then introduce y_i，i＝33...35。

And then a cofactor x associated with each element of the added row, respectively₉₂，x₉₃，x₉₄Is added to the system of equations eq.49. The resulting system of equations eq.50 is shown in fig. 62.

Solving the equation System Eq.50 provides t_im，i＝0，1

Second line of fourth cross-section (line n)

The second line of the fourth cross-section is selected according to step 244 of fig. 33.

A pair of points is selected according to step 246.

The poles and non-poles of the second line of the fourth cross section are then applied to equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄The new equation eq.51 results in 0.

The new equation Eq.51 and the previous equation t_2mCombine to produce the system of equations Eq.52 having Ψ_i ^IVAnd i is 1 and 2 as an unknown number.

And then a cofactor x associated with each element of the added row, respectively₉₅，x₉₆，x₉₇Is added to the system of equations eq.52. The resulting system of equations eq.53 is shown in fig. 63.

Providing Ψ by the resulting system of equations Eq.53_i ^IVAnd i is an expression of 1, 2.

Then may use Ψ₁ ⁱIV and Ψ₂ ^jIV calculating z₄₃₃，z₄₃₀，z₄₂₄，z₄₁₄，z₄₃₂，z₄₂₉，z₄₂₃，z₄₁₃。

By means of psi₁ ⁱIV yields the equation system eq.54. Then add each element of the row separatelyRelative cofactor x₉₈，x₉₉，x₁₀₀，x₁₀₁，x₁₀₂Is added to the system of equations eq.54. The resulting system of equations Eq.55 is shown in FIG. 64A.

Solving the equation System Eq.55 provides z₄₃₃，z₄₃₀，z₄₂₄，z₄₁₄。

By means of psi₂ ⁱIV generates the equation system eq.56. And then a cofactor x associated with each element of the added row, respectively₁₀₃，x₁₀₄，x₁₀₅，x₁₀₆，x₁₀₇Is added to the system of equations eq.56. The resulting system of equations eq.57 is shown in fig. 64B.

Solving the equation System Eq.57 provides x₁₀₃，x₁₀₄，x₁₀₅，x₁₀₆，x₁₀₇，z₄₃₂，z₄₂₉，z₄₂₃，z₄₁₃。

A new section is selected according to step 254 of fig. 33 because no other lines are available in the current section. In a preferred embodiment and as described above, the fifth cross-section is selected after the fourth cross-section is solved.

First line of fifth section (line o)

Those skilled in the art will appreciate that Ψ is now known_i ^V，i＝1...14。

A pair of points is selected according to step 246.

The poles and non-poles are then applied to equation 4 α t₀+βt₁+2γt₂+δt₃+4εt₄The new equation eq.58 results in 0.

Now introduce y_i，i＝48...51。

By t_0oExpression of (2) and Ψ₀ ⁱIV may yield an equation system eq.59 having z₄₃₅，z₄₃₄，z₄₃₁，z₄₂₅，z₄₁₅As an unknown. And then a cofactor x associated with each element of the added row, respectively₁₀₇，x₁₀₈，x₁₀₉，x₁₁₀，x₁₁₁，x₁₁₂Is added to the system of equations eq.59. The resulting system of equations eq.60 is shown in fig. 65.

Solving this equation system provides z₄₃₅，z₄₃₄，z₄₃₁，z₄₂₅，z₄₁₅Is described in (1).

Those skilled in the art will now understand that all coefficients for the quadric surface have now been determined.

Polar relationship

It is understood that the polarity condition expressed by equation 0 may be expressed otherwise.

As described above, algebraic expressions depend on x₀，x₁，x₂，x₃I.e. F (x)₀，x₁，x₂，x₃) 0. It will be appreciated that this can be done by first calculating F (λ Y + μ Z), where Z is another point, and then doing so in the resulting equation by λ^n-1·μ_iFactorization (i ═ 0.. n) can yield a Taylor expansion of the algebraic surface with respect to the first point Y. R ═ λ Y + μ Z is defined as a non-pole.

Taylor expansion of an algebraic surface about a first point Y into

Then:

a pole may be defined by C ═ ρ Y + σ Z.

By replacing Y with R and Z with C in the latter equation, Δ can be determined^pcF (R). By mixing of^pcF (R) is identified asWhereinCan provide alpha^p _n，iWhere p is the number of polarities. The relationship between the poles and non-poles can then be generated.

Referring now to FIG. 66, a plot for z is shown in FIG. 66^pAlpha of n^p _n，1，α^p _n，2，α^p _n，3，α^p _n，4The graph represents an equation characterizing a straight line for p-th polarity. Those skilled in the art will appreciate that many expressions of polarity may be used accordingly.

Referring now to FIG. 67, a view of a 1 st pole surface is shown. More specifically, the figure shows a quadric surface with a pole and a non-pole located on the pole surface.

Referring now to fig. 68, a view of a 2 nd polar plane is shown. More specifically, the figure shows a quadric surface with two poles, each of which lies in the 2 nd-order polar plane, and a non-pole.

Referring now to fig. 69, a view of a 3-dimensional polar plane is shown. More specifically, the figure shows a quadric surface with one and three poles and one non-pole located on a 3 rd-order polar plane.

Deformation of an existing surface

In another embodiment, the invention is used for the deformation of an existing object by an operator. The object may or may not have been created using the present invention.

According to a first step, the operator selects a number of times a figure depicting the object or the part of the object under investigation is depicted. Depending on the number of times chosen, a number of control points, as described above, must be placed to define a figure depicting the object or the part of the object under study.

According to a second step, the operator provides a reference cell. The reference cell is placed in an advantageous position depending on the type of deformation the operator wishes to produce on the object.

According to a third step, the operator places at least one control point in an advantageous position, depending on the type of deformation the operator wishes to produce on the object.

According to a fourth step, the operator performs a deformation of the graph generated with the control points.

In an alternative embodiment, the operator directly performs the deformation on the object by placing the reference cell in an advantageous position and placing the control points of the object, and by manipulating at least one of the control points according to the desired deformation.

Claims (12)

1. A method of generating an m-degree form using algebraic coefficients of an equation defining the m-degree form in an n-dimensional space using a plurality of points defining the form, the method comprising the steps of:

providing a frame comprising more than one geometric surface, each of the more than one geometric surfaces comprising at least one line, each of the at least one line comprising at least one of the plurality of points;

providing a solution equation with equations defining the m-degree pattern, the solution equation being dependent on at least three points;

for each of a plurality of points, generating a corresponding equation by applying the solution equation to each of the plurality of points and to two selected points, the two selected points being located on one of the plurality of geometric surfaces, the two selected points defining a line of the skeleton, each of the plurality of points being on the line;

solving the generated corresponding equations in an interactive manner to provide algebraic coefficients of the m-degree graph, the interactive manner existing among processing the generated equations with each of the plurality of points in an order existing among selecting the generated corresponding equations, the corresponding equations corresponding to a point located on a line having a highest number of points thereon, the line being located on at least one geometric surface of the framework, the geometric surface having the highest number of lines thereon;

generating an m-degree graph using the provided algebraic coefficients of the m-degree graph.

2. The method of claim 1, wherein: wherein one point may be a pole or a non-pole, the solution equation being generated using equations defining the m-degree pattern and a polar relationship, the solution equation being dependent on four points, two of the four points being a pole and its associated non-pole, the other two of the four points defining a line of the framework on which the pole and its associated non-pole are located.

3. The method of claim 1, wherein: wherein more than one of the geometric surfaces is planar.

4. The method of claim 1, wherein: wherein the n-dimensional space is a projection space.

5. The method of claim 1, wherein: wherein the geometric space is an (n +1) -dimensional Euclidean space.

6. The method of claim 1, wherein: further comprising the step of storing the algebraic coefficients of the m-degree form together with an identifier.

7. The method of claim 4, wherein: wherein the projection space is a 4-dimensional space, wherein the graph is in a 3-dimensional geometric space.

8. The method of claim 1, wherein: wherein the generation of the m-degree graph using the algebraic coefficients of the provided m-degree graph is performed using a ray tracing technique.

9. An apparatus for generating an m-degree form using algebraic coefficients of an equation defining the m-degree form in an n-dimensional space using a plurality of points defining the form, the apparatus comprising:

an object store, the object comprising more than one geometric surface, each of the more than one geometric surfaces comprising at least one line, each of the at least one line comprising at least one of the plurality of points;

a control point to algebraic coefficient converter unit receiving each of the plurality of points and a solution equation defined by the equation defining the m-degree graph and dependent on at least three points, the control point to algebraic coefficient converter unit generating a corresponding equation by applying the solution equation to each of the plurality of points and to two points located on one of more than one geometric surface, the two selected points defining a line of the skeleton on which each of the plurality of points is located, and solving the generated corresponding equation in an interactive manner to provide algebraic coefficients for the m-degree graph; the manner of interaction residing in the equations generated by processing each of the plurality of points in an order that resides in selecting a corresponding equation generated corresponding to a point located on a line having a highest number of points thereon, the line being located on at least one geometric surface of the framework on which the highest number of lines reside; and

an output unit receives the algebraic coefficients of the supplied m-degree patterns and forms the m-degree patterns.

10. The apparatus of claim 9, wherein: wherein the output unit comprises a ray tracing unit, which further receives a view signal point.

11. A method of monitoring the deformation of an m-degree form using algebraic coefficients of an equation defining the m-degree form in an n-dimensional space using a plurality of points defining the form, the method comprising the steps of:

providing a skeleton according to the planned modification of the graph, the skeleton comprising more than one geometric surface, each of the more than one geometric surfaces comprising at least one line, each of the at least one line comprising at least one of the plurality of points;

modifying at least one portion of the provided framework based on the planned graphical modification;

providing a solution equation with equations defining the m-degree pattern, the solution equation being dependent on at least three points;

generating, for each of the plurality of points, a corresponding equation by applying the solution equation to each of the plurality of points and to two points located on one of more than one geometric surface, the two selected points defining a line of the skeleton, each of the plurality of points being on the line;

solving the generated corresponding equations in an interactive manner to provide algebraic coefficients of the m-degree graph; the manner of interaction residing in the equations generated by processing each of the plurality of points in an order that resides in selecting a corresponding equation generated corresponding to a point located on a line having a highest number of points thereon, the line being located on at least one geometric surface of the framework on which the highest number of lines reside;

outputting the algebraic coefficients of the modified graph.

12. The method of claim 11, further comprising the step of generating the modified m-degree form using the output algebraic coefficients of the modified m-degree form.