US5402532A — Direct display of CSG expression by use of depth …

Claim 1

Independent Claim 178 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects, said one or more primitive objects each being comprised of a plurality of faces; means for repetitively evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; and means, responsive to the operation of the evaluating means, for detecting a condition wherein a plurality of consecutive evaluations occur without causing a change in a tentative image of the product, the detecting means including means for counting consecutive evaluations that result in no change in the tentative image of the product and for generating a signal for causing an operation of the evaluating means to terminate when a number of consecutive evaluations occur that are sufficient to construct the correct image of the product, wherein the number is less than the square of the number of primitive objects in the product.

Claim 2

Dependent Claim 178 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects, said one or more primitive objects each being comprised of a plurality of faces; means for repetitively evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; and means, responsive to the operation of the evaluating means, for detecting a condition wherein a plurality of consecutive evaluations occur without causing a change in a tentative image of the product, the detecting means including means for counting consecutive evaluations that result in no change in the tentative image of the product and for generating a signal for causing an operation of the evaluating means to terminate when a number of consecutive evaluations occur that are sufficient to construct the correct image of the product, wherein the number is less than the square of the number of primitive objects in the product.

Claim 3

Dependent Claim 100 words

Apparatus as set forth in claim 1 wherein the evaluating means includes a plurality of depth buffer means and further includes means for comparing a depth, relative to a reference plane, of a point on a face of the primitive object being evaluated to a depth of a first point stored within a first depth buffer means (Z3) and also to a depth of a second point stored within a second depth buffer means (Z2), wherein the depths of the first point and the second point are determined during a prior evaluation of the same or of another primitive object.

Claim 4

Dependent Claim 38 words

Apparatus as set forth in claim 2 wherein the evaluating means includes means, responsive to the operation of the comparing means, for replacing the depth stored within Z3 by the depth of the current point (Z) when Z2<Z<Z3.

Claim 5

Dependent Claim 67 words

Apparatus as set forth in claim 2 wherein the evaluating means includes means, responsive to the operation of the comparing means, for replacing the depth stored within Z3 with the depth of the current point (Z) when Z2<Z<Z3 and further including means, responsive to the operation of the replacing means, for resetting the counting means to a value that is less than the number of primitive objects.

Claim 6

Dependent Claim 44 words

Apparatus as set forth in claim 2 wherein the evaluating means includes means, responsive to the operation of the comparing means, for replacing the depth stored within Z3 with the depth of the current point (Z) when Z2+eps<Z<Z3+eps, wherein eps is a tolerance value.

Claim 7

Dependent Claim 76 words

Apparatus as set forth in claim 2 wherein the evaluating means includes means, responsive to the operation of the comparing means, for replacing the depth stored within Z3 with the depth of the current point (Z) when Z2<Z<Z3, and wherein the evaluating means further includes means for recording, for a line projected from the reference plane through the current point, a number of faces, of the primitive object being evaluated, that are intersected by the line.

Claim 8

Dependent Claim 44 words

Apparatus as set forth in claim 6 and further comprising second means for replacing the depth stored within Z2 with the depth stored within Z3 only when the evaluating means indicates that the current point lies on or within the primitive object being evaluated.

Claim 9

Dependent Claim 56 words

Apparatus as set forth in claim 7 wherein Z3 has an associated intensity buffer I3, wherein Z2 has an associated intensity buffer I2, and wherein the second replacing means also replaces the intensity stored within I2 with the intensity stored within I3 only when the current point lies on a front face of the primitive object.

Claim 10

Dependent Claim 44 words

Apparatus as set forth in claim 7 and further comprising a third depth buffer means (Z1), and also comprising third means, responsive to the generation of the signal, for replacing the depth stored within Z1 with the depth stored within Z2 only where Z2<Z1.

Claim 11

Dependent Claim 92 words

Apparatus as set forth in claim 2 wherein the the evaluating means includes means for projecting from points on a face of the primitive object to the reference plane, the reference plane being partitioned into a plurality of pixels, the evaluating means further including means, responsive to the operation of the projecting means, for determining a location of a pixel or pixels covered by the projection of the face and for selecting only those pixels having a center that lies within the projection from points on the face of the primitive object.

Claim 12

Dependent Claim 63 words

Apparatus as set forth in claim 2 wherein the reference plane is partitioned into a plurality of pixels and wherein the evaluating means includes means for projecting from a point on a face of the primitive object to the reference plane to determine a depth of the point from a pixel, and wherein the depth is referenced to a center of the pixel.

Claim 13

Dependent Claim 51 words

Apparatus as set forth in claim 2 and further comprising shadow depth buffer means (ZS) for storing a depth, referenced to a plane of a light source that illuminates the solid, of a projection from a point on a face of the primitive object to the plane of the light source.

Claim 14

Dependent Claim 40 words

Apparatus as set forth in claim 12 and further comprising means for comparing a depth of a point, referenced to the reference plane, to the depth stored within ZS for determining if the point is visible from the light source.

Claim 15

Dependent Claim 239 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and means for repetitively evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; wherein the evaluating means includes means for projecting from points on a face of the primitive object to a reference plane, the reference plane being partitioned into a plurality of pixels, the evaluating means further including means, responsive to the operation of the projecting means, for determining a location of a pixel or pixels covered by the projection of the face and for selecting only those pixels having a consistent point, independent of an inclination of the face with respect to the reference plane, that lie within the projection from the points on the face of the primitive object, the evaluating means further including means for determining a reference depth to an initially evaluated pixel that is covered by the projection of the face, the reference depth being determined from the consistent point of the initially evaluated pixel to a point on the face that projects onto the initially evaluated pixel, and wherein the reference depth is subsequently employed by the evaluating means in determining a depth to points on the face that project onto other pixels.

Claim 16

Dependent Claim 239 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and means for repetitively evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; wherein the evaluating means includes means for projecting from points on a face of the primitive object to a reference plane, the reference plane being partitioned into a plurality of pixels, the evaluating means further including means, responsive to the operation of the projecting means, for determining a location of a pixel or pixels covered by the projection of the face and for selecting only those pixels having a consistent point, independent of an inclination of the face with respect to the reference plane, that lie within the projection from the points on the face of the primitive object, the evaluating means further including means for determining a reference depth to an initially evaluated pixel that is covered by the projection of the face, the reference depth being determined from the consistent point of the initially evaluated pixel to a point on the face that projects onto the initially evaluated pixel, and wherein the reference depth is subsequently employed by the evaluating means in determining a depth to points on the face that project onto other pixels.

Claim 17

Dependent Claim 185 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and means for evaluating the one or more primitive objects to determine displayable faces thereof; wherein the evaluating means includes means for comparing a depth, relative to a reference plane, of a point on a face of the primitive object being evaluated to a depth of a first point stored within a first depth buffer means and also to a depth of a second point stored within a second depth buffer means, and wherein the comparing means makes the depth comparison using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to compensate for a resolution of the first and the second depth buffer means, said predetermined depth tolerance value enabling a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 18

Dependent Claim 185 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and means for evaluating the one or more primitive objects to determine displayable faces thereof; wherein the evaluating means includes means for comparing a depth, relative to a reference plane, of a point on a face of the primitive object being evaluated to a depth of a first point stored within a first depth buffer means and also to a depth of a second point stored within a second depth buffer means, and wherein the comparing means makes the depth comparison using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to compensate for a resolution of the first and the second depth buffer means, said predetermined depth tolerance value enabling a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 19

Dependent Claim 229 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and means for evaluating the one or more primitive objects to determine displayable faces thereof; wherein the evaluating means includes first means for repetitively comparing depths of points, that lie on faces of primitive objects being evaluated, to previously determined depths and for storing depths, within a depth buffer means (Z1), of each point determined to lie on displayable faces of the primitive objects, the depth being relative to a reference plane, wherein the evaluating means includes second means, responsive to a termination of operation of the first means, for comparing points on front-facing faces of the primitive objects and for storing within an intensity buffer means (I1) intensities of points on the front-facing surfaces that have a depth determined to be equal to a corresponding depth stored within Z1; and wherein, the first and the second comparing means each make depth comparisons using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to enable a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 20

Dependent Claim 229 words

Apparatus for generating displayable information that is expressive of a three dimensional solid, comprising: means for expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and means for evaluating the one or more primitive objects to determine displayable faces thereof; wherein the evaluating means includes first means for repetitively comparing depths of points, that lie on faces of primitive objects being evaluated, to previously determined depths and for storing depths, within a depth buffer means (Z1), of each point determined to lie on displayable faces of the primitive objects, the depth being relative to a reference plane, wherein the evaluating means includes second means, responsive to a termination of operation of the first means, for comparing points on front-facing faces of the primitive objects and for storing within an intensity buffer means (I1) intensities of points on the front-facing surfaces that have a depth determined to be equal to a corresponding depth stored within Z1; and wherein, the first and the second comparing means each make depth comparisons using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to enable a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 21

Independent Claim 167 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects, the one or more primitive objects each being comprised of a plurality of faces; repetitively evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; for each repetitive evaluation, incrementing a count; detecting a condition wherein the count indicates that a plurality of consecutive evaluations occur without causing a change in a tentative image of the product; and generating a signal for causing the step of repetitively evaluating to terminate when a number of consecutive evaluations occur that result in no change in the tentative image of the product, wherein the number is sufficient to construct the correct image of the product and is less than the square of the number of primitive objects in the product.

Claim 22

Independent Claim 167 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects, the one or more primitive objects each being comprised of a plurality of faces; repetitively evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; for each repetitive evaluation, incrementing a count; detecting a condition wherein the count indicates that a plurality of consecutive evaluations occur without causing a change in a tentative image of the product; and generating a signal for causing the step of repetitively evaluating to terminate when a number of consecutive evaluations occur that result in no change in the tentative image of the product, wherein the number is sufficient to construct the correct image of the product and is less than the square of the number of primitive objects in the product.

Claim 23

Dependent Claim 94 words

A method as set forth in claim 17 wherein the step of evaluating includes a step of comparing a depth, relative to a reference plane, of a point on a face of the primitive object being evaluated to a depth of a first point stored within a first depth buffer means (Z3) and also to a depth of a second point stored within a second depth buffer means (Z2), wherein the depths of the first point and the second point are determined during a prior evaluation of the same or of another primitive object.

Claim 24

Dependent Claim 33 words

A method as set forth in claim 18 wherein the step of evaluating includes a step of replacing the depth stored within Z3 with the depth (z) of a current point when Z2<Z<Z3.

Claim 25

Dependent Claim 55 words

A method as set forth in claim 18 wherein the step of evaluating includes a step of replacing the depth stored within Z3 with the depth (Z) of a current point when Z2<Z<Z3, and further includes a step of resetting the counting means to a value that is less than the number of primitive objects.

Claim 26

Dependent Claim 39 words

A method as set forth in claim 18 wherein the step of evaluating includes a step of replacing the depth stored within Z3 with the depth of the current point (Z) when Z2+eps<Z<Z3+eps, wherein eps is a tolerance value.

Claim 27

Dependent Claim 63 words

A method as set forth in claim 18 wherein the step of evaluating includes the steps of: replacing the depth stored within Z3 with the depth (Z) of a current point when Z2<Z<Z3; and recording, for a line projected from the reference plane through the current point, a number of faces, of the primitive object being evaluated, that are intersected by the line.

Claim 28

Dependent Claim 47 words

A method as set forth in claim 22 and further comprising a step of replacing the depth stored within Z2 with the depth stored within Z3 only when the recorded number of faces indicates that the current point lies on or within a primitive object being evaluated.

Claim 29

Dependent Claim 56 words

A method as set forth in claim 23 wherein Z3 has an associated intensity buffer I3, wherein Z2 has an associated intensity buffer I2, and further including a step of replacing the intensity stored within I2 with the intensity stored within I3 only when the current point lies on a front face of the primitive object.

Claim 30

Dependent Claim 38 words

A method as set forth in claim 23 wherein the step of generating a signal includes a step of replacing a depth stored within a third z-buffer means (Z1) with the depth stored within Z2, only if Z2<Z1.

Claim 31

Dependent Claim 83 words

A method as set forth in claim 18 wherein the reference plane is partitioned into a plurality of pixels and wherein the step of evaluating includes the steps of projecting from points on a face of the primitive object to the reference plane; and determining which pixel or pixels are covered by the face, wherein a pixel is determined to be covered only if a center of the pixel lies within the projection from the points on the face of the primitive object.

Claim 32

Dependent Claim 67 words

A method as set forth in claim 18 wherein the reference plane is partitioned into a plurality of pixels and wherein the step of evaluating includes a step of projecting from a point on a face of the primitive object to the reference plane to determine a depth of the point from a pixel, and wherein the determined depth is referenced to a center of the pixel.

Claim 33

Dependent Claim 60 words

A method as set forth in claim 18 and further comprising a step of storing a depth, referenced to a plane of a light source that illuminates the solid, of a projection from a point on a face of the primitive object to the plane of the light source, the depth being stored within a shadow depth buffer means (ZS).

Claim 34

Dependent Claim 42 words

A method as set forth in claim 28 and further comprising a step of comparing a depth of a point, referenced to the reference plane, to the depth stored within ZS for determining if the point is visible from the light source.

Claim 35

Independent Claim 236 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; wherein the step of evaluating includes a step of projecting from points on a face of the primitive object to a reference plane, the reference plane being partitioned into a plurality of pixels, to determine which pixel or pixels are covered by the face, the step of projecting including a step of determining that a given pixel is covered only if a consistent point of the pixel lies within the projection from the points on the face of the primitive object, the consistent point being independent of an inclination of the face with respect to the reference plane, wherein the step of evaluating includes an initial step of determining a reference depth to an initially evaluated pixel that is covered by the projection of the face, the reference depth being determined from the consistent point of the initially evaluated pixel to a point on the face that projects onto the initially evaluated pixel, and wherein the reference depth is subsequently employed in determining a depth to points on the face that project onto other pixels.

Claim 36

Independent Claim 236 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and evaluating a sequence of one or more primitive objects to determine a correct image of a product of the primitive objects; wherein the step of evaluating includes a step of projecting from points on a face of the primitive object to a reference plane, the reference plane being partitioned into a plurality of pixels, to determine which pixel or pixels are covered by the face, the step of projecting including a step of determining that a given pixel is covered only if a consistent point of the pixel lies within the projection from the points on the face of the primitive object, the consistent point being independent of an inclination of the face with respect to the reference plane, wherein the step of evaluating includes an initial step of determining a reference depth to an initially evaluated pixel that is covered by the projection of the face, the reference depth being determined from the consistent point of the initially evaluated pixel to a point on the face that projects onto the initially evaluated pixel, and wherein the reference depth is subsequently employed in determining a depth to points on the face that project onto other pixels.

Claim 37

Independent Claim 170 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and evaluating the one or more primitive objects to determine displayable faces thereof; wherein the step of evaluating includes a step of comparing a depth, relative to a reference plane, of a point on a face of the primitive object being evaluated to a depth of a first point stored within a first depth buffer means and also to a depth of a second point stored within a second depth buffer means, and wherein the step of comparing compares the depths to using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to enable a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 38

Independent Claim 170 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and evaluating the one or more primitive objects to determine displayable faces thereof; wherein the step of evaluating includes a step of comparing a depth, relative to a reference plane, of a point on a face of the primitive object being evaluated to a depth of a first point stored within a first depth buffer means and also to a depth of a second point stored within a second depth buffer means, and wherein the step of comparing compares the depths to using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to enable a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 39

Independent Claim 218 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and evaluating the one or more primitive objects to determine displayable faces thereof; wherein the step of evaluating includes the steps of repetitively comparing depths of points on faces of primitive objects being evaluated to previously determined depths; storing depths, within a depth buffer means (Z1), of each point determined to lie on displayable faces of the primitive objects, the depth being relative to a reference plane; comparing a depth of points on front-facing faces of the primitive objects to a corresponding depth stored within Z1; and storing within an intensity buffer means (I1) intensities of points on the front-facing surfaces having a depth determined to be equal to the corresponding depth stored within Z1; wherein the step of repetitively comparing and the step of comparing each compare the depths using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to enable a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 40

Independent Claim 218 words

A method for generating displayable information that is expressive of a three dimensional solid, comprising the steps of: expressing the solid in a Constructive Solid Geometry representation thereof so as to be comprised of one or more primitive objects; and evaluating the one or more primitive objects to determine displayable faces thereof; wherein the step of evaluating includes the steps of repetitively comparing depths of points on faces of primitive objects being evaluated to previously determined depths; storing depths, within a depth buffer means (Z1), of each point determined to lie on displayable faces of the primitive objects, the depth being relative to a reference plane; comparing a depth of points on front-facing faces of the primitive objects to a corresponding depth stored within Z1; and storing within an intensity buffer means (I1) intensities of points on the front-facing surfaces having a depth determined to be equal to the corresponding depth stored within Z1; wherein the step of repetitively comparing and the step of comparing each compare the depths using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors that occur at least during the determination of the displayable faces and to enable a correct interpretation, consistent with a definition of a solid, that eliminates dangling faces and edges.

Claim 41

Independent Claim 196 words

A method for generating displayable information for representing a solid that is expressed in a Constructive Solid Geometry representation that includes one or more primitives, the one or more primitives each being comprised of a plurality of faces, comprising the steps of: repetitively evaluating a sequence of one or more primitives to determine a correct image of a product of the primitives, the step of repetitively evaluating including a step of determining depths to faces of the primitives, referenced to a pixel plane, using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors and to reduce a probability of an occurrence of a singularity in the displayable information; for each repetitive evaluation, incrementing a count; detecting a condition wherein the count indicates that a plurality of consecutive evaluations occur without causing a change in a tentative image of the product; and generating a signal for causing the step of repetitively evaluating to terminate when a number of consecutive evaluations occur that are sufficient to construct the correct image of the product, wherein the number is less than the square of the number of primitives in the product.

Claim 42

Independent Claim 196 words

A method for generating displayable information for representing a solid that is expressed in a Constructive Solid Geometry representation that includes one or more primitives, the one or more primitives each being comprised of a plurality of faces, comprising the steps of: repetitively evaluating a sequence of one or more primitives to determine a correct image of a product of the primitives, the step of repetitively evaluating including a step of determining depths to faces of the primitives, referenced to a pixel plane, using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors and to reduce a probability of an occurrence of a singularity in the displayable information; for each repetitive evaluation, incrementing a count; detecting a condition wherein the count indicates that a plurality of consecutive evaluations occur without causing a change in a tentative image of the product; and generating a signal for causing the step of repetitively evaluating to terminate when a number of consecutive evaluations occur that are sufficient to construct the correct image of the product, wherein the number is less than the square of the number of primitives in the product.

Claim 43

Dependent Claim 86 words

A method as set forth in claim 33 wherein the step of repetitively evaluating includes an initial step of determining a reference depth to an initially evaluated pixel that is covered by the projection of a face, the reference depth being determined from a geometric center of the initially evaluated pixel to a point on the face that projects onto the initially evaluated pixel, and wherein the reference depth is subsequently employed in determining a depth to points on the face that project onto other pixels.

Claim 44

Dependent Claim 203 words

Apparatus for generating displayable information for representing a solid that is expressed in a Constructive Solid Geometry representation that includes one or more primitives, the one or more primitives each being comprised of a plurality of faces, comprising: means for repetitively evaluating a sequence of one or more primitives to determine a correct image of a product of the primitives, said means for repetitively evaluating including means for determining depths to faces of the primitives, referenced to a pixel plane, using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors and to reduce a probability of an occurrence of a singularity in the displayable information; means, responsive to each repetitive evaluation, for incrementing a counter means; means for detecting a condition wherein said counter means indicates that a plurality of consecutive evaluations occur without causing a change in a tentative image of the product; and means for generating a signal for causing the operation of said evaluating means to terminate when a number of consecutive evaluations occur that are sufficient to construct the correct image of the product, wherein the number is less than the square of the number of primitives in the product.

Claim 45

Dependent Claim 203 words

Apparatus for generating displayable information for representing a solid that is expressed in a Constructive Solid Geometry representation that includes one or more primitives, the one or more primitives each being comprised of a plurality of faces, comprising: means for repetitively evaluating a sequence of one or more primitives to determine a correct image of a product of the primitives, said means for repetitively evaluating including means for determining depths to faces of the primitives, referenced to a pixel plane, using a predetermined depth tolerance value, the predetermined depth tolerance value being selected to compensate for computation-induced round-off errors and to reduce a probability of an occurrence of a singularity in the displayable information; means, responsive to each repetitive evaluation, for incrementing a counter means; means for detecting a condition wherein said counter means indicates that a plurality of consecutive evaluations occur without causing a change in a tentative image of the product; and means for generating a signal for causing the operation of said evaluating means to terminate when a number of consecutive evaluations occur that are sufficient to construct the correct image of the product, wherein the number is less than the square of the number of primitives in the product.

Claim 46

Dependent Claim 84 words

Apparatus as forth in claim 35 wherein said evaluating means includes means for determining a reference depth to an initially evaluated pixel that is covered by the projection of a face, the reference depth being determined from a geometric center of the initially evaluated pixel to a point on the face that projects onto the initially evaluated pixel, and wherein the reference depth is subsequently employed by said evaluating means in determining a depth to points on the face that project onto other pixels.

Annuity Stage	Window Opens	Payment Deadline	Surcharge Period Until	Estimated Fee	Status
3.5 Years (Year 4)	1998-03-27	1998-09-26	1999-03-28	$2000.00	Window Closed / Lapsed
7.5 Years (Year 8)	2002-03-27	2002-09-26	2003-03-28	$3760.00	Window Closed / Lapsed
11.5 Years (Year 12)	2006-03-27	2006-09-26	2007-03-28	$7700.00	Window Closed / Lapsed

Direct display of CSG expression by use of depth buffers

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