Computer aided modeling technique for 3D free-form objects and apparatus thereof

Claim 3

A computer aided modeling technique for three-dimensional free-form objects according to claim 1, wherein n=2, and the sets Ξ q (q=1, 2, . . . . n) each having a finite number of points in the three-dimensional space comprise two sets Ξ 1 and Ξ 2 expressed by the following equations with respect to three positive numbers L X , L Y , L Z : ##EQU28##

Claim 4

A computer aided modeling technique for three-dimensional free-form objects according to claim 1, comprising in a configuration of the field generating primitive includes the steps of: determining the scalar quantity a which satisfies relation a>0; and determining the density distribution c(x,y,z) of the field generating primitive from equation c(x,y,z)=φ(f(x,y,z)-b+a) using a smooth function f(x,y,z) and a smooth monotone increasing function φ(ξ), said function f(x,y,z) defining a set of points (x,y,z) satisfying relation f(x,y,z)=b as the surface of object, a set of points (x,y,z) satisfying relation f(x,y,z)>b as the inside of object and a set of points (x,y,z) satisfying relation f(x,y,z)<b as the outside of object, with regard to a scalar quantity b, and said monotone increasing function φ(ξ) satisfying equations 0≦φ(ξ)≦2a and φ(a)=a.

Claim 5

A computer aided modeling technique for three-dimensional free-form objects according to claim 1, comprising the steps of: determining the scalar quantity a satisfing relation a>0; defining the function c(x,y,z) as follows using a function d associated with a smooth bijection (one-to-one onto-mapping) d: R 1 →R 2 from region R 1 to R 2 in the three-dimensional space and smooth inverse mapping d -1 : R 2 →R 1 c(x,y,z)=F(d -1 (x,y,z))(points (x,y,z) included in the region R 2 ) c(x,y,z)=0(points (x,y,z) not included in the region R 2 ); regarding said function c(x,y,z) as the density distribution of said field generating primitive; updating the density distribution Fq(x,y,z) as Fq(x,y,z)-{Fq(x,y,z)·D(x,y,z)/(2a)} using density distribution D(x,y,z) of the field generating primitive approximating the region R 1 in surface shape; and blending the updated density distribution Fq(x,y,z) and the field generating primitive c(x,y,z) according to the following equation, thereby generating density distribution Gq(x,y,z) and realizing a deformation of the object: G.sub.q (x,y,z)=F.sub.q (x,y,z)+c(x,y,z)-{Fq(x,y,z)·c(x,y,z)/(2a)}.

Claim 6

A computer aided modeling technique for three-dimensional free-form objects according to claim 1, comprising the step of modifying the sampling data Λ q ={μ p (i,j,k)|i+j+k=0,1, . . . , m, i≧0, j≧0, k≧0} directly and thus modifying the density distribution Fq(x,y,z) thereby to correct the density function F(x,y,z) for expressing the object.

Claim 7

A computer aided modeling technique for three-dimensional free-form objects, comprising the steps of: defining density distribution F(x,y,z) in a three-dimensional space x-y-z; determining a scalar quantity a; defining a set of points (x,y,z) satisfying the relation F(x,y,z)=a as a surface of object; defining a set of points (x,y,z) satisfying the relation F(x,y,z)>a as an inside of object; defining a set of points (x,y,z) satisfying the relation F(x,y,z)<a as an outside of object; producing density distribution c(x,y,z) by arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position; generating density distribution Gq(x,y,z) by blending the produced density distribution (x,y,z) and density Fq(x,y,z) (q=1, 2, . . . , n) already existing in thespace; determining data Λ q by sampling up to the mth-order partial derivative of the generated density distribution Gq(x,y,z) on the points included in the set Ξ q having a finite number of points in the three-dimensional space; generating smooth density distribution function Fq(x,y,z) for interpolating the sampling data An, and expressing new density distribution Fq(x,y,z) with the generated function thereby to update said density distribution Fq(x,y,z); assuming that density distribution F(x,y,z) obtained by convex combination of a number of n of smooth functions Fq(x,y,z) (q=1, 2, . . . ,n) at each coordinate point is density distribution F(x,y,z) for expressing an object; displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density function which is comprised of a convex combination of said updated Fq(x,y,z) density functions; and saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 8

A computer aided modeling technique for three-dimensional free-form objects, comprising the steps of: defining density distribution F(x,y,z) in a three-dimensional space x-y-z; determining a scalar quantity a; defining a set of points (x,y,z) satisfying the relation F(x,y,z)=a as a surface of object; defining a set of points (x,y,z) satisfying the relation F(x,y,z)>a as an inside of object; defining a set of points (x,y,z) satisfying the relation F(x,y,z)<a as an outside of object; producing density distribution c(x,y,z) by arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position; generating density distribution Gq(x,y,z) by blending the produced density distribution (x,y,z) and density Fq(x,y,z) (q=1, 2, . . . , n) already existing in thespace; determining data Λ q by sampling up to the mth-order partial derivative of the generated density distribution Gq(x,y,z) on the points included in the set Ξ q having a finite number of points in the three-dimensional space; generating smooth density distribution function Fq(x,y,z) for interpolating the sampling data An, and expressing new density distribution Fq(x,y,z) with the generated function thereby to update said density distribution Fq(x,y,z); assuming that density distribution F(x,y,z) obtained by convex combination of a number of n of smooth functions Fq(x,y,z) (q=1, 2, . . . ,n) at each coordinate point is density distribution F(x,y,z) for expressing an object; displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density function which is comprised of a convex combination of said updated Fq(x,y,z) density functions; and saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 9

A computer aided modeling technique for three-dimensional free-form objects according to claim 6, wherein n=2, and two sets Ξ 1 and Ξ 2 expressed as follows with respect to three positive numbers L X , L Y , L Z configure the sets Ξ q (q=1, 2, . . . , n) each having a finite number of points in the three-dimensional space: ##EQU29## .

Claim 10

A computer aided modeling technique for three-dimensional free-form objects, comprising the steps of: defining density distribution F(x,y,z) in a three-dimensional space x-y-z; determining a scalar quantity defining a set of points (x,y,z) satisfying relation F(x,y,z)=a as a surface of object; defining a set of points (x,y,z) satisfying relation F(x,y,z)>a as an inside of object; defining a set of points (x,y,z) satisfying relation F(x,y,z)<a as an outside of object; producing density distribution c(x,y,z) by arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position; blending the produced density distribution c(x,y,z) and density distribution Fq(x,y,z) (q=1, 2, . . . , n) already existing in the space using a smooth function Ψ(ξ,ζ) having two variables ξ,ζ according to equation Gq(x,y,z)=Ψ(Fq(x,y,z), c(x,y,z)) thereby to generate density distribution Gq(x,y,z); determining sampling data Λ q by sampling up to the mth-order partial derivative of the generated density distribution Gq(x,y,z) on the points included in a set Ξ q having a finite number of points in the three-dimensional space; generating a smooth function Fq(x,y,z) for interpolating the sampling data Λ q and expressing new density distribution Fq(x,y,z) by said function thereby to update the density distribution Fq(x,y,z); and assuming that density distribution F(x,y,z) obtained by convex combination of a number n of smooth functions Fq(x,y,z) (q=1, 2, . . . , n) at each coordinate point is density distribution F(x,y,z) for expressing an object; displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density function which is comprised of a convex combination of said updated Fq(x,y,z) density functions; and saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 11

A computer aided modeling technique for three-dimensional free-form objects, comprising the steps of: defining density distribution F(x,y,z) in a three-dimensional space x-y-z; determining a scalar quantity defining a set of points (x,y,z) satisfying relation F(x,y,z)=a as a surface of object; defining a set of points (x,y,z) satisfying relation F(x,y,z)>a as an inside of object; defining a set of points (x,y,z) satisfying relation F(x,y,z)<a as an outside of object; producing density distribution c(x,y,z) by arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position; blending the produced density distribution c(x,y,z) and density distribution Fq(x,y,z) (q=1, 2, . . . , n) already existing in the space using a smooth function Ψ(ξ,ζ) having two variables ξ,ζ according to equation Gq(x,y,z)=Ψ(Fq(x,y,z), c(x,y,z)) thereby to generate density distribution Gq(x,y,z); determining sampling data Λ q by sampling up to the mth-order partial derivative of the generated density distribution Gq(x,y,z) on the points included in a set Ξ q having a finite number of points in the three-dimensional space; generating a smooth function Fq(x,y,z) for interpolating the sampling data Λ q and expressing new density distribution Fq(x,y,z) by said function thereby to update the density distribution Fq(x,y,z); and assuming that density distribution F(x,y,z) obtained by convex combination of a number n of smooth functions Fq(x,y,z) (q=1, 2, . . . , n) at each coordinate point is density distribution F(x,y,z) for expressing an object; displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density function which is comprised of a convex combination of said updated Fq(x,y,z) density functions; and saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 12

A computer aided modeling apparatus for three-dimensional free-form objects in which a set of points satisfying relation F(x,y,z)=a is defined as a surface of object, a set of points (x,y,z) satisfying relation F(x,y,z)>a as an inside of object, and a set of points (x,y,z) satisfying relation F(x,y,z)<a as an outside of object, F(x,y,z) being density distribution in a three-dimensional space x-y-z and a being a scalar quantity, comprising: producing means for arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position thereby to produce density distribution c(x,y,z); generating means for blending the produced density distribution c(x,y,z) and density distribution Fq(x,y,z) (q=1, 2, . . . , n) already existing in the space using a smooth function Ψ(ξ,ζ) having two variables ξ, ζ according to equation Gq(x,y,z)=Ψ(Fq(x,y,z), c(x,y,z)) thereby to generate density distribution Gq(x,y,z); sampling data computing means for sampling up to the mth-order partial derivative of the generated density distribution G q (x,y,z) on the points included in a set Ξ q having a finite number of points in the three-dimensional space and determining sampling data Λ q ={μ p (i,j,k)={1/(i!j!k!)}∂ i+j+k G q (x p , y p , z p )/∂x i ∂y j ∂z k |i+j+k=0, 1, . . . , m, i≧0, j≧0, k≧0; all the points P=(x p ,y p ,z p ) included in Ξ q }) ; and updating means for generating a smooth function Fq(x,y,z) for interpolating the determined sampling data Λ q and expressing new density distribution Fq(x,y,z) with the generated function thereby to update said density distribution Fq(x,y,z), and assuming that the convex combination of a number n of smooth functions Fq(x,y,z) (q=1, 2 . . . . , n) at each coordinate point as density distribution F(x,y,z) representing for an object; means for displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density distribution functions which is comprised of a convex combination of said updated Fq(x,y,z) density distribution functions; means for altering the parameters of the c(x,y,z) density function while viewing said isosurfaces; and means for saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 13

A computer aided modeling apparatus for three-dimensional free-form objects in which a set of points satisfying relation F(x,y,z)=a is defined as a surface of object, a set of points (x,y,z) satisfying relation F(x,y,z)>a as an inside of object, and a set of points (x,y,z) satisfying relation F(x,y,z)<a as an outside of object, F(x,y,z) being density distribution in a three-dimensional space x-y-z and a being a scalar quantity, comprising: producing means for arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position thereby to produce density distribution c(x,y,z); generating means for blending the produced density distribution c(x,y,z) and density distribution Fq(x,y,z) (q=1, 2, . . . , n) already existing in the space using a smooth function Ψ(ξ,ζ) having two variables ξ, ζ according to equation Gq(x,y,z)=Ψ(Fq(x,y,z), c(x,y,z)) thereby to generate density distribution Gq(x,y,z); sampling data computing means for sampling up to the mth-order partial derivative of the generated density distribution G q (x,y,z) on the points included in a set Ξ q having a finite number of points in the three-dimensional space and determining sampling data Λ q ={μ p (i,j,k)={1/(i!j!k!)}∂ i+j+k G q (x p , y p , z p )/∂x i ∂y j ∂z k |i+j+k=0, 1, . . . , m, i≧0, j≧0, k≧0; all the points P=(x p ,y p ,z p ) included in Ξ q }) ; and updating means for generating a smooth function Fq(x,y,z) for interpolating the determined sampling data Λ q and expressing new density distribution Fq(x,y,z) with the generated function thereby to update said density distribution Fq(x,y,z), and assuming that the convex combination of a number n of smooth functions Fq(x,y,z) (q=1, 2 . . . . , n) at each coordinate point as density distribution F(x,y,z) representing for an object; means for displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density distribution functions which is comprised of a convex combination of said updated Fq(x,y,z) density distribution functions; means for altering the parameters of the c(x,y,z) density function while viewing said isosurfaces; and means for saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 14

A computer aided modeling apparatus for three-dimensional free-form objects according to claim 9, wherein n=2, and two sets Ξ 1 , Ξ 2 expressed by the following equations with respect to three positive numbers L X , L Y , L Z configure sets Ξq (q=1, 2, . . . , n) each having a finite number of points in the three-dimensional space: ##EQU30##

Claim 15

A computer aided modeling apparatus for three-dimensional free-form objects in which, a set of points (x,y,z) satisfying relation F(x,y,z)=a, where F(x,y,z) is density distribution in a three-dimensional space x-y-z, and a is a scalar quantity, is defined as a surface of object, a set of points (x,y,z) satisfying relation F(x,y,z)>a as an inside of object and a set of points (x,y,z) satisfying relation F(x,y,z)<a as an outside of object, comprising: producing means for producing density distribution c(x,y,z) by arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position; generating means for generating density distribution Gq(x,y,z) by blending the produced density distribution c(x,y,z) and density distribution Fq(x,y,z) (q=1, 2, . . . , n) already existing in the space; sampling data computing means for determining sampling data Λ q by sampling up to the mth-order partial derivative of the generated density distribution Gq(x,y,z) on the points included in a set Ξ q having a finite number of points in the three-dimensional space; and updating means for generating a smooth function Fq(x,y,z) for interpolating the determined sampling data Λ q and expressing new density distribution Fq<x,y,z) by the generated function thereby to update said density distribution Fq(x,y,z), and assuming that the convex combination of a number n of smooth functions Fq(x,y,z) (q=1, 2, . . . , n) at each coordinate point is density distribution F(x,y,z) for expressing an object; means for displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density distribution functions which is comprised of a convex combination of said updated Fq(x,y,z) density distribution functions; means for altering the parameters of the c(x,y,z) density function while viewing said isosurfaces; and means for saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.

Claim 16

A computer aided modeling apparatus for three-dimensional free-form objects in which, a set of points (x,y,z) satisfying relation F(x,y,z)=a, where F(x,y,z) is density distribution in a three-dimensional space x-y-z, and a is a scalar quantity, is defined as a surface of object, a set of points (x,y,z) satisfying relation F(x,y,z)>a as an inside of object and a set of points (x,y,z) satisfying relation F(x,y,z)<a as an outside of object, comprising: producing means for producing density distribution c(x,y,z) by arranging a field generating primitive prepared by a system or a user in a desired orientation and at a desired position; generating means for generating density distribution Gq(x,y,z) by blending the produced density distribution c(x,y,z) and density distribution Fq(x,y,z) (q=1, 2, . . . , n) already existing in the space; sampling data computing means for determining sampling data Λ q by sampling up to the mth-order partial derivative of the generated density distribution Gq(x,y,z) on the points included in a set Ξ q having a finite number of points in the three-dimensional space; and updating means for generating a smooth function Fq(x,y,z) for interpolating the determined sampling data Λ q and expressing new density distribution Fq<x,y,z) by the generated function thereby to update said density distribution Fq(x,y,z), and assuming that the convex combination of a number n of smooth functions Fq(x,y,z) (q=1, 2, . . . , n) at each coordinate point is density distribution F(x,y,z) for expressing an object; means for displaying isosurfaces of said object to evaluate its shape represented by said Fq(x,y,z) density distribution functions which is comprised of a convex combination of said updated Fq(x,y,z) density distribution functions; means for altering the parameters of the c(x,y,z) density function while viewing said isosurfaces; and means for saving the sampling data in memory when said displayed isosurfaces are satisfactory to the user.