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

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 .LAMBDA..sub.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 .XI..sub.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.

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 .PSI.(.xi.,.zeta.) having two variables .xi.,.zeta. according to equation Gq(x,y,z)=.PSI.(Fq(x,y,z), c(x,y,z)) thereby to generate density distribution Gq(x,y,z);
• determining sampling data .LAMBDA..sub.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 .XI..sub.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 .LAMBDA..sub.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.

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 .PSI.(.xi.,.zeta.) having two variables .xi., .zeta. according to equation Gq(x,y,z)=.PSI.(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.sub.q (x,y,z) on the points included in a set .XI..sub.q having a finite number of points in the three-dimensional space and determining sampling data .LAMBDA..sub.q ={.mu..sub.p (i,j,k)={1/(i!j!k!)}.differential..sup.i+j+k G.sub.q (x.sub.p, y.sub.p, z.sub.p)/.differential.x.sup.i .differential.y.sup.j .differential.z.sup.k .vertline.i+j+k=0, 1, . . . , m, i.gtoreq.0, j.gtoreq.0, k.gtoreq.0;
• all the points P=(x.sub.p,y.sub.p,z.sub.p) included in .XI..sub.q }) ;
• and
• updating means for generating a smooth function Fq(x,y,z) for interpolating the determined sampling data .LAMBDA..sub.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.

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 .LAMBDA..sub.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 .XI..sub.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 .LAMBDA..sub.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.