CGAL 4.7 - 3D Mesh Generation: Mesh_3/implicit_functions.cpp

\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.7 - 3D Mesh Generation

All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
Mesh_3/implicit_functions.cpp
// Sphere (r=1)
double sphere_function (double x, double y, double z) {
double x2=x*x, y2=y*y, z2=z*z;
return x2+y2+z2-1;
}
// Ellipsoid (r=1)
double ellipsoid_function (double x, double y, double z) {
double x2=x*x, y2=y*y, z2=z*z;
return x2+2*y2+4*z2-1;
}
// Torus (r=2)
double torus_function (double x, double y, double z) {
double x2=x*x, y2=y*y, z2=z*z;
double x4=x2*x2, y4=y2*y2, z4=z2*z2;
return x4 + y4 + z4 + 2 *x2* y2 + 2*
x2*z2 + 2*y2* z2 - 5 *x2 + 4* y2 - 5*z2+4;
}
// "Chair" (r=6)
double chair_function (double x, double y, double z) {
double x2=x*x, y2=y*y, z2=z*z;
double x4=x2*x2, y4=y2*y2, z4=z2*z2;
return x4-1.2*x2*y2+3.6*x2*z2-7.50*x2+y4+3.6*y2*z2-7.50*y2+.2*z4-7.50*z2+64.0625-16.0*z*y2+16.0*x2*z;
}
// "Tanglecube" (r=4)
double tanglecube_function (double x, double y, double z) {
double x2=x*x, y2=y*y, z2=z*z;
double x4=x2*x2, y4=y2*y2, z4=z2*z2;
return x4 - 5*x2 + y4 - 5*y2 + z4 - 5*z2 + 11.8;
}
double cube_function (double x, double y, double z){
if( x < 1 && -x < 1 &&
y < 1 && -y < 1 &&
z < 1 && -z < 1 )
return -1.;
return 1.;
}
// Barth's octic surface (degree 8)
double octic_function (double x, double y, double z) { // r=2
double x2=x*x, y2=y*y, z2=z*z;
double x4=x2*x2, y4=y2*y2, z4=z2*z2;
double x6=x4*x2, y6=y4*y2, z6=z4*z2;
double x8=x4*x4, y8=y4*y4, z8=z4*z4;
return 43.30495169 *x2* y2 + 43.30495169 *x2* z2 + 43.30495169 *y2 * z2 + 44.36067980 *x6* y2 + 44.36067980* x6 * z2 + 66.54101970* x4* y4 + 66.54101970* x4 * z4 + 44.36067980 *x2 * y6 - 11.70820393* x2 - 11.70820393* y2 - 11.70820393* z2 + 37.65247585 *x4 + 37.65247585 *y4 + 37.65247585* z4 + 11.09016995* x8 + 11.09016995 *y8 + 11.09016995* z8 + 133.0820394 *x2* y4* z2 + 133.0820394 *x2 * y2 * z4 + 44.36067980* x2 * z6 + 44.36067980 *y6 * z2 + 66.54101970 *y4 * z4 + 44.36067980 *y2 * z6 + 133.0820394* x4* y2 * z2 - 91.95742756 *x4 * y2 - 91.95742756 *x4 *z2 - 91.95742756* x2 * y4 - 91.95742756 *x2 *z4 - 91.95742756* y4 * z2 - 183.9148551* x2 *y2 *z2 - 30.65247585 *x6 - 30.65247585* y6 - 91.95742756 *y2 * z4 - 30.65247585* z6 + 1.618033988;
}
// "Heart"
double heart_function (double x, double y, double z) { // radius = 2
return (2*x*x+y*y+z*z-1)*(2*x*x+y*y+z*z-1)*(2*x*x+y*y+z*z-1) - (0.1*x*x+y*y)*z*z*z;
}
// Klein's bottle
double klein_function (double x, double y, double z) { // radius = 4
return (x*x+y*y+z*z+2*y-1)*((x*x+y*y+z*z-2*y-1)*(x*x+y*y+z*z-2*y-1)-8*z*z)+16*x*z*(x*x+y*y+z*z-2*y-1);
}
// Ring
double ring_function (double x, double y, double z) { // radius = ?
double e=0.1;
double f1 = x*x+y*y+z*z-1;
double f2 = x;
f1 = f1*f1-e*e;
f2 = f2*f2-e*e;
if (f1 < 0 && f2 < 0)
return -1.;
else if (f1 > 0 || f2 > 0)
return 1.;
else
return 0.;
}
// False knot
double false_knot_function (double x, double y, double z) { // radius = 1
double d=1.2, e=0.1;
double f1 = x*(x*x-z*z)-2*x*z*z-y*y+d*d-x*x-z*z-y*y;
double m2 = z*(x*x-z*z)+2*x*x*z;
double f2 = 4*y*y*(d*d-x*x-z*z-y*y) - m2*m2;
f1 = f1*f1-e*e;
f2 = f2*f2-e*e;
if (f1 < 0 && f2 < 0)
return -1.;
else if (f1 > 0 || f2 > 0)
return 1.;
else
return 0.;
}
// Knot 1
void puiss(double& x, double& y, int n) {
double xx = 1, yy = 0;
while(n>0) {
if (n&1) {
double xxx = xx, yyy = yy;
xx = xxx*x - yyy*y;
yy = xxx*y + yyy*x;
}
double xxx = x, yyy = y;
x=xxx*xxx-yyy*yyy;
y=2*xxx*yyy;
n/=2;
}
x = xx;
y = yy;
}
double knot1_function (double a, double b, double c) { // radius = 4
double e=0.1;
double x, y, z, t, den;
den=1+a*a+b*b+c*c;
x=2*a/den;
y=2*b/den;
z=2*c/den;
t=(1-a*a-b*b-c*c)/den;
double x3=x, y3=y, z2=z, t2=t;
puiss(x3,y3,3);
puiss(z2,t2,2);
double f1 = z2-x3;
double f2 = t2-y3;
f1 = f1*f1;
f2 = f2*f2;
e=e*e/(den-1);
if (f1+f2-e < 0)
return -1.;
else if (f1+f2-e > 0)
return 1.;
else
return 0.;
}
/*
double knot1_function (double x, double y, double z) { // radius = 4
double e=0.1;
double c1, c2, c3, c4, den;
den=1+x*x+y*y+z*z;
c1=2*x/den;
c2=2*y/den;
c3=2*z/den;
c4=(1-x*x-y*y-z*z)/den;
double f1 = c1*(c1*c1-c2*c2)-2*c1*c2*c2-c3*c3+c4*c4;
double f2 = c2*(c1*c1-c2*c2)+2*c1*c1*c2-2*c3*c4;
f1 = f1*f1;
f2 = f2*f2;
e=e*e/(den-1);
if (f1+f2-e < 0)
return -1.;
else if (f1+f2-e > 0)
return 1.;
else
return 0.;
}
*/
double knot1_surf1_function (double x, double y, double z) { // radius = 4
double c1, c2, c3, c4, den;
den=1+x*x+y*y+z*z;
c1=2*x/den;
c2=2*y/den;
c3=2*z/den;
c4=(1-x*x-y*y-z*z)/den;
return c1*(c1*c1-c2*c2)-2*c1*c2*c2-c3*c3+c4*c4;
}
double knot1_surf2_function (double x, double y, double z) { // radius = 4
double c1, c2, c3, c4, den;
den=1+x*x+y*y+z*z;
c1=2*x/den;
c2=2*y/den;
c3=2*z/den;
c4=(1-x*x-y*y-z*z)/den;
return c2*(c1*c1-c2*c2)+2*c1*c1*c2-2*c3*c4;
}
// Knot 2
double knot2_function (double a, double b, double c) { // radius = 4
double e=0.025;
double x, y, z, t, den;
den=1+a*a+b*b+c*c;
x=2*a/den;
y=2*b/den;
z=2*c/den;
t=(1-a*a-b*b-c*c)/den;
double x7=x, y7=y, x13=x, y13=y;
puiss(x7,y7,7);
puiss(x13,y13,13);
double z3=z, t3=t;
puiss(z3,t3,3);
double f1t = (z3-x7)*(z3-x7+100*x13) - (t3-y7)*(t3-y7+100*y13);
double f2t = (z3-x7)*(t3-y7+100*y13) + (t3-y7)*(z3-x7+100*x13);
double f1 = f1t*(z3-x7-100*x13) - f2t*(t3-y7-100*y13);
double f2 = f1t*(t3-y7-100*y13) + f2t*(z3-x7-100*x13);
f1 = f1*f1;
f2 = f2*f2;
e=e*e/(den-1);
if (f1+f2-e < 0)
return -1.;
else if (f1+f2-e > 0)
return 1.;
else
return 0.;
}
// Knot 3
double knot3_function (double a, double b, double c) { // radius = 4
double e=0.025;
double x, y, z, t, den;
den=1+a*a+b*b+c*c;
x=2*a/den;
y=2*b/den;
z=2*c/den;
t=(1-a*a-b*b-c*c)/den;
double x19=x, y19=y, z17=z, t17=t;
puiss(x19,y19,19);
puiss(z17,t17,17);
double f1 = z17-x19;
double f2 = t17-y19;
f1 = f1*f1;
f2 = f2*f2;
e=e*e/(den-1);
if (f1+f2-e < 0)
return -1.;
else if (f1+f2-e > 0)
return 1.;
else
return 0.;
}