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Well that should be more than an appetizer for those eager to use Super
Patch functions.
I take it "Clien" is actually Klien, perhaps? And you're really Cen T.?
Ken wrote:
>
> Just a random sampling of equasions I have collected form various sources.
> Accuracy of these equations are not gauranteed and may indeed be totaly
> wrong so please don't shoot me if they do not work. There are also references
> to graphic examples which I cannot provide for you. Sorry.
>
> If anybody does play with these and produces working code I would appreciate
> a copy of the working Pov example so I can fill out this collection as an
> aid to others that may follow in your steps.
>
> The Triaxial tritorus is defined parametrically as
>
> x = sin(u) (1+cos(v))
> y = sin(u+2 PI/3) (1+cos(v+2 PI/3))
> z = sin(u+4 PI/3) (1+cos(v+4 PI/3)
>
> Where -PI <= u <= PI and -PI <= v <= PI
>
> unknow if the following works -
> sin(x)(1+cos(x))sin(y+2 12/3)(1+cos(y+2 12/3))sin(z+4 12/3)(1+cos(z+4 12/3)
>
> ---------------------------------------------------------------------------
>
> A Clien Cycloid Shape
>
> x= cos(u/c)*cos(u/b)*(a+cos(v))+sin(u/b)*sin(v)*cos(v);
> y= sin(u/c)*cos(u/b)*(a+cos(v))+sin(u/b)*sin(v)*cos(v);
>
> z=-sin(u/b)*(a+cos(v))+cos(u/b)*sin(v)*cos(v);
>
> Where: 0<=u<=2 b c PI and 0<=v<=4 PI
>
> a=10;
> b=3;
> c=2;
> X=Cos[u/c]*Cos[u/b]*(a+Cos[v])+Sin[u/b]*Sin[v]*Cos[v];
> Y=Sin[u/c]*Cos[u/b]*(a+Cos[v])+Sin[u/b]*Sin[v]*Cos[v];
> Z=-Sin[u/b]*(a+Cos[v])+Cos[u/b]*Sin[v]*Cos[v];
>
> ParametricPlot3D[{X, Y, Z}, {u, 0, 2*b*c*Pi}, {v, 0, 4 Pi},
> PlotPoints -> {120,10}, Axes -> False, Boxed -> False,
> ViewPoint->{5.265, -6.828, 2.580}];
>
> ---------------------------------------------------------------------------
>
> To apply a random scale to the x demension and correct for rand scale
> as object is translated.
>
> #declare R = seed(pi);
>
> #declare NextX = 0;
> #while( NextX < 20 )
> #delare Scale = 1+rand(R);
> box{0,1 scale Scale}
> translate <NextX,0,0>
> #declare NextX = NextX + Scale
> #end
>
> -----------------------------------------------------------------------------
>
> Multifaceted sphere equations
>
> The equations of the points on the surface of the sphere are
>
> x = cos(theta) cos(phi)
> y = cos(theta) sin(phi)
> z = sin(theta)
>
> where -90 <= theta <= 90
> 0 <= phi <= 360
>
> To create a facet approximation, theta and phi are stepped in
> small angles between their respective bounds. So if we take
> the angle step size to be dtheta and dphi, the four vertices
> of any facet correspond to
>
> (theta,phi)
> (theta+dtheta,phi)
> (theta+dtheta,phi+dphi)
> (theta,phi+dphi)
>
> void CreateUnitSphere(dtheta,dphi)
> int dtheta,dphi;
> {
> int n;
> int theta,phi;
> XYZ p[4];
>
> for (theta=-90;theta<=90-dtheta;theta+=dtheta) {
> for (phi=0;phi<=360-dphi;phi+=dphi) {
> n = 0;
> p[n].x = cos(theta*DTOR) * cos(phi*DTOR);
> p[n].y = cos(theta*DTOR) * sin(phi*DTOR);
> p[n].z = sin(theta*DTOR);
> n++;
> p[n].x = cos((theta+dtheta)*DTOR) * cos(phi*DTOR);
> p[n].y = cos((theta+dtheta)*DTOR) * sin(phi*DTOR);
> p[n].z = sin((theta+dtheta)*DTOR);
> n++;
> p[n].x = cos((theta+dtheta)*DTOR) * cos((phi+dphi)*DTOR);
> p[n].y = cos((theta+dtheta)*DTOR) * sin((phi+dphi)*DTOR);
> p[n].z = sin((theta+dtheta)*DTOR);
> n++;
> if (theta > -90 && theta < 90) {
> p[n].x = cos(theta*DTOR) * cos((phi+dphi)*DTOR);
> p[n].y = cos(theta*DTOR) * sin((phi+dphi)*DTOR);
> p[n].z = sin(theta*DTOR);
> n++;
> }
>
> /* Do something with the n vertex facet p */
>
> }
> }
> }
>
> --------------------------------------------------------------------
>
>
> This short note describes the parametric equations which give
> rise to an approximate model of a drop of water, for example, a
> tear drop.
>
> The equations as functions of longitute phi and lattitude theta are:
>
> x = 0.5 *(1-cos(8)) sin(8) cos(circle with verticle line through it)
> y = 0.5 *(1-cos(8)) sin(8) sin(circle with verticle line through it)
> z = cos(8)
>
> where 0 <= circ w/line tru/it <= 2pi
> and 0 <= 8 <= pi
>
> When theta is 0 there is a discontinuity at the apex where
> x = 0 y = 0 z = 1
>
> An implicit equation for the aforementioned tear drop is:
>
> 1 - 4x^2 - 4y^2 - 2z + 2z^3 - z^4 = 0,
>
> or, it simplifies a bit as 4(x^2+y^2)=(1+z)(1-z)^3, which is a surface
> of revolution bounded by -1 <= z <= 1. The POV command is:
>
> quartic{ <
> 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0,
> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0,
> -1, 2, 0, -2, 1> }
>
> Sometimes, a parametric map involving trig functions can be converted
> to an algebraic implicit equation by using new variables:
>
> u=cos(t),
> v=sin(t),
>
> and introducing a new implicit relation:
>
> u^2+v^2=1.
>
> The idea is to make the parametric equations into polynomials, at the
> cost of increasing the number of variables and equations. Then u and
> v can (sometimes) be eliminated from the system of polynomial
> equations to get an implicit equation in x,y,z.
>
> ---------------------------------------------------------------------
>
> Parametric function that mathmaticaly discribes a shape quite similar
> in appearence to a cork screw blade.
>
> v*cos(4*u), u, v*sin(4*u)
>
> <-PI/2,-1>, <PI/2, 1>
>
> ---------------------------------------------------------------------
>
> Parametric function that mathmaticaly discribes a shape quite similar
> in appearence to a cylinder or truncated cone.
>
> sin(v), u, cos(v)
>
> <-1,-PI>, <1,PI>
>
> ---------------------------------------------------------------------
>
> A goniometric function implicitly defined by the following mathmatical
> expression
>
> function_xy
>
> function cos(10*(sqrt(sqr(x)+sqr(y))))
>
> ---------------------------------------------------------------------
>
> 6th order polnominal function describing a heart shaped surface
>
> cub(2*sqr(x)+sqr(y)+sqr(z)-1) - (0.1*sqr(x)+sqr(y))*cub(z)
>
> ---------------------------------------------------------------------
>
> A torus shape ?
>
> function{sqrt(sqr(sqrt(sqr(x)+sqr(z))-R0)+sqr(y)) -R1 }
>
> ---------------------------------------------------------------------
>
>
> ---------------------------------------------------------------------
> a different surface
> cos(10*(sqrt(sqrt(x)+sqrt(y))))
> ---------------------------------------------------------------------
>
> =========================
>
> Borromean rings are three interlinked rings such that no two rings are linked.
> In other words, the three rings cannot be separated and yet no two of them are
> linked.
>
> It can be shown that there is no solution for flat rings, it can only be
> constructed by perturbing the rings in a third dimension.
>
> The rings can be formed with various base geometries including rectangles
> and triangles as shown below.
>
> --------------------
>
> One way (as shown below) to create the rings mathematically is with the
> following three parametric equations, one for each ring.
>
> (cos(u) , sin(u) + r , cos(3u) / 3)
> (cos(u) + 0.5 , sin(u) - r/2 , cos(3u) / 3)
> (cos(u) - 0.5 , sin(u) - r/2 , cos(3u) / 3)
>
> Where u = 0 -> 2pi
>
> This is illustrated below for r = sqrt(3)/3, the radius of the spheres
> placed along the path is 0.2
> ----------
>
> It is possible to make the rings out of elliptical rings, it does
> require 3 dimensions. The parametric equations for the three "rings"
> are
> ( 0 , r1 cos(u) , r2 sin(u) )
> ( r2 cos(u) , 0 , r1 sin(u) )
> ( r1 cos(u) , r2 sin(u) , 0 )
>
> Where u = 0 -> 2pi
> This is illustrated below for r1 = 2, r2 = 1, and the radius of
> the tubes = 0.2
> --------
> And finally, using box geometry...
> --------
>
> ===========================
>
> The Kuen Surface
>
> 2(cos(s)+s sin(s)) sin(t)
> x = -------------------------
> 1+ s^2sin(t)^2
>
> 2(sin(s)-s cos(s)) sin(t)
> y = -------------------------
> 1+ s^sin(t)^2
>
> log(tan(t/2))+2 cos(t)
> z = -------------------------
> 1+ s^2sin(t)^2
>
> =======================
> Maeder's Owl
>
> Maeder's Minimal Surface
>
> x = v cos(u) - 0.5 v2 cos(2 u);
> y = - v sin(u) - 0.5 v2 sin(2 u);
> z = 4 v1.5 cos(3 u / 2) / 3;
>
> 0 <= u <= 4 PI and 0 <= v <= 1
>
> =========================
> C source for Meader's Owl
> =========================
>
> #include "stdio.h"
> #include "stdlib.h"
> #include "math.h"
> #include "paulslib.h"
>
> int main(int argc,char **argv)
> {
> int i,j,nu=100,nv=20;
> double u,v;
> XYZ p;
> COLOUR colour;
> COLOUR black = {0.0,0.0,0.0};
>
> printf("CMESH\n%d %d\n",n+1,n+1);
> for (i=0;i<=n;i++) {
> for (j=0;j<=n;j++) {
>
> u = i * 4 * PI / nu;
> v = j / (double)nv;
>
> p.x = v * cos(u) - 0.5 * v * v * cos(2 * u);
> p.y = - v * sin(u) - 0.5 * v * v * sin(2 * u);
> p.z = 4 * pow(v,1.5) * cos(1.5 * u) / 3;
>
> colour = GetColour(u,0.0,4*PI,4);
> printf("%g %g %g %g %g %g 0.5\n",p.x,p.y,p.z,
> colour.r,colour.g,colour.b);
> } } }
>
> ===========================
> Mobius Strip
>
> The mobius stip is the simplest geometric shape which has only one
> surface and only one edge. It can be created by taking a strip of
> paper, giving it a half twist along its long axis, and then joining
> the two narrow ends together.
>
> The mobius strip in 3 dimensions can be represented parameterically f(s,t) as
follows
>
> \ /
> x | | cos(s)+t*cos(s/2)*cos(s)
> \ /
> y | = | sin(s)+t*cos(s/2)*sin(s)
> / \
> z | | t*sin(s/2)
> / \
>
> where s ranges typically from 0.0 to 2.0*pi
> and t ranges typically from -0.4 to 0.4
>
> ===============================================
>
> Knot Equasions
>
> Knot 4
> There are a whole family of curves including knots which are
> formed by the equations:
>
> x = r * cos(phi) * cos(theta)
> y = r * cos(phi) * sin(theta)
> z = r * sin(phi)
>
> which are the equations for converting from polar to cartesian coordinates
> except that we make r, theta, phi a function of a parameter beta which ranges
> from 0 to pi.
>
> For the following
>
> r(beta) = 0.8 + 1.6 * sin(6 * beta)
> theta(beta) = 2 * beta
> phi(beta) = 0.6 * pi * sin(12 * beta)
>
> Knot 5
> Using the same systems of equations as in knot 4, with
>
> r(beta) = 1.2 * 0.6 * sin(0.5 * pi + 6 * beta)
> theta(beta) = 4 * beta
> phi(beta) = 0.2 * pi * sin(6 * beta)
>
> ---------------------------------------------------------------
>
> SUPERTOID
>
> The supertoroid is a family of geometric primitives based on the torus. Let
> r0 the radius of the inner ring
> r1 the radius of the outer ring
> as illustrated below
>
> Where theta and phi range from 0 to 2pi. The equation for the supertoroid is
> the same as that for the torus except that the sin and cosine terms are raised
> to powers.
>
> n1 n2
> x = cos (u) (r0+r1 cos (v)
>
> n1 n2
> y = sin (u) (r0+r1 cos (v)
>
> n2
> z = r1 sin (v)
>
> It is different values of these powers which give rise to a family
> of 3D shapes all basically toroidal in shape. The value of n1 determines
> the shape of the torus ring, n2 determines the shape of the cross
> section of the ring.
>
> Examples of the supertoroid generated for different values of n1 and
> n2 are shown below, of course, the legal values of n1 and n2 form a
> continuum of values from 0 to infinity (although there are representation
> issues near 0 and above 4).
>
> ================================================
>
> This last is the SUPERTOID object I believe converted from the above
> equation to work with the isosurface patch. I seem to recall that it
> does not work 100% as expected and may indeed contain critical errors.
>
> /*
> #macro Supertoroid(_Rj, _Rn, _E1, _E2)
> union
> {
> parametric
> {
> function (cos(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(v) ^ _E2) * _Rn
> <0, 0>, <2 * pi, 2 * pi>
> <-(_Rj + _Rn + .1), -(_Rj + _Rn + .1), -(_Rj +_Rn +.1)>,
> <_Rj +_Rn + .1, _Rj + _Rn + .1, _Rj + _Rn + .1>
> accuracy .001
> rotate <90, 0, 0>
> }
> parametric
> {
> function (cos(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(v) ^ _E2) * _Rn
> <0, 0>, <2 * pi, 2 * pi>
> <-(_Rj + _Rn + .1), -(_Rj + _Rn + .1), -(_Rj +_Rn +.1)>,
> <_Rj +_Rn + .1, _Rj + _Rn + .1, _Rj + _Rn + .1>
> accuracy .001
> rotate <90, 0, 0>
> rotate <0, 90, 0>
> }
> parametric
> {
> function (cos(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(v) ^ _E2) * _Rn
> <0, 0>, <2 * pi, 2 * pi>
> <-(_Rj + _Rn + .1), -(_Rj + _Rn + .1), -(_Rj +_Rn +.1)>,
> <_Rj +_Rn + .1, _Rj + _Rn + .1, _Rj + _Rn + .1>
> accuracy .001
> rotate <90, 0, 0>
> rotate <0, 180, 0>
> }
> parametric
> {
> function (cos(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(u) ^ _E1) * (_Rj + _Rn * (cos(v) ^ _E2)),
> (sin(v) ^ _E2) * _Rn
> <0, 0>, <2 * pi, 2 * pi>
> <-(_Rj + _Rn + .1), -(_Rj + _Rn + .1), -(_Rj +_Rn +.1)>,
> <_Rj +_Rn + .1, _Rj + _Rn + .1, _Rj + _Rn + .1>
> accuracy .001
> rotate <90, 0, 0>
> rotate <0, 270, 0>
> }
> }
> #end
>
> // End POV code
>
> //You call it like this:
>
> // Begin POV code
>
> object
> {
> Supertoroid(8, 3, 2, .5)
> texture
> {
> pigment { color Red }
> normal { dents 2 scale <.1, 3, .2> }
> finish { specular 1 roughness .002 }
> }
> rotate <-30. -30, 0>
> }
>
> */
>
> --
> Ken Tyler
>
> mailto://tylereng@pacbell.net
--
omniVERSE: beyond the universe
http://members.aol.com/inversez/homepage.htm
mailto://inversez@aol.com?Subject=PoV-News
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