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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
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