

So, I think I started this around 10 last night and [mostly] solved it by
midnight.
I did it in parametric form, but then I couldn't get the stupid parametric to
render, so I somehow was able to see my way to doing it in implicit form for an
isosurface.
We already know the equation for the ellipsoid
https://en.wikipedia.org/wiki/Ellipsoid
In order to derive a normal vector for each point on its surface, we need to
know the equation of the plane tangent to the ellipsoid at those points.
http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32A/chap12/section6/811d45/811_45.html
and that says that the coefficients for the polynomial are the scalars of the
normal vector.
Then ya just plug that into vnormalize to get a unit normal vector and tack that
onto the ellipsoid formula.
Hit the documentation to see what the formulae for vnormalize, vlength, and vdot
are, and you have:
#declare R = 1;
#declare a = 1;
#declare b = 0.7;
#declare c = 1;
#declare F_ellipsoid = function {(pow(x,2)/pow(a,2)) + (pow(y,2)/pow(b,2))
+(pow(z,2)/pow(c,2))  R}
#declare F_Normal = function {
((pow(x,2)/pow(a,2)) / pow(a,2)) +
((pow(y,2)/pow(b,2)) / pow(b,2)) +
((pow(z,2)/pow(c,2)) / pow(c,2)) }
#declare FVdot = function {pow(((pow(x,2)/pow(a,2)) / pow(a,2)), 2) +
pow(((pow(y,2)/pow(b,2)) / pow(b,2)), 2) + pow(((pow(z,2)/pow(c,2)) / pow(c,2)),
2)}
#declare Normalized = function {F_Normal (x, y, z) / sqrt (FVdot (x, y, z))}
My full scene is here.
Perhaps someone better versed in the parametric object can determine why I can't
get it to render the full surface, since the exact same equations are used in
the nested loop of spheres to correctly approximate the surface.
I think the only additional thing I'd do is find some way to verify the
correctness of this solution by verifying that the distance between the offset
curve and the ellipsoid is constant over the entire surface.
For that, I'd likely use a trace() method for the offset and the ellipsoid, and
then calculate the shortest Euclidean distance.
Might be able to do that and rewrite my trace() based curve making script to
plot out an approximation, and then either use a triangular grid or a series of
rectangles to make a smooth triangle approximation of the surface like Nylander,
Loney, TOK, and Jaap Frank.
########################################################################
#version 3.8;
global_settings {assumed_gamma 1.0}
// Offset / Parallel surface of an ellipsoid.
// Bill Walker "Bald Eagle" 7/20/2018
// for Mike Horvath "posfan12" at
http://news.povray.org/povray.general/thread/%3C5b513fd2%241%40news.povray.org%3E/
// parametric only renders a small section  not functional yet, and SLOW
#include "colors.inc"
//#include "shapes.inc"
//#include "shapes2.inc"
#include "shapes3.inc"
#declare Zoom = 128;
camera {
orthographic
location <0, 0, 20> // position & direction of view
look_at <0, 0, 0>
right x*image_width/Zoom // horizontal size of view
up y*image_height/Zoom // vertical size of view
}
camera {
location <0, 0, 4> // position & direction of view
look_at <0, 0, 0>
right x*image_width/image_height // horizontal size of view
up y // vertical size of view
}
sky_sphere {pigment {rgb <0.5, 0.5, 1>}}
plane {y, 3 pigment {checker}}
light_source {<5, 5, 30> color White}
#declare Ellipse = torus {1, 0.01 rotate x*90 pigment {Red} scale <0.5, 1, 1> }
#declare n=2;
//object {Ellipse}
//object {Ellipse scale <n, n, 1>}
#declare R = 1;
#declare a = 1;
#declare b = 0.7;
#declare c = 1;
#declare F_ellipsoid = function {(pow(x,2)/pow(a,2)) + (pow(y,2)/pow(b,2))
+(pow(z,2)/pow(c,2))  R}
#declare F_Normal = function {
((pow(x,2)/pow(a,2)) / pow(a,2)) +
((pow(y,2)/pow(b,2)) / pow(b,2)) +
((pow(z,2)/pow(c,2)) / pow(c,2)) }
#declare FVdot = function {pow(((pow(x,2)/pow(a,2)) / pow(a,2)), 2) +
pow(((pow(y,2)/pow(b,2)) / pow(b,2)), 2) + pow(((pow(z,2)/pow(c,2)) / pow(c,2)),
2)}
#declare Normalized = function {F_Normal (x, y, z) / sqrt (FVdot (x, y, z))}
// for dynamic adapting of the max_gradient value
#declare Min_factor = 0.6; // between 0 and 1
#declare MaxGradient = 4;
#declare P0 = MaxGradient*Min_factor;
#declare P1 = sqrt(MaxGradient/(MaxGradient*Min_factor));
#declare P2 = 0.7; // between 0 and 1
#declare Ellipsoid =
isosurface {
function {F_ellipsoid (x, y, z)}
accuracy 0.001
max_gradient 3
//evaluate P0, P1, min (P2, 1)
contained_by {sphere {0, R}}
//contained_by {box {<R, R, R>, <R, R, R>}}
pigment {rgb <0, 0, 1>}
}
// for dynamic adapting of the max_gradient value
#declare Min_factor = 0.6; // between 0 and 1
#declare MaxGradient = 3;
#declare P0 = MaxGradient*Min_factor;
#declare P1 = sqrt(MaxGradient/(MaxGradient*Min_factor));
#declare P2 = 0.7; // between 0 and 1
#declare PEllipsoid =
isosurface {
function {F_ellipsoid (x, y, z)  Normalized (x, y, z)/5 }
accuracy 0.001
max_gradient 5
//evaluate P0, P1, min (P2, 1)
contained_by {sphere {0, R*2}}
//contained_by {box {<R, R, R>*2, <R, R, R>*2}}
pigment {rgbt <1, 1, 0, 0.8>}
}
object {Ellipsoid}
object {PEllipsoid} // translate x*R*2}
#declare EllipseX = function (u, v) {a*cos(u)*sin(v)}
#declare EllipseY = function (u, v) {b*sin(u)*sin(v)}
#declare EllipseZ = function (v) {c*cos(v)}
#declare FNormalX = function (u, v) {EllipseX (u, v) / pow(a,2)}
#declare FNormalY = function (u, v) {EllipseY (u, v) / pow(b,2)}
#declare FNormalZ = function (v) {EllipseZ (v) / pow(c,2)}
#declare FVDot = function (u, v)
{pow(FNormalX(u,v),2)+pow(FNormalY(u,v),2)+pow(FNormalZ(v),2)}
#declare FVnormalizeX = function (u, v) {FNormalX (u, v) / sqrt (FVDot (u, v))}
#declare FVnormalizeY = function (u, v) {FNormalY (u, v) / sqrt (FVDot (u, v))}
#declare FVnormalizeZ = function (u, v) {FNormalZ (v) / sqrt (FVDot (u, v))}
#declare Step1 = pi/18;
#declare Step2 = pi/36;
/*
#for (V, 0, tau, Step2)
#for (U, 0, pi, Step1)
//#declare X = a*cos(U)*sin(V);
#declare X = EllipseX (U, V);
//#declare Y = b*sin(U)*sin(V);
#declare Y = EllipseY (U, V);
//#declare Z = c*cos(V);
#declare Z = EllipseZ (V);
sphere {<X, Y, Z> 0.01 pigment {Blue}} //point at <x, y, z> on the ellipsoid
#declare Normal = <X/pow(a,2), Y/pow(b,2), Z/pow(c, 2)>;
sphere {<EllipseX (U, V) + FVnormalizeX (U, V)/10, EllipseY (U, V) +
FVnormalizeY (U, V)/10, EllipseZ (V) + FVnormalizeZ (U, V)/10> 0.01 pigment
{Red}} // surface normal at <X, Y, Z>
#end
#end
*/
//  parametric surface 
#declare Parallel = parametric {
function {EllipseX (u, v)}
function {EllipseY (u, v)}
function {EllipseZ (v)}
<0, pi>, <0, 2*pi> // start, end (u,v)
contained_by {sphere {0, R}}
max_gradient 50
accuracy 0.005
precompute 5 x,y,z
texture {pigment{ color rgb <0, 1, 0>}}
}
//object {Parallel}
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