Why Obj? Because it is simple, and can be read by many programs and converters.
Why Matlab? Because it is a great environment for doing math, simulation and numerical hacking even if the language itself is crude, from a computer science standpoint.
[u,v]=meshgrid(-6:0.5:6,-6:0.5:6); x=u; y=v; z=u.^2+v.^2; mesh(x,y,z) saveobjmesh('parab.obj',x,y,z)
This can then be imported into Bryce (File->Import Object). In this case a bit of scaling might be needed.
The default version does not include any normal information, making the polygon edges look hard. Saveobjmesh allows you to add normals, although the smoothing in Bryce makes a nice job of calculating them anyway.
[u,v]=meshgrid(-6:0.5:6,-6:0.5:6); x=u; y=v; z=u.^2+v.^2; mesh(x,y,z) nx=-2*x; ny=-2*y; nz=1+nx*0; saveobjmesh('parab.obj',x,y,z,nx,ny,nz)
Sometimes one wants a literal mesh rather than a polygonal surface. I wrote a version of saveobjmesh, saveobjmeshgrid.m, that instead of each quadrilateral draws four smaller quadrilaterals around a hole, producing a nice mesh-like surface.
a=2*pi; s=2*pi/60; [x,y,z]=meshgrid(-a:s:a,-a:s:a,-a:s:a); cx = cos(2*x); cy = cos(2*y); cz = cos(2*z); u = 10.0*(cos(x).*sin(y) + cos(y).*sin(z) + cos(z).*sin(x))- 0.5*(cx.*cy + cy.*cz + cz.*cx) - 14.0; [f,v]=isosurface(u,0); vertface2obj(v,f,'gyroid.obj')
The first lines creates a function with a nice gyroid lattice structure (formula from The Scientific Graphics Project). Then the u=0 isosurface is calculated, which produces a list of vertices and faces. These are saved with vertface2obj. In this case the mesh resolution is rather high in order not to miss any pieces.
Another example is the Barth sextic
s=0.05; [x,y,z]=meshgrid(-2:s:2,-2:s:2,-2:s:2); t = 0.5*(1 + sqrt(5)); u= 4*(t^2* x.^2 - y.^2).*(t^2.* y.^2 - z.^2).*(t^2 .* z.^2 - x.^2) - (1+2*t)*(x.^2 + y.^2 + z.^2 - 1).^2; [f,v]=isosurface(u,0); vertface2obj(v,f,'barth.obj')
Here it is placed on a verandah overlooking a canal.
Made using a parametric mesh:
s=2*pi/100; [u,v]=meshgrid(0:s:(2*pi),0:s:(2*pi)); k=0.5; t=.05*(2*pi-u).*(1+sin(v*2+u*3)./(1+sin(u*4).^2)); x=cos(u*k).*(1+t.*cos(v)); y=sin(u*k).*(1+t.*cos(v)); z= t.*sin(v);
In this case I made one "horn" mesh and a smaller straight mesh and imported them into Bryce. Then I multireplicated the smaller straight mesh in a circle around the foot of the big horn to make the base. Then I grouped them and made a mirrored copy.
Plotting knots becomes relatively easy, at least toroidal knots:
su=2*pi/150; sv=2*pi/20; [u,v]=meshgrid(0:su:(2*pi),0:sv:(2*pi)); r1=2; % Main radius r2=0.4; % Radius of torus r3=0.3; % Radius of tube p1=2; % Turns around origin p2=3; % Twists around itself x=r1*cos(u*p1) + r2*cos(u*p1).*cos(u*p2) +r3*cos(u*p1).*sin(v); y=r1*sin(u*p1) + r2*sin(u*p1).*cos(u*p2) +r3*sin(u*p1).*sin(v); z=r2*sin(u*p2)+r3*cos(v);
A threefoil knot with the Bryce default metal grid texture, mapped parametrically. Since the u,v coordinates of the above script are not locally orthogonal, a nice twistiness is apparent in the texture.
This time the mesh is a real grid (made by saveobjmeshgrid) of a (3,7) knot.
A knot where the tube radius produces self-intersections. The globular cluster is imagemapped on a square, and then the glass-textured knot placed in front. No postprocessing was needed.
Here it is applied to Boy's surface:
ss=pi/100; [s,t]=meshgrid(0:ss:pi,0:ss:pi); x=cos(t).*sin(s); y=sin(t).*sin(s); z=cos(s); f=1/2*((2*x.^2-y.^2-z.^2) + 2*y.*z.*(y.^2-z.^2) + z.*x.*(x.^2-z.^2)+x.*y.*(y.^2-x.^2)); g=sqrt(3)/2*((y.^2-z.^2) + z.*x.*(z.^2-x.^2) + x.*y.*(y.^2-x.^2)); h=((x+y+z).*((x+y+z).^3 + 4*(y-x).*(z-y).*(x-z)))/8; explodemesh('b.obj',h/8,f,g,0.5);This way the somewhat tricky interconnections of the surface becomes visible:
By giving a factor larger than one to explodemesh the polygons instead becomes scales:
As an aside, the parametrization allows one to map the Earth onto the surface. Obviously there has to be something odd going on since it is non-orientable and the Earth's surface certainly is, but it is not easy to see where it is.
An Obj file is an ascii file with information organized into lines. The important kinds of lines are:
v 0 0 0 v 1 0 0 v 0 1 0 v 1 1 0 g square f 1 2 3 f 2 4 3Which defines a square shape consisting of two triangular faces. The normal is assumed to point upwards if the vertices are passed counterclockwise.
See the links below for further descriptions and specifications.