matlab Rendering a torus from vertices - matlab

Question

I have an assignment to render a torus. This is my first time with matlab and i've managed to struggle along and get 2/3 parts done with some horrible kludged code.
The first step of the assignment is to render a circle as a set of 20 points. For which i produced:
circle (IMG)
Then the next step is to rotate and translate that circle and draw it 20 times to represent a torus shape, so i got this:
torus points (IMG)
The next step is to render a 3d representation of this torus from the list of vertices.
What I have is massive a list of a vertices in a 400x3 matrix like so:
7.66478245119846 -1.84059939326890 0.292371704722737
7.53434247103331 -1.79821687453702 0.573576436351046
7.32764268084884 -1.73105604149887 0.798635510047293
7.06491629627043 -1.64569106442929 0.945518575599317
6.77188080634298 -1.55047806205660 0.999847695156391
6.47722056651889 -1.45473714644104 0.956304755963036
... ... ...
where each subsequent 20 rows is another circle.
The assignment recommends that I use the surf function in order to render this but I can't figure out how. All the examples I've seen use surf to represent 2 dimensional planes that get distorted by a height value. Which doesn't seem appropriate for rendering this kind of 3 dimensional shape at all.
The approach I'm trying is to build a list of faces and then using the patch function to render the circle. Where the first 2 points of each circle makes a square with the corresponding 2 points of the next circle, then rendering.
Using something like this:
for i=1:400
face = [(i) (i+1) (i+21) (i+20)];
patch('Faces',face,'Vertices',torus_vertices,'FaceColor','r'); %Should do this at the end
end
For which i get something like this:
3d Torus (IMG)
It twists and and some of the side and inner side faces are messed up. I think it might have something to do with the ordering of the vertices flipping around at some point.
What would be the best way to approach this problem? If at all possible i want to do it with the surf function.
Ex1.m
%Initial positions
position = [2 0 0];
normal = [0 1 0];
%Rotation matrix
rotate18 = [cos(todeg(18)) -sin(todeg(18)) 0;
sin(todeg(18)) cos(todeg(18)) 0;
0 0 1];
% translate along the x axis by 5
translate = [5 0 0];
%% iterate 20 times to get a list of all the vertices
taurus_vertices = zeros(0, 3);
for i=0:20
%rotate translation by 18 degrees
translate = translate * rotate18;
%translate
position = position + translate;
%rotate the normal so it faces the right direction
normal = normal * rotate18;
%Get vertices for the circle and append to vertices list
circle_vertices = circle_3D(1, position, normal);
taurus_vertices = cat(1, taurus_vertices, circle_vertices);
%translate back to original position
position = position - translate;
end
%scatter3(taurus_vertices(1:end, 1), taurus_vertices(1:end, 2), taurus_vertices(1:end, 3));
%% Render each face
for i=1:400
face = [(i) (i+1) (i+21) (i+20)];
patch('Faces',face,'Vertices',taurus_vertices,'FaceColor','r');
end
circle.m
function h_circle=circle_3D(r, M, n)
%% Prepare input parameters
if size(n,2)>size(n,1)
n=n';
end
if size(M,2)>size(M,1)
M=M';
end
%% Define unit vectors u and v
% u and v define a new coordinate system in a plane perpendicular to n
a=[1;0;0];
b=[0;1;0];
if isempty(find(cross(a,n), 1))==1
a=[0;0;1];
elseif isempty(find(cross(b,n), 1))==1
b=[0;0;1];
end
alpha=dot(n,a)/dot(n,n);
u=a-alpha*n;
v=cross(u,n);%b-beta*n-gamma*u;
u=u/sqrt(sum(u.*u));
v=v/sqrt(sum(v.*v));
%% Plot the circle
hold on
axis equal
degs = 0;
points = 0;
verts = zeros(20, 3);
for phi=0: pi()/180 : 2*pi()
degs=degs+1;
if (mod(degs,18) == 0 )
points = points + 1;
verts(points,1)=M(1,1)+r*cos(phi)*u(1,1)+r*sin(phi)*v(1,1);
verts(points,2)=M(2,1)+r*cos(phi)*u(2,1)+r*sin(phi)*v(2,1);
verts(points,3)=M(3,1)+r*cos(phi)*u(3,1)+r*sin(phi)*v(3,1);
end
end
h_circle= verts;

Answer 1

Your problem is perfect for trisurf- given a set of points, you need to build a triplet to connect a mesh to.
For your problem, you can use:
%inner circle points and radius
N1=20;
r1=1;
%outer circle points and radius
N2=30;
r2=5;
%inner cicle angles
thC=linspace(0,2*pi*(1-1/N1),N1)';
%inner cicle points
xyzC=[r1*sin(thC), zeros(N1,1),r1*cos(thC)]';
%torus points
xyzT = zeros(3,N1*N2);
for i=1:N2
%circle transformation
thT = 2*pi*i/N2;
T = [1 0 0 r2*cos(thT); 0 1 0 r2*sin(thT);0 0 1 0]*[cos(thT) -sin(thT) 0 0;sin(thT) cos(thT) 0 0 ; 0 0 1 0; 0 0 0 1];
%add points
xyzT(:,(i-1)*N1+1:i*N1)=T*[xyzC ;ones(1,N1)];
end
%build patch triples
tri=[];
for i=1:N2
for j=1:N1
%get three points:
% jth from ith circle
% j+1th from ith circle
% jth from i+1th circle
tri(end+1,:)=[(i-1)*N1+j (i-1)*N1+j+1 i*N1+j];
%get three points:
% j+1th from ith circle
% j+1th from i+1th circle
% jth from i+1th circle
tri(end+1,:)=[ i*N1+j (i-1)*N1+j+1 i*N1+j+1];
end
end
tri=mod(tri-1,N1*N2)+1;
trisurf(tri,xyzT(1,:),xyzT(2,:),xyzT(3,:));axis equal
%fancy
shading interp
camlight left
and get:

Answer 2

Thank you for the example. It would of been nicer to render the torus using triangle instead of rectangles but deadline is like an hour away and I managed to get my current one working!
I figured out what was causing my issues. Rotating the translation matrix was having an adverse effect on the orientation of each circles points (they don't line up), causing it to twist.
After looking through the notes more thoroughly (that is going several years worth of online notes back), I managed to find some psuedo code for a sweep function that I used to completely rewrite. I now generate 20 circles in the same spot, and rotate each one around the origin by increasing amounts instead. Which led to this:
%% Settings
points = 20; %Number of points in each circle
circles = 20; %Number of circles making up the torus
radius = 1; %Radius of the circle
scale = 0.75; %Scale to apply to the whole torus
center = [2 0 0]; %Center point of the first circle to sweep into a torus
%% Create (circles+1) circles after the other in an array at point [2 0 0]
%The extra circle overlaps the first, this is to make face generation much
%simpler.
V = zeros(circles*points, 3);
for i=0:points:points*circles
for k=1:points
V(i+k,1) = center(1) + radius * cosd((k-1)*(360/points));
V(i+k,2) = center(2) + 0;
V(i+k,3) = center(3) + radius * sind((k-1)*(360/points));
end
end
%% Sweep the circles, rotate each circle 18 degrees more than the previous
for n=0:points:circles*points
%Calculate degrees for current circle
D = (n/points) * 360/circles;
%Create a Z-rotation matrix
Rz = [
cosd(D) sind(D) 0;
-sind(D) cosd(D) 0;
0 0 1;
];
%Rotate each point of the circle
for i=1:points
V(n+i, :) = Rz * V(n+i, :)';
end
end
%% Scale the torus
%Create a scalar matrix
S = [
scale 0 0;
0 scale 0;
0 0 scale
];
%Scale each point
for n=0:points:circles*points
for i=1:points
V(n+i, :) = S * V(n+i, :)';
end
end
%% Generate faces
F = zeros(circles*points, 4);
for n=1:points:circles*points
for k=1:points
%If it's an endface then have destination face vertices wrap around to the first face of the current circle
if(mod(k, points) == 0)
F((n-1)+k,2)= (n-1)+k+1 - points;
F((n-1)+k,3)= n+points+k - points;
else
%otherwise use the next faces starting vertices
F((n-1)+k,2)= (n-1)+k+1;
F((n-1)+k,3)= n+points+k;
end
%Set the points coming from the previous face
F((n-1)+k,1)= (n-1)+k;
F((n-1)+k,4)= n+points+k-1;
end
end
%% Render
%Configure renderer
axis equal;
hold on;
%Render points
scatter3(V(1:end, 1), V(1:end, 2), V(1:end, 3), 'MarkerEdgeColor', 'b');
%Render faces
patch('Faces', F, 'Vertices', V, 'FaceColor', 'g');
Which makes: