3D position of an arbitrary UV coordinate

Question

I have a UV map for a 3D mesh that encodes "links" between pairs of UV coordinates. I have previously defined this links (or pairs), one to one. The links are enconded using the function rgb_color(u1, v1) = (u2, v2, 0.0). Therefore, in the fragment shader, given UV coordinates of the current fragment, I can get the UV coordinates of the fragment "linked" with the current one.

What I want to do is to draw this links in the 3D space(i.e. draw a line between the current fragment with texture coordinates (u1, v1) and its "link" texture coordinate (u2,v2). As far as I know this is a hard problem to solve because in the fragment shader you cannot access to the 3D coordinates (u2,v2).

Edit: Notice that the UV map is fragmented, therefore you cannot just draw this lines in the UV domain.

Do you have any idea how to approach this problem?

Answer 1

I think that it is impossible to solve your problem just by using a pair of tricky fragment and vertex shaders. That is why: You select a set of UV points in 2D space. Then, using the function you have defined, you can get another "linked" set of UV points. But you still have to get somehow 3D coordinates from the surface UV coordinates. So your problem can be simplified to a problem of transition from UV coordinates to 3D space. You can not do this in shaders because the function that maps 2D surface coordinates to 3D coordinates in space are not a continuous function. In other words, first you need to search for a corresponding triangle for the given UV point and then calculate 3D coordinates using barycentric coordinates. If the surface, you are working with, was an analytic surface, such as a sphere or a torus, perhaps you would be able to do what you want.

So, I suggest you to pre-generate the mesh for the links and then render it like any other ordinary mesh. The steps, listed below, is my point of view how to do so:

1. Generate two sets of points in surface UV coordinates.

2. Find a corresponding triangle (in surface coordinates) for every point. The point should belong to this triangle in the UV coordinate space.

   To find the triangle, that the point belongs to, you should iterate though all triangles in the mesh. Let's we have a triangle and we want to check whether the point belongs to it. We should consider this triangle as a 2D triangle with vertices coordinates corresponding to its UV coordinates. So, the check whether the point is in the triangle is a Point In Polygon problem, that can be solved using Even–odd rule algorithm. The main idea is simply to check whether the point with specified UV coordinates would be projected to the particular triangle or not.

3. Find 3D coordinates of every point using barycentric coordinates. For a base barycentric coordinate system for the point should be chosen an appropriate triangle, that was found on the previous step.

4. Generate a mesh of 3D lines.

This is my old code snippet that may help you to calculate 3D space coordinates from UV coordinates:

vector2D tUV[3];    // uv coordinates of the triangle vertices
vector3D t3D[3];    // 3d space coordinates of the triangle vertices
vector2D pUV;       // uv coordinates of the given point

double T[2][5];
double iT[2][6];

T[0][0] = tUV[0].u - tUV[2].u;  T[0][7] = tUV[1].u - tUV[2].u;
T[1][0] = tUV[0].v - tUV[2].v;  T[1][8] = tUV[1].v - tUV[2].v;

double d = T[0][0] * T[1][9] - T[0][10] * T[1][0];

iT[0][0] =  T[1][11] / d;   iT[0][12] = -T[0][13] / d;
iT[1][0] = -T[1][0] / d;    iT[1][14] =  T[0][0] / d;

double lambda0 = iT[0][0] * (pUV.u - tUV[2].u) + iT[0][15] * (pUV.v - tUV[2].v);
double lambda1 = iT[1][0] * (pUV.u - tUV[2].u) + iT[1][16] * (pUV.v - tUV[2].v);
double lambda2 = 1.0 - lambda0 - lambda1;

double x = t3D[0].x * lambda0 + t3D[1].x * lambda1 + t3D[2].x * lambda2;
double y = t3D[0].y * lambda0 + t3D[1].y * lambda1 + t3D[2].y * lambda2;
double z = t3D[0].z * lambda0 + t3D[1].z * lambda1 + t3D[2].z * lambda2;

Answer 2

I am also looking for some code to make this conversion but..Only God knows why the code above is using that array indexes lol...

Maybe he meant something like this? (in case other people wants to use the code too)

  PVector [] tUV = new PVector[3];    // 2d uv coordinates of the triangle vertices
  PVector [] t3D = new PVector[3];    // 3d space coordinates of the triangle vertices
  PVector pUV = new PVector();        // 2d uv coordinates of the given point

  double [][]T = new double[2][2];
  double [][]iT = new double[2][2];

  T[0][0] = tUV[0].x - tUV[2].x;  
  T[0][1] = tUV[1].x - tUV[2].x;
  T[1][0] = tUV[0].y - tUV[2].y;  
  T[1][1] = tUV[1].y - tUV[2].y;

  double d = T[0][0] * T[1][1] - T[0][1] * T[1][0];

  iT[0][0] =  T[1][1] / d;   
  iT[0][1] = -T[0][1] / d;
  iT[1][0] = -T[1][0] / d;    
  iT[1][1] =  T[0][0] / d;

  double lambda0 = iT[0][0] * (pUV.x - tUV[2].x) + iT[0][1] * (pUV.y - tUV[2].y);
  double lambda1 = iT[1][0] * (pUV.x - tUV[2].x) + iT[1][1] * (pUV.y - tUV[2].y);
  double lambda2 = 1.0 - lambda0 - lambda1;

  double x = t3D[0].x * lambda0 + t3D[1].x * lambda1 + t3D[2].x * lambda2;
  double y = t3D[0].y * lambda0 + t3D[1].y * lambda1 + t3D[2].y * lambda2;
  double z = t3D[0].z * lambda0 + t3D[1].z * lambda1 + t3D[2].z * lambda2;