How to generate a rotation matrix or a quaternion (properly rotated in regards to texture coordinates) for each triangle of a mesh?

Question

I'm trying to build a uniform grid made of 3D points over the surface of an arbitrary mesh (we have texture-coordinates for each vertex...)

Or to reformulate: imagine a mesh textured with a texture featuring equally spaced dots: I want to obtain the 3D coordinates of these dots.

So far, I'm able to generate a normal for each triangle of the mesh but it is not enough.

If I had the right matrix, or quaternion or even orthonormal-basis for each triangle, I guess that I could transform any 2D coordinates (on the triangle's plane) to 3D coordinates (in object space...)

The problem is how to rotate properly in regards to texture coordinates?

To be more specific: for each triangle, I can define a normal. It could serve as the x,y,z part of a quaternion. But how could I generate the w part?

Answer 1

Edit new answer:

Ok this shows you how to calculate the rotation-matrix that rotates the triangle from its local space into world-space.

Given normalLocal (the normal of the triangle in object-space) and normalWorld (the normal of the triangle in world-space)

1. Find the rotation-axis by taking the cross-product of the 2 normals

   axis = normalLocal cross normalWorld

2. Find the rotation angle by taking arccosine of the dot-product of both normals

   angle = acos(normalLocal dot normalWorld)

3. Now create a rotation-matrix from those values .

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Old obsolete answer:

You can just use the texture-coordinates to define the triangle points. Just put them into x and y of the point and leave z at zero.

Example:

texUV {u, v} are the 3 texture-coordinates for 1 trangle

p are the 3 associated euclidean points of the triangle

texUV0 = {0.0, 0.0}
texUV1 = {0.5, 0.5}
texUV2 = {0.5, 0.0}

p0 = {texUV0.u, texUV0.v, 0.0}
p1 = {texUV1.u, texUV1.v, 0.0}
p2 = {texUV2.u, texUV2.v, 0.0}

p0 = {0.0, 0.0, 0.0}
p1 = {0.5, 0.5, 0.0}
p2 = {0.5, 0.0, 0.0}

The normal for this triangle is {0.0, 0.0, 1.0} or {0.0, 0.0, -1.0} depending on your definition of a front face, clockwise or counterclockwise.

Answer 2

I found this answer by user S1CA in the gamedev.net forum:

A, B, C are the vertices of a particular triangle.
Auv, Buv, Cuv are the UV coordinates at the vertices.
Axyz, Bxyz, Cxyz are the positions of the vertices.
Puv are the UV coordinates of the point.
Pabc are the barycentric coordinates of the point [for that triangle]
Pxyz is the object space position of the point

aa = (Bu-Au) * (Cv-Av) - (Cu-Au) * (Bv-Av)
Pa = ((Bu-Pu) * (Cv-Pv) - (Cu-Pu) * (Bv-Pv)) / aa
Pb = ((Cu-Pu) * (Av-Pv) - (Au-Pu) * (Cv-Pv)) / aa
Pc = ((Au-Pu) * (Bv-Pv) - (Bu-Pu) * (Av-Pv)) / aa

The above are just a bunch of 2D cross products. The 2D cross product happens to give you the area of a triangle. If you look at the Wolfram reference posted above, it notes that the areas of the sub triangles ABP ACP and BCP are proportional to the barycentric coordinates.

The division of each sub triangle by the area of the main triangle is so we can get the useful property of areal coordinates that for any point which is inside the triangle, Pa+Pb+Pc=1

Once you have barycentric coordinates, you can use those to convert between texture units and object units by doing:

Pxyz = Axyz*Pa + Bxyz*Pb + Cxyz*Pc