What property the "vertex texcoord" should have when calculating tangent Space

Question

    I'm using OpenMesh to handle triangle meshes.
    I have done mesh parametrization to set vertex texcoord and my whole understanding about vertex texcoord is getting from there. It should be a changeable value to vertex if I'm not getting it wrong.
    But now I want to calculate the tangent space for every vertex and all tutorials talk about "vertex texcoord" like it's a fixed property to vertex.
    Here's one tutorials I read, and it says

If the mesh we are working on doesn't have texcoord we'll skip the Tangent Space phase, because is not possible to create an arbitrary UV Map in the code, UV Maps are design dependents and change the way as the texture is made.

So, what property the "texcoord" should have when calculating tangent Space
Thank you!

What do you mean by changeable value? When you have choosen a parametrization, then each vertex will have a specific texture coordinate which can then be used to calculate the tangent space. — BDL, Sep 01 '16 at 12:33
It doesn't matter where the texture coordinates come from. Be it a parameterization (where you need mutable texture coordinates, indeed) or a modeling step. When you want to calculate the tangent space, these coordinates have to be fixed. If you change them, the tangent space will change, too. — Nico Schertler, Sep 01 '16 at 15:02
Thanks for answering and sorry for the vague question. When I did mesh parametrization, using different method can get different result and then different texcoord. But when I'm learning tangent space, it makes me feel texcoord is a build-in attribute. It's clear now. So I can use whatever parametrization result to calculate tangent space. — Hugo.C, Sep 02 '16 at 01:06

Yakov Galka · Accepted Answer · 2020-07-01T13:28:16.003

It's unclear what you're asking exactly, so hopefully this will help your understanding.

The texture-coordinates (texcoord) of each vertex are set during the model design phase and are loaded with the mesh. They contains the UV coordinates that the vertex is mapped to within the texture.

The tangent space is formed of the tangent, bitangent and normal (TBN) vectors at each point. The normal is either loaded with the mesh or can be calculated by averaging the normals of triangles meeting at the vertex. The tangent is the direction in which the U coordinate of the texcoord changes the most, i.e. the partial derivative of the model-space position by U. Similarily the bitangent is the partial derivative of position by V. Tangent and bitangent can be calculated together with the normals of each face and then averaged at the vertices, just like normals do.

For flat faces, tangent and bitangent are perpendicular to the normal by construction. However, due to the averaging at the vertices they may not be perpendicular anymore. Also even for flat faces, the tangent may not be perpendicular to the bitangent (e.g. imagine a skewed checker board texture mapping). However, to simplify the inversion of the TBN matrix, it is sometimes approximated with an orthogonal matrix, or even a quaternion. Even though this approximation isn't valid for skewed-mapped textures, it may still give plausible results. When orthogonality is assumed, the bitangent can be calculated as a cross product between the tangent and the normal.

Thanks for answering and sorry for the vague question. When I did mesh parametrization, using different method can get different result and then different texcoord. But when I'm learning tangent space, it makes me feel texcoord is a build-in attribute. It's clear now. So I can use whatever parametrization result to calculate tangent space. — Hugo.C, Sep 02 '16 at 01:13

What property the "vertex texcoord" should have when calculating tangent Space

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