Figuring out UVs for a circle mesh created by code

Question

I'm trying to generate a circular mesh made up of triangles with a common center at the center of the circle.

I want the UVs to point outward, so that a moving shader applied to this will seem like it's flowing from the center to the edges. This already happens naturally on the sphere primitive, but I only need a flat surface without an underside.

mesh

I use this code to generate the mesh. I thought the UV points would be similar to the vertexes, but I'm not getting any good results.

 private void _MakeMesh(int sides, float radius = 0.5f)
 {
 m_LiquidMesh.Clear();
 float angleStep = 360.0f / (float) sides;
 List<Vector3> vertexes      = new List<Vector3>();
 List<int>       triangles   = new List<int>();
 List<Vector2>   uvs         = new List<Vector2>();
 Quaternion      rotation  = Quaternion.Euler(0.0f, angleStep, 0.0f);
 
 // Make first triangle.
 vertexes.Add(new Vector3(0.0f, 0.0f, 0.0f));  
 vertexes.Add(new Vector3(radius, 0.0f, 0.0f));    
 vertexes.Add(rotation * vertexes[1]);        
 
 // First UV ??
 uvs.Add(new Vector2(0, 0));
 uvs.Add(new Vector2(1, 0));
 uvs.Add(rotation * uvs[1]);
 
 // Add triangle indices.
 triangles.Add(0);
 triangles.Add(1);
 triangles.Add(2);
 for (int i = 0; i < sides - 1; i++)
 {
     triangles.Add(0); 
     triangles.Add(vertexes.Count - 1);
     triangles.Add(vertexes.Count);
 
     // UV ??
 
     vertexes.Add(rotation * vertexes[vertexes.Count - 1]);
 }
 m_LiquidMesh.vertices   = vertexes.ToArray();
 m_LiquidMesh.triangles  = triangles.ToArray();
 m_LiquidMesh.uv         = uvs.ToArray();
 
 m_LiquidMesh.RecalculateNormals();
 m_LiquidMesh.RecalculateBounds();
 }

Please, where am I going wrong with this?

Answer 1

This is an old post, but maybe someone else will benefit from my solution.

So basically I gave my center point the center of the uv (0.5, 0.5) and then used the used circle formula to give every other point the uv coordinate. But of course I had to remap the cos and sin results from -1..1 to 0..1 and everything is working great.

     Vector2[] uv = new Vector2[vertices.Length];
     uv[uv.Length - 1] = new Vector2(0.5f, 0.5f);

     for (int i = 0; i < uv.Length - 1; i++)
     {
         float radians = (float) i / (uv.Length - 1) * 2 * Mathf.PI;
         uv[i] = new Vector2(Mathf.Cos(radians).Remap(-1f, 1f, 0f, 1f), Mathf.Sin(radians).Remap(-1f, 1f, 0f, 1f));
     }
         
     mesh.uv = uv;

Where the remap is an extension like this and it basically take a value in a range and remaps it to another range (in this case from -1..1 to 0..1):

 public static float Remap(this float value, float from1, float to1, float from2, float to2) {
     return (value - from1) / (to1 - from1) * (to2 - from2) + from2;
 }

Figuring out UVs for a circle mesh created by code

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