How to get UV coordinates for sphere (Cylindrical Projection)

Question

I have created one sphere and I want to map onto it a texture map. But I want first to project my map texture to cylinder and then to sphere.

So I want to create a function which takes as parameter a 3D point from sphere and calculate a uv coordinate of this point using cylindrical coordinates. My guess is that I have to transform the spherical coordinates to cylindrical but I am not sure.

Is this possible and how can I implement it ?

Answer 1

Assuming atan2 returns an angle in radian between -pi and pi (π) you do something like:

n = Normalize(sphere_surface_point - sphere_center);
u = atan2(n.x, n.z) / (2*pi) + 0.5;
v = n.y * 0.5 + 0.5;

Where sphere_surface_point is the point on the sphere surface.

/ (2*pi) is there to convert the returned angle to a value between -0.5 and 0.5

Add 0.5 to shift it to the range 0..1 for texture mapping that gives you the u texture coordinate.

n.y is already your v texture coordinate.

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It's more of a computer graphics (https://computergraphics.stackexchange.com/) question even though texture projections comes up a lot in the context of game dev.

Answer 2

Leaving this here for myself:

static const float PI = 3.14159265f;

// assumes p is a point on unit sphere

// https://en.wikipedia.org/wiki/File:Equirectangular_projection_SW.jpg
float2 sphere2mapUV_Equirectangular(float3 p)
{
    return float2(
        atan2(p.x, -p.z) / (2 * PI) + .5,
        p.y * .5 + .5
    );
}

// https://en.wikipedia.org/wiki/File:Lambert_cylindrical_equal-area_projection_SW.jpg
float2 sphere2mapUV_EqualArea(float3 p)
{
    return float2(
        (atan2(p.x, -p.z) / PI + 1) / 2,
        asin(p.y) / PI + .5
    );
}

I added links to example images in case you need a visual reference for checking which projection your map is made with.

Usage:

float2 uv = sphere2map(normalize(position));

That assumes position is in model space.

In case you're using this in vertex shader the position might already be normalized, so normalization could be skipped.

Your sphere model will usually have center at float3(0,0,0), but if that's not the case you'll have to subtract the sphere_center_position from position.