# How to get UV coordinates for sphere (Cylindrical Projection)

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 ?

• @DMGregory I want to project the image (texture) along the side of a cylinder which is subsequently projected onto the sphere orthogonally to the "up" vector. – john john Jan 7 '16 at 0:38
• The image you've shown looks like equirectangular projection (ie. latitude & longitude). Is this representative of what you need? If so, you don't need to use a cylinder as an intermediate. – DMGregory Jan 7 '16 at 0:43
• Ah, or it may be Lambert's Equal-Area Projection (found by reverse image search). This is effectively the same as equirectangular in the horizontal axis, but compresses the vertical axis close to the poles (which might not be desirable for texture mapping, since it will tend to cause loss of detail there. Its aspect ratio of pi:1 is also not as convenient for mipmapping as equirectangular's 2:1) – DMGregory Jan 7 '16 at 0:53

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.

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.

• could you please explain to me the reason why in the function atan2 you put as numerator the x value and in the denominator the z value ? – john john Jan 7 '16 at 10:13
• The angle around the sphere's equator is the texture's u coordinate. atan2 gives us that angle. Assuming y is the vertical (north-south) axis of your globe. Otherwise swap y and z in the formula if z is the vertical axis in your sphere's coordinate system. – Stephane Hockenhull Jan 7 '16 at 13:12