Spherical UV coordinates map latitude/longitude angles to texture image coordinates for sphere texturing

To apply a 2D image texture to a sphere, each surface point is mapped to UV coordinates (0–1 range) using spherical geometry. The 3D surface normal is converted to angles: theta (polar/latitude angle from south pole) and phi (azimuthal/longitude angle around the Y-axis). These are normalized: u = phi/(2*pi), v = theta/pi. This maps the sphere’s surface onto the rectangular texture image, with north pole at v=1, south pole at v=0, and the prime meridian at u=0. Seams appear at u=0/1 and the poles are pinched — familiar UV distortions on spherical objects in any 3D tool.

Examples

For a sphere at origin, given surface hit point p (unit vector): phi = atan2(-z, x) + pi, theta = acos(-y). Then u = phi/(2*pi), v = theta/pi.

Assessment

Describe the UV distortion artifact that appears at the poles of a sphere-mapped texture. Why does it occur?