3D Graphics, VR & AR Math

Convert a unit quaternion (x, y, z, w) to roll/pitch/yaw Euler angles, with gimbal-lock detection.

Convert roll/pitch/yaw (degrees) to a unit quaternion (x, y, z, w) for engines, IMUs and animation.

Convert a rotation axis and angle to a unit quaternion — the most intuitive way to specify a 3D rotation.

Compose two rotations by multiplying quaternions (Hamilton product) — order matters, and this shows why.

Dot product, cross product, the angle between two 3D vectors and whether they're parallel or perpendicular.

Convert between field of view and focal length for cameras, game engines and lenses, for any sensor/film size.

Convert a 3×3 rotation matrix to a unit quaternion using Shepperd's numerically-stable method.

Spherical linear interpolation between two orientations at parameter t — smooth, constant-speed rotation blending.

Compute the (normalized) surface normal of a triangle from its three vertices — for lighting and culling.

Express a 2D point as weights of a triangle's vertices — for interpolation, hit-testing and rasterization.

From a camera position and target, get the forward direction and the yaw/pitch to aim there.

Solve missing width/height for a target aspect ratio, and get megapixels and the simplified ratio.