ForRotation

ForRotation is a term used to describe the operation of rotating geometric objects in space, typically within mathematics, computer graphics, and related fields. In mathematical terms, a rotation is a linear isometry that preserves distances and orientation, represented by a matrix in the special orthogonal group SO(n). In practice, ForRotation can be realized through several common representations.

In two dimensions, a rotation about the origin by angle θ is given by the 2x2 matrix [ [cos θ, -sin θ], [sin θ, cos θ] ], which maps a point (x, y) to (x', y'). In three dimensions, rotations can be expressed as 3x3 rotation matrices, axis–angle representations, Euler angles, or unit quaternions. When applying ForRotation to a point or vector, one multiplies the coordinate by the chosen rotation representation, for example v' = Rv in matrix form or v' = qvq⁻¹ in quaternion form.

Key properties include preservation of vector norms and inner products, and a determinant of +1. Rotations are invertible, with inverses corresponding to the rotation by the opposite angle (or the conjugate of a unit quaternion). Composition of rotations is achieved by matrix multiplication or quaternion multiplication, yielding a new rotation.

Computationally, quaternions are often preferred in 3D contexts to avoid gimbal lock and to enable efficient concatenation and normalization. ForRotation is central to algorithms in computer graphics, robotics, virtual reality, and simulations, where precise and stable orientation changes are required.

See also: rotation matrix, axis-angle, Euler angles, quaternion, Rodrigues’ formula.