rperp2

rperp2 is a two-dimensional vector operation used in mathematics, computer graphics, and physics to produce a vector that is perpendicular to a given 2D vector. In many conventions, it represents a rotation by 90 degrees, converting a vector into its normal or tangential partner depending on the context.

Definition and convention: for a 2D vector r = (x, y) in R^2, rperp2(r) is commonly defined as (-y, x). This result is perpendicular to r, satisfies r · rperp2(r) = 0, and preserves magnitude, since |rperp2(r)| = |r|. Some libraries or texts may use the alternative form (y, -x), which corresponds to a clockwise rotation by 90 degrees; the choice depends on the adopted orientation convention.

Mathematical properties: rperp2 acts as a linear operator on R^2 and can be represented by the rotation matrix [ [0, -1], [1, 0] ]. Squaring the operator yields rperp2(rperp2(r)) = -r, reflecting a 180-degree flip after two applications. It is frequently used to generate perpendicular directions, normals to curves, or tangents in planar geometry.

Applications: rperp2 appears in constructing normal vectors for rendering and collision detection, in finite difference schemes for computing tangential components, and in physics simulations where perpendicular directions are needed to model constrained motion or rotational effects. It also serves as a building block in algorithms that require a quick perpendicular transformation without explicit angle calculations.

Notes: implementations may vary in sign conventions, so it is advisable to consult the specific library documentation to confirm the exact form of rperp2 used. See also rperp, perp, and rotation operations in 2D.