scalareflect

Scalareflect is a term used in mathematics and computer graphics to describe a compound linear transformation that combines uniform scaling with reflection. In its common formulation, a scalareflect acts on a vector v in R^n by applying a reflection R (an involution with R^2 = I and det(R) = -1) and then scaling by a nonzero scalar s, so that the transformation is represented by the matrix T = s R. Because scalar multiplication commutes with matrix multiplication, this definition is equivalent to applying the same scalar to the result of the reflection, or to performing the two steps in reverse order.

In two dimensions, a reflection across a line with unit normal n is given by R = I − 2 n n^T. A 2D scalareflect is then T = s (I − 2 n n^T). In three dimensions, the same idea uses a unit normal to define a plane reflection, with T = s R where R is the corresponding plane reflection matrix. More generally, scalareflects are linear maps T: R^n → R^n of the form T = s R with s ≠ 0 and R a reflection matrix.

Key properties include invertibility (T is invertible when s ≠ 0) and a determinant det(T) = s^n det(R) = −s^n, since det(R) = −1 for a reflection. The inverse is T⁻¹ = (1/s) R. Applications appear in geometric modeling, shader programming, and analysis of compounded transformations where a uniform scaling is combined with a mirror-like symmetry.

Note that scalareflect is not a standard, universally defined term in all mathematical references; it is a descriptive label for the specific composition of scaling and reflection used in various contexts.