UnitDiagonal

UnitDiagonal is a term used in geometry to denote the diagonal of a unit-length square, and by extension the diagonal of a unit hypercube in higher dimensions. In the common 2D case, a square with side length 1 has a diagonal length d = sqrt(2). This result follows from the Pythagorean theorem: d^2 = 1^2 + 1^2, so d = sqrt(2) (approximately 1.4142).

Vector form: The diagonal runs from (0,0) to (1,1) and has the direction vector (1,1). The corresponding unit vector along the diagonal is (1/√2, 1/√2).

Higher dimensions: In n dimensions, the diagonal length of a unit hypercube is sqrt(n). For a unit cube (n=3), the diagonal is sqrt(3).

Properties: The 2D UnitDiagonal is irrational; its square equals 2. It scales with the Euclidean norm and is often used to measure distance along diagonal directions.

Applications: UnitDiagonal appears in problems involving line length and distance calculations on grids, collision detection, and algorithms that require normalized diagonal directions, such as texture mapping and bounding-box computations.

See also: Diagonal, Euclidean norm, Pythagoras theorem, Unit vector.