transformationsscaling

Scaling transformation, often called a dilation, is a geometric transformation that enlarges or reduces objects with respect to a fixed point known as the center of scaling. When the center is the origin and the scale factor is uniform, all coordinates are multiplied by the same number, expanding or contracting the object while preserving its shape. If the center is not the origin, the same idea applies but the shift to and from the center is included, producing an affine transformation.

In two dimensions, a uniform scale by a factor k about the origin maps a point (x, y) to (kx, ky). About a center c = (cx, cy), the mapping is x' = cx + k(x − cx) and y' = cy + k(y − cy). Non-uniform scaling uses different factors along axes, for example kx along x and ky along y, so (x, y) maps to (kxa, k y) in appropriate coordinates. In three dimensions, non-uniform scaling uses factors along x, y, and z. When the center is not at the origin, scaling is an affine transformation rather than a linear one.

Key properties include that scaling preserves straight lines and parallelism; a line remains a line, and parallel lines remain parallel after scaling. Uniform scaling preserves angles and the overall shape up to a change in size, while non-uniform scaling can distort angles and shape. The operation is invertible with inverse scaling by the reciprocal factor, and composing two scalings multiplies their factors.

Applications of scaling transformations appear in computer graphics, where objects are resized, in image processing for resizing images, and in engineering and design to create proportional models. They form a basic, widely used tool in transformation geometry and CAD systems.