Dilations

Dilations are geometric transformations that produce a scaled version of a figure without changing its shape. Each dilation is determined by a center point O and a dilation factor k. A point P is moved to P' along the line OP so that OP' = k · OP. The image is similar to the original; lengths are multiplied by |k|, areas by k^2, and angles are preserved. If k > 1 the figure enlarges; if 0 < k < 1 it reduces; if k = 1 there is no change. The center O is fixed, and every other point moves along its radius from O. If k < 0 the image lies on the opposite side of O, effectively a reflection through O combined with dilation.

In the coordinate plane, if O is the origin, the dilation is simply P' = kP; for a different center O, P' = O + k(P − O). Dilations are a form of similarity transformation, together with translations and rotations, and map lines to lines while preserving the center-based direction of rays through O.

In image processing, dilation refers to a related but distinct concept known as morphological dilation. For binary images, A ⊕ B = { z | B translated by z intersects A }. With grayscale images, dilation uses a structuring element to propagate bright values into neighboring pixels, expanding regions, filling small holes, and joining nearby features. This operation is widely used in object detection and image cleanup, often in combination with erosion and other morphological operations.