Enlargements

Enlargement is a geometric transformation that maps every point of a figure to a new point along the line from a fixed center to the original point, with distances from the center scaled by a constant factor. The fixed point, called the center of enlargement, remains unchanged. The transformation is defined by a nonzero scaling factor k.

For a center C and a scale factor k, each original point P is mapped to P' on the line CP such that CP' = |k| · CP and the direction from C to P is scaled by k. If k > 1 the image is larger than the original; if 0 < k < 1, the image is smaller; if k = 1 there is no change. If k < 0, the image is rotated by 180 degrees around C and scaled by |k|, effectively combining enlargement with a half-turn.

Enlargement preserves straight lines and the ratios of distances along lines through the center. It preserves angles between corresponding lines, so figures remain similar to the original. Orientation is preserved when k is positive and reversed when k is negative. The center of enlargement is a fixed point for all points in the plane, and successive enlargements combine by multiplying their scale factors.

In practice, enlargements appear in mathematics to study similarity, in art and design to scale diagrams, and in photography and printing to produce larger copies from negatives or digital images. Modern digital tools perform computer-based enlargements by resampling pixels, which can affect image sharpness.

Related concepts include dilation and homothety, which describe similarity transformations with a chosen center. Enlarge­ments are a foundational example of how geometric figures can be transformed while preserving shape.