heartsimilar

heartsimilar is a geometric term describing a class of plane figures that are related to a canonical heart shape by similarity transformations. A similarity is a composition of translation, rotation, uniform scaling, and optionally reflection. Under this relation, any image of the standard heart curve is considered heartsimilar to the others.

The canonical heart is the well-known heart-shaped algebraic curve defined by the equation (x^2 + y^2 - 1)^3 - x^2 y^3 = 0. This curve is often cited in mathematical art and geometry as a simple explicit example of a heart form.

Therefore, a figure is heartsimilar to the canonical heart if it can be obtained from points on the heart curve by a similarity map, typically written as z -> s R z + t, where s > 0 is a scale, R is a rotation or reflection, and t is a translation. All such figures share the same shape, differing only in size, orientation, and position. In particular, they have the same curvature distribution up to a uniform scale.

Applications include shape analysis in computer vision, pattern recognition, and educational demonstrations of similarity. The term serves as a convenient label for a family of heart-like shapes and can be extended conceptually to three-dimensional heart-like surfaces.

See also: Similarity (geometry); Heart curve; Shape analysis.