equidistants

Equidistants refers to loci related to equal distances from given geometric objects. In plane geometry, the concept covers points, lines, or curves that share the same distance to two or more fixed elements, such as points, lines, or curves. Equidistant sets arise in classical constructions as well as in more advanced curve theory, including offsetting and locus problems.

A fundamental example is the locus of points equidistant from two fixed points A and B. This locus is the perpendicular bisector of segment AB: every point on it is equally distant from A and B, and conversely any point equidistant from A and B lies on this line.

For two fixed lines, the set of points equidistant from the lines is the pair of angle bisectors formed by the lines. These bisectors are the internal and external loci where the distances to the two lines are equal.

A more general case is the locus of points equidistant from a fixed point P and a fixed line l. This set is a parabola with focus at P and directrix l; by definition, any point on the parabola has equal distance to P and to l.

Equidistant curves, or offset curves, are the sets of points at a fixed distance d from a given curve. Parallel or offset curves are used in design and manufacturing, robotics, and computer graphics, and they may develop features such as cusps or self-intersections depending on the original curve.

In triangle geometry, the incenter is the unique point equidistant from all three sides, lying at the intersection of the internal angle bisectors. More generally, lines that are equidistant from two sides are the angle bisectors.

Applications of equidistants include geometric constructions, characterization of conic sections, and various design algorithms that rely on precise distance relations.