Orthoptic (geometry)

In geometry, the orthoptic of a curve is the locus of points from which two tangents to the curve meet at right angles. That is, a point lies on the orthoptic if and only if the tangents from that point to the curve are perpendicular.

The orthoptic is sometimes also called the director circle, especially when referring to the orthoptic of a conic section. For example, the orthoptic of a parabola is its directrix, and the orthoptic of an ellipse is a circle.

The concept of orthoptics can be extended to three-dimensional objects, though the term is not as commonly used. In 3D, the orthoptic would represent the set of points from which perpendicular tangent planes can be drawn to the object.

The calculation of the orthoptic for a given curve often involves finding the equation of the tangent lines to the curve and then determining the condition under which two such tangent lines are perpendicular. This typically leads to an implicit equation defining the locus of points which constitutes the orthoptic.