Quadratrix of Dinostratus
The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve, which is created by a uniform motion. It is one of the oldest examples for a kinematic curve, that is a curve created through motion. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem (hence trisectrix). Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle (hence quadratrix).
Consider a square ABCD with an inscribed quarter circle centered in A, such that the side of the square is the circle's radius. Let E be a point that travels with a constant angular velocity on the quarter circle arc from D to B. In addition the point F travels with a constant velocity from D to A on the line segment AD, in such a way that E and F start at the same time at D and arrive at the same time in B and A. Now the quadratrix is defined as the locus of the intersection of the parallel to AB through F and the line segment AE.
If one places such a square ABCD with side length a in a (cartesian) coordinate system with the side AB on the x-axis and vertex A in the origin, then the quadratix is described by a planar curve with:
![\gamma :(0,{\tfrac {\pi }{2}}]\rightarrow \mathbb {R} ^{2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04d8a0ec9f976fbd161120f0ec294c717a520e80)
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