quadratrix of Hippias
The quadratrix of Hippias, also called the quadratrix of Dinostratus, is the first curve in recorded history that was not part of a line or a circle, and the first curve known that is not constructible in the classical sense; in other words, it can't be drawn using a straightedge and a compass alone, but instead has to be plotted point by point.
The quadratrix can be thought of as the intersection of two lines moving with constant velocity: the first line rotates (for example, counterclockwise) while the second line moves along (say, in the direction of the positive y-axis). It has the Cartesian equation y = x cot(πx/2a).
The quadratrix was discovered by Hippias of Elis in about 430 BC and was used by him in his work on trisecting an angle and squaring the circle. In fact, its name refers to its use in turning curvilinear space into a rectangular area.