The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two lines, one moving by translation at a uniform speed, and the other moving by rotation around one of its points at a uniform speed. An alternative definition as a parametric curve leads to an equivalence between the quadratrix, the image of the Lambert W function, and the graph of the function y = x cot x {\displaystyle y=x\cot x} . The discovery of this curve is attributed to the Greek sophist Hippias of Elis, around 420 BC. Historians of mathematics have suggested that Hippias used it to solve the angle trisection problem, hence its name as a trisectrix. Later around 350 BC Dinostratus used it to solve the problem of squaring the circle, hence its name as a quadratrix. Dinostratus's theorem, used by Dinostratus to square the circle, relates an endpoint of the curve to the value of π. Both angle trisection and squaring the circle can be solved using a compass, a straightedge, and a given copy of this curve; however, they cannot be solved with compass and straightedge alone. Although a dense set of points on the curve can be constructed by compass and straightedge, allowing these problems to be approximated, the whole curve cannot be constructed in this way. The quadratrix of Hippias is a transcendental curve. It is one of several curves used in Greek mathematics for squaring the circle.
Definitions
By moving lines Consider a square A B C D {\displaystyle ABCD} , and an inscribed quarter circle arc centered at A {\displaystyle A} with radius equal to the side of the square. Let E {\displaystyle E} be a point that travels with a constant angular velocity along the arc from D {\displaystyle D} to B {\displaystyle B} , and let F {\displaystyle F} be a point that travels simultaneously with a constant velocity from D {\displaystyle D} to A {\displaystyle A} along line segment A D ¯ {\displaystyle {\overline {AD}}} , so that E {\displaystyle E} and F {\displaystyle F} start at the same time at D {\displaystyle D} and arrive at the same time at B {\displaystyle B} and A {\displaystyle A} . Then the quadratrix is defined as the locus of the intersection of line segment A E ¯ {\displaystyle {\overline {AE}}} with the parallel line to A B ¯ {\displaystyle {\overline {AB}}} through F {\displaystyle F} .
Helicoid section Suppose that a plane contains two lines, one moving by rotation and the other moving by translation within the plane, and that this whole ensemble is also moving linearly, in a direction perpendicular to the plane, through three-dimensional space. Then, the combined motion of the plane with the translating line will trace out an inclined plane, while the combined motion of the plane with the rotating line will trace out a helicoid. The point where the two lines intersect will trace out a space curve whose projection onto the original plane is the quadratrix. Pappus of Alexandria observed that this construction can be reversed: intersecting a helicoid with an appropriately chosen inclined plane, and then projecting the curve of intersection onto a plane, can form a quadratrix.
Parametric equation
Choose Cartesian coordinates so that the defining square A B C D {\displaystyle ABCD} of the quadratrix lies in the positive quadrant, with A {\displaystyle A} at the origin ( 0 , 0 ) {\displaystyle (0,0)} and B {\displaystyle B} at point ( a , 0 ) {\displaystyle (a,0)} where a {\displaystyle a} is the side length of the square. For this definition, it is convenient to reverse the motion of the two defining lines of the quadratrix so that they start together on the x {\displaystyle x} -axis at time t = 0 {\displaystyle t=0} and end intersecting at D {\displaystyle D} at time t = π 2 {\displaystyle t={\tfrac {\pi }{2}}} ; this reversal does not change the curve that the lines trace out. Then the quadratrix can be described by parametric equations that give the coordinates of each point on the curve as a function of the time parameter t {\displaystyle t} , as
x ( t ) = 2 a π t cot ( t ) {\displaystyle x(t)={\frac {2a}{\pi }}t\cot(t)}
and
y ( t ) = 2 a π t , {\displaystyle y(t)={\frac {2a}{\pi }}t,}
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