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It lets one turn a quarter circle into square of the same area, hence a square with twice the side length has the same area as the full circle. According to Dinostratus' theorem the quadratrix divides one of the sides of the associated square in a ratio of 2 π {\displaystyle {\tfrac {2}{\pi }}} . [ 1 ]
The cruciform curve, or cross curve is a quartic plane curve given by the equation = where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a 2 x 2 + b 2 y 2 = 1, and is therefore a rational plane algebraic curve of genus zero.
A circular sector is shaded in green. Its curved boundary of length L is a circular arc. A circular arc is the arc of a circle between a pair of distinct points.If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle ...
Curvature is usually measured in radius of curvature.A small circle can be easily laid out by just using radius of curvature, but degree of curvature is more convenient for calculating and laying out the curve if the radius is as large as a kilometer or mile, as is needed for large scale works like roads and railroads.
In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral.
The curve represents a circle exactly, but it is not exactly parametrized in the circle's arc length. This means, for example, that the point at t {\displaystyle t} does not lie at ( sin ( t ) , cos ( t ) ) {\displaystyle (\sin(t),\cos(t))} (except for the start, middle and end point of each quarter circle, since the representation is ...
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The point where the curve crosses the y-axis has y = 2a/π; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of 1/π, leading to a solution of the classical problem of squaring the circle.