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Since C = 2πr, the circumference of a unit circle is 2π. In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. [1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.
Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square. Equivalently, the problem is to arrange n points in a unit square aiming to get the greatest minimal separation, d n , between points. [ 1 ]
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 upper half of the unit circle can be parameterized as =.
The Furman is a unit of angular measure equal to 1 ⁄ 65,536 of a circle, or just under 20 arcseconds. It is named for Alan T. Furman, the American mathematician who adapted the CORDIC algorithm for 16-bit fixed-point arithmetic sometime around 1980. [81] 16 bits give a resolution of 2 16 = 65,536 distinct angles.
where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width. = where A is the area of a circle. More generally, =
The unit vector ^ has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of (). If the particle displacement rotates through an angle dθ in time dt , so does u ^ R ( t ) {\displaystyle {\hat {\mathbf {u} }}_{R}(t)} , describing an arc on the unit circle ...
Consider the unit circle circumscribed by a square of side length 2. The transformation sends the circle to an ellipse by stretching or shrinking the horizontal and vertical diameters to the major and minor axes of the ellipse. The square gets sent to a rectangle circumscribing the ellipse. The ratio of the area of the circle to the square is ...
Arc distances on a great circle are the same as the distance between the same points on a sphere, and on the hemispheres into which the circle divides the sphere.. The Riemannian unit circle of length 2 π can be embedded, without any change of distance, into the metric of geodesics on a unit sphere, by mapping the circle to a great circle and its metric to great-circle distance.