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SageMath is designed partially as a free alternative to the general-purpose mathematics products Maple and MATLAB. It can be downloaded or used through a web site. SageMath comprises a variety of other free packages, with a common interface and language. SageMath is developed in Python.
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, =
A circle of radius 23 drawn by the Bresenham algorithm. In computer graphics, the midpoint circle algorithm is an algorithm used to determine the points needed for rasterizing a circle. It's a generalization of Bresenham's line algorithm. The algorithm can be further generalized to conic sections. [1] [2] [3]
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that
McCullough and Wade [18] extended this approach, which produces all Pythagorean triples when k > h √ 2 /d: Write a positive integer h as pq 2 with p square-free and q positive. Set d = 2pq if p is odd, or d= pq if p is even. For all pairs (h,k) of positive integers, the triples are given by
Viète obtained his formula by comparing the areas of regular polygons with 2 n and 2 n + 1 sides inscribed in a circle. [ 1 ] [ 2 ] The first term in the product, 2 / 2 {\displaystyle {\sqrt {2}}/2} , is the ratio of areas of a square and an octagon , the second term is the ratio of areas of an octagon and a hexadecagon , etc.
The implicit function theorem describes conditions under which an equation (,) = can be solved implicitly for x and/or y – that is, under which one can validly write = or = (). This theorem is the key for the computation of essential geometric features of the curve: tangents , normals , and curvature .
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes.Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.