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Euler's theorem: = | | = In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
R: Radius of curvature R c: Radius of circular curve at the end of the spiral θ: Angle of curve from beginning of spiral (infinite R) to a particular point on the spiral. This can also be measured as the angle between the initial tangent and the tangent at the concerned point. θ s: Angle of full spiral curve L, s
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal p c (the contrapedal coordinate ) even though it is not an independent quantity and it relates to ( r , p ) as p c := r 2 − p ...
By dividing out e r 1 x, it can be seen that = = Therefore, the general case for u(x) is a polynomial of degree k − 1, so that u(x) = c 1 + c 2 x + c 3 x 2 + ⋯ + c k x k −1. [6] Since y(x) = ue r 1 x, the part of the general solution corresponding to r 1 is
This diagram gives the route to find the Schwarzschild solution by using the weak field approximation. The equality on the second row gives g 44 = −c 2 + 2GM/r, assuming the desired solution degenerates to Minkowski metric when the motion happens far away from the blackhole (r approaches to positive infinity).
Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers. In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted R n or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors.
The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a 0 = 1 ( 665,857 / 470,832 ) is too large by about 1.6 × 10 −12; its square is ≈ 2.000 000 000 0045.