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Problems involving tangent circles are often generalized to spheres. For example, the Fermat problem of finding sphere(s) tangent to four given spheres is a generalization of Apollonius' problem , whereas Soddy's hexlet is a generalization of a Steiner chain .
The internal and external tangent lines are useful in solving the belt problem, which is to calculate the length of a belt or rope needed to fit snugly over two pulleys. If the belt is considered to be a mathematical line of negligible thickness, and if both pulleys are assumed to lie in exactly the same plane, the problem devolves to summing ...
Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to: three points (denoted PPP, generally 1 solution)
The extension of Apollonius' problem to three dimensions, namely, the problem of finding a fifth sphere that is tangent to four given spheres, can be solved by analogous methods. [9] For example, the given and solution spheres can be resized so that one given sphere is shrunk to point while maintaining tangency. [ 38 ]
For example, if two circular arcs ... or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century.
The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. [2] Leonhard Euler used it to evaluate the integral ∫ d x / ( a + b cos x ) {\textstyle \int dx/(a+b\cos x)} in his 1768 integral calculus textbook , [ 3 ] and Adrien-Marie Legendre described ...
For example, for Newton's method as applied to a function f to oscillate between 0 and 1, it is only necessary that the tangent line to f at 0 intersects the x-axis at 1 and that the tangent line to f at 1 intersects the x-axis at 0. [19] This is the case, for example, if f(x) = x 3 − 2x + 2.
These are an infinite family of circles tangent to the -axis of the Cartesian coordinate system at its rational points. Each fraction / (in lowest terms) has a circle tangent to the line at the point (/,) with curvature . Three of these curvatures, together with the zero curvature of the axis, meet the conditions of Descartes' theorem whenever ...