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P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion. An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original ...
[1] [6] Although less is known about the existence of inversive distance circle packings than for tangent circle packings, it is known that, when they exist, they can be uniquely specified (up to Möbius transformations) by a given maximal planar graph and set of Euclidean or hyperbolic inversive distances.
Or, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [5] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin, acos, atan. [6]
where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n are perpendicular vectors of length ρ (that is, a · a = b · b = ρ 2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t. The relevant derivatives of g work out to be
Thus, by the properties of inversive geometry, since the figure traced by point D is the inverse of the figure traced by point B, if B traces a circle passing through the center of inversion O, then D is constrained to trace a straight line. But if B traces a straight line not passing through O, then D must trace an arc of a circle passing ...
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement.
The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative. In fact, we have the following relation, the inverse function theorem: (()) ′ = ′ where . However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is ...