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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 ...
In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k 2. The inverse of the curve C is then the locus of P as Q runs over C.
Toggle Mathematics (Geometry) subsection. 1.1 Algebraic curves. 1.1.1 Rational curves. ... Inverse curve; Involute; Isoptic including Orthoptic; Negative pedal curve.
Inverse curve; Inversive distance; L. Limiting point (geometry) P. Pappus chain; S. Steiner chain; T. A Treatise on the Circle and the Sphere
Some authors modify this formula by taking the inverse hyperbolic cosine of the value given above, rather than the value itself. [ 2 ] [ 4 ] [ 5 ] That is, rather than using the number I {\displaystyle I} as the inversive distance, the distance is instead defined as the number δ {\displaystyle \delta } obeying the equation
Sometimes, this multivalued inverse is called the full inverse of f, and the portions (such as √ x and − √ x) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch , and its value at y is called the principal value of f −1 ( y ) .
In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius R.His work initiated a large body of publications, now called inversive geometry.
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem [1] or the upside down Pythagorean theorem [2]) is as follows: [3] Let A, B be the endpoints of the hypotenuse of a right triangle ABC. Let D be the foot of a perpendicular dropped from C, the vertex of the right angle, to the hypotenuse. Then