In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845).Curves and Their Properties by Robert C. Yates, National Council of Teachers of Mathematics, Inc., Washington, D.C., p. 127: "Geometrical inversion seems to be due to Jakob Steiner who indicated a knowledge of the subject in 1824. He was closely followed by Adolphe Quetelet (1825) who gave some examples. Apparently independently discovered by Giusto Bellavitis in 1836, by Stubbs and Ingram in 1842–3, and by Lord Kelvin in 1845.)" The concept of inversion can be generalized to higher-dimensional spaces.
Inversion in a circle
Inverse of a point
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 such that
This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P also takes P back to P, so the result of applying the same inversion twice is the identity transformation which makes it a self-inversion (i.e. an involution). To make the inversion a total function that is also defined for O, it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity.
It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, for a point inside the circle, the nearer the point to the center, the further away its transformation. While for any point (inside or outside the circle), the nearer the point to the circle, the closer its transformation.
Compass and straightedge construction
Point outside circle To construct the inverse P of a point P outside a circle Ø: Draw the segment from O (center of circle Ø) to P. Let M be the midpoint of OP. (Not shown) Draw the blue circle c with center M going through P. Let N and N be the points where Ø and c intersect. Draw segment NN. P is where OP and NN intersect.
Point inside circle To construct the inverse P of a point P inside a circle Ø: Draw ray r (the horizontal line) from O (center of circle Ø) through P. Draw line s (the vertical line) through P perpendicular to r. Let N be one of the points where Ø and s intersect. Draw the segment ON. Draw line t through N perpendicular to ON. P is where ray r and line t intersect.
Dutta's construction There is a construction of the inverse point to A with respect to a circle Ø that is independent of whether A is inside or outside Ø.Dutta, Surajit (2014) A simple property of isosceles triangles with applications , Forum Geometricorum 14: 237–240
Consider a circle Ø with center O and a point A which may lie inside or outside the circle Ø. Take the intersection point C of the ray OA with the circle Ø. Connect the point C with an arbitrary point B on the circle Ø (different from C and from the point on Ø antipodal to C) Let h be the reflection of ray BA in line BC. Then h cuts ray OC in a point A. A is the inverse point of A with respect to circle Ø.
Properties
The inversion of a set of points in the plane with respect to a circle is the set of inverses of these points. The following properties make circle inversion useful. A circle that passes through the center O of the reference circle inverts to a line not passing through O, but parallel to the tangent to the original circle at O, and vice versa; whereas a line passing through O is inverted into itself (but not pointwise invariant). A circle not passing through O inverts to a circle not passing through O. If the circle meets the reference circle, these invariant points of intersection are also on the inverse circle. A circle (or line) is unchanged by inversion if and only if it is orthogonal to the reference circle at the points of intersection.
Additional properties include: If a circle q passes through two distinct points A and A' which are inverses with respect to a circle k, then the circles k and q are orthogonal. If the circles k and q are orthogonal, then a straight line passing through the center O of k and intersecting q, does so at inverse points with respect to k. Given a triangle OAB in which O is the center of a circle k, and points A' and B' inverses of A and B with respect to k, then
The points of intersection of two circles p and q orthogonal to a circle k, are inverses with respect to k. If M and M' are inverse points with respect to a circle k on two curves m and m', also inverses with respect to k, then the tangents to m and m' at the points M and M' are either perpendicular to the straight line MM' or form with this line an isosceles triangle with base MM'. Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.
Examples in two dimensions
Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center Inversion of a circle is another circle; or it is a line if the original circle contains the center Inversion of a parabola is a cardioid Inversion of hyperbola is a lemniscate of Bernoulli
Application For a circle not passing through the center of inversion, the center of the circle being inverted and the center of its image under inversion are collinear with the center of the reference circle. This fact can be used to prove that the Euler line of the intouch triangle of a triangle coincides with its OI line. The proof roughly goes as below:
Invert with respect to the incircle of triangle ABC. The medial triangle of the intouch triangle is inverted into triangle ABC, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ABC are collinear.
Any two non-intersecting circles may be inverted into concentric circles. Then the inversive distance (usually denoted δ) is defined as the natural logarithm of the ratio of the radii of the two concentric circles.
In addition, any two non-intersecting circles may be inverted into congruent circles, using circle of inversion centered at a point on the circle of antisimilitude.
The Peaucellier–Lipkin linkage is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion.
Pole and polar
If point R is the inverse of point P then the lines perpendicular to the line PR through one of the points is the polar of the other point (the pole).
Poles and polars have several useful properties:
If a point P lies on a line l, then the pole L of the line l lies on the polar p of point P. If a point P moves along a line l, its polar p rotates about the pole L of the line l. If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points. If a point lies on the circle, its polar is the tangent through this point. If a point P lies on its own polar line, then P is on the circle. Each line has exactly one pole.
Other circle inversions There are special inversions. They have special centers and radiuses, so the image after it, is convinient and helps to solve a problem.
inversion Here we consider triangle a ABC, where |AC| = b and |AB| = c
Most of the times in articles it is called inversion, although it is actually a combination of inversion in point A with radius, and reflection across A-angle bisector.
Its main properties are:
Points B and C "swap" places Circle (ABC) becames line BC and vice versa Incircle bacames A-mixtilinear excircle A-excirlce becames A-mixtilinear incircle swaps with swaps with Line l through A becames line k through A, and they are isogonals w.r.t. angle . This implies that atlitude AA' becames AO, or that median swaps with symmedian.
Those facts and many other can be proved and produced from properties of inversion that were disscussed in article before.
Inversion in Miquel point of quadrilateral This is also very useful inversion, that can be used for solving geometry problems.
Consider quadrilateral ABCD and let lines AB CD intersect at E and let line AD BC intersect at F. Let M be Miqeul point of this quadrilateral. Then combination of inversion at M with radius and reflection across angle bissector of angle (which is same line as angle bissectors of angles and ) will be called Miquel inversion.
Its main properties are:
Points A C swaps Points B D swaps Points E F swaps Circle (ABF) becames line DCE
Same as before, other useful facts can be produced from properties of inversion and deeper investigation on the subject.
In three dimensions
Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point P in 3D with respect to a reference sphere centered at a point O with radius R is a point P ' on the ray with direction OP such that . As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center O of the reference sphere, then it inverts to a plane. Any plane passing through O, inverts to a sphere touching at O. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through O it inverts into a line. This reduces to the 2D case when the secant plane passes through O, but is a true 3D phenomenon if the secant plane does not pass through O.
Examples in three dimensions
Sphere The simplest surface (besides a plane) is the sphere. The first picture shows a non trivial inversion (the center of the sphere is not the center of inversion) of a sphere together with two orthogonal intersecting pencils of circles.
Cylinder, cone, torus The inversion of a cylinder, cone, or torus results in a Dupin cyclide.
Spheroid A spheroid is a surface of revolution and contains a pencil of circles which is mapped onto a pencil of circles (see picture). The inverse image of a spheroid is a surface of degree 4.
Hyperboloid of one sheet A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles. The picture shows one such line (blue) and its inversion.
Stereographic projection as the inversion of a sphere
A stereographic projection usually projects a sphere from a point (north pole) of the sphere onto the tangent plane at the opposite point (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane. If the sphere (to be projected) has the equation (alternately written ; center , radius , green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point . The lines through the center of inversion (point ) are mapped onto themselves. They are the projection lines of the stereographic projection.
6-sphere coordinates The 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the Cartesian coordinates.
Axiomatics and generalization