Inversive geometry
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In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles, where by a circle one may also mean a line (a circle with infinite radius).
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Circle inversion
Inverse of a point
Image:Inversion illustration1.png In the plane, the inverse of a point P in respect to a circle of center O and radius R is a point P' such that P and P' are on the same ray going from O, and OP times OP' equals the radius squared,
- <math>OP\times OP'=R^2.</math>
This circle in respect to which inversion is performed will be called the reference circle.
Image:Inversion illustration2.png Image:Inversion illustration3.png Image:InvertedCircle.png
One can check that the inverse of a point inside the reference circle is outside the reference circle and vice-versa. A point on the circle stays in the same place under inversion. The center of the circle gets transformed to infinity, and the infinity gets transformed to the circle center. This all can be summarized by saying that the closer a point is to the center, the further it goes when inverted, and the other way around, with the points on the circle staying where they are.
Properties
One may invert a set of points in respect to a circle by inverting each of the points which make it up. The following properties is what makes circle inversion important.
- A line not passing though the center of the reference circle is inverted into a circle passing through the center of the reference circle, and vice versa; whereas a line passing through the center of the reference circle is inverted into itself.
- A circle not passing through the center of the reference circle is inverted into a circle not passing through the center of the reference circle. The circle (or line) after inversion stays as before iff it is orthogonal to the reference circle at their points of intersection.
Application
Note that the center of a circle being inverted and the center of the circle as result of inversion are collinear with the center of the reference circle. This fact could be useful in proving 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.
In addition, two dimensional inversion can be extended to 3-dimensional by making use of a sphere instead.
The Erlangen program
In the spirit of the Erlangen program, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversions, which in coordinate form, basically are conjugate to
- <math>x_i\rightarrow \frac{r^2 x_i}{\sum_j x_j^2}</math>
where r is the radius of the inversion.
In 2 dimensions, with r = 1, this is circle inversion with respect to the unit circle. In the complex plane this corresponds to taking the reciprocal of the conjugate.
As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less than a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.
Duality of inversive geometry and hyperbolic geometry
This is usually referred to as the (Euclidean) conformal geometry/hyperbolic geometry duality. An (n − 1)-dimensional inversive geometry is dual to an n-dimensional hyperbolic geometry. There is now a one-to-one correspondence between (n − 2)-dimensional circles (or hyperspheres if you wish) and (n − 1)-dimensional hyperplanes in hyperbolic geometry. An inversion about this hypersphere corresponds to a reflection about the hyperplane. This can be seen in the Poincaré disc model and/or the Poincaré half-plane model where the (n − 2)-dimensional hypersphere is the boundary of the (n − 1)-dimensional hyperplane.
This is the Wick-rotated version of the AdS/CFT duality. In fact, since most calculations are performed in the Wick-rotated model, this is the duality which is really being used.
See also AdS/CFT.
Computer program
The following QBasic program can be used to interactively explore the nature of inversive transformations:
'PROGRAM INVGEOM.BAS pi = 4 * ATN(1) SCREEN 13 WINDOW (0, 0)-(300, 3 / 4 * 300) 'diatessaron newx = 143 newy = 98 newrad = 14 cx = 150 cy = 100 R = 30 main: CLS 'given circle FOR thet = 0 TO 2 * pi STEP .01 PSET (newx + newrad * COS(thet), newy - newrad * SIN(thet)) NEXT COLOR 2 'identity circle which defines inversive transformation FOR thet = 0 TO 2 * pi STEP .01 PSET (cx + R * COS(thet), cy - R * SIN(thet)) NEXT COLOR 15 'transformed version of given circle FOR thet = 0 TO 2 * pi STEP .01 x = newx + newrad * COS(thet) y = newy - newrad * SIN(thet) xdif = x - cx ydif = y - cy rho = SQR(xdif ^ 2 + ydif ^ 2) rhop = R ^ 2 / rho xdifp = xdif / rho * rhop 'xdifp = xdif / rho^2 * R^2 ydifp = ydif / rho * rhop xp = cx + xdifp yp = cy + ydifp PSET (xp, yp) NEXT waiting: k$ = INKEY$ IF k$ = "" THEN GOTO waiting 'move given circle one step to the left IF LCASE$(k$) = "q" THEN newx = newx - 1 'move given circle one step to the right IF LCASE$(k$) = "w" THEN newx = newx + 1 'move given circle one step up IF LCASE$(k$) = "a" THEN newy = newy - 1 'move given circle one step down IF LCASE$(k$) = "s" THEN newy = newy + 1 'decrease radius of given circle by one unit IF LCASE$(k$) = "z" THEN newrad = newrad - 1 'increase radius of given circle by one unit IF LCASE$(k$) = "x" THEN newrad = newrad + 1 'move identity circle one step to the left IF LCASE$(k$) = "e" THEN cx = cx - 1 'move identity circle one step to the right IF LCASE$(k$) = "r" THEN cx = cx + 1 'move identity circle one step up IF LCASE$(k$) = "d" THEN cy = cy - 1 'move identity circle one step down IF LCASE$(k$) = "f" THEN cy = cy + 1 'decrease the radius of the identity circle by one unit IF LCASE$(k$) = "c" THEN R = R - 1 'increase the radius of the identity circle by one unit IF LCASE$(k$) = "v" THEN R = R + 1 GOTO main
The inversive transformation is defined by an "identity circle" which is shown in green. A pair of white circles represent the "given circle" and the "transformed version of the given circle". While the program is running, the given circle can be moved with the keys: Q (left), W (right), A (up), D (down), Z (decrease radius), X (increase radius). The identity circle can be moved with the keys: E (left), R (right), D (up), F (down), C (decrease radius), V (increase radius).