Inversive geometry

From Academic Kids

In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. By circle, we mean both circles and straight lines. This is an angle-preserving (or "conformal") geometry. This is also the same as conformal geometry with reflections (i.e. inversions). For two dimensions, conformal geometry with reflections aka inversive geometry is simply the Riemann sphere with antimeromorphic transformations.

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. Note that 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.

Inversive geometry/hyperbolic geometry duality

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).

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