Inversion (geometry)
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In geometry, an inversion is a transformation that maps 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
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.
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 through 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 if and only if 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.
Inversions in three dimensions
Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point <math>P</math> in 3D with respect to a reference sphere centered at a point <math>O</math> with radius <math>R</math> is a point <math>P'</math> such that <math>OP\times OP'=R^2</math> and the points <math>P</math> and <math>P'</math> are on the same ray going from <math>O</math>.
As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center <math>O</math> of the reference sphere, then it inverts to a plane. Any plane not passing through <math>O</math>, inverts to a sphere touching at <math>O</math>.
Stereographic projection is a special case of sphere inversion. Indeed, consider a sphere <math>B</math> of radius 1 and a plane <math>P</math> touching <math>B</math> at the South Pole <math>S</math> of <math>B</math>. Then <math>P</math> is the stereographic projection of <math>B</math> in respect to the North Pole <math>N</math> of <math>B</math>. Consider a sphere <math>B_2</math> of radius 2 centered at <math>N</math>. The inversion in respect to <math>B_2</math> transforms <math>B</math> into its stereographic projection <math>P</math>.
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.
Inversion of an algebraic curve
We may invert a plane algebraic curve given by a single polynomial equation f(x, y) = 0 by setting
- <math>u = \frac{x}{x^2+y^2},\ v=\frac{y}{x^2+y^2}.</math>
Clearing denominators, we have the polynomial equations <math>ux^2+uy^2-x = 0, vx^2+vy^2-y=0</math>, and eliminating x and y from the system of three equations in four unknowns consisting of these two equations and f (for instance, by using resultants) we can readily find the equation of the curve inverted in the unit circle. Now <math>x=u/(u^2+v^2),\ y=v/(u^2+v^2)</math> and applying the transformation again leads back to the original curve.
For example, applying the above transformation to the lemniscate
- <math>(x^2 + y^2)^2 = a^2 (x^2 - y^2)</math>
gives us
- <math>a^2(u^2-v^2) = 1,</math>
the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero. If we apply it to the Fermat curve xn + yn = 1, where n is odd, we obtain
- <math>(u^2+v^2)^n = u^n+v^n.</math>
Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem.
Conformal mapping property
A transformation is conformal, or angle-preserving, if at every point the Jacobian is a scalar times an orthogonal matrix. This means that if J is the Jacobian, then JJT = kI. Computing the Jacobian in the case zi = xi/||x||2, where ||x||2 = x12 + ... + xn2 gives JJT = kI, with k = 1/||x||4; hence the inversive map is conformal.
Inversive geometry and hyperbolic geometry
The (n − 1)-sphere with equation
- <math>x_1^2 + \cdots + x_n^2 + 2a_1x + \cdots + 2a_nx + c = 0</math>
will have a positive radius so long as a12 + ... + an2 is greater than c, and on inversion gives the sphere
- <math>x_1^2 + \cdots + x_n^2 + 2\frac{a_1}{c}x + \cdots + 2\frac{a_n}{c}x + \frac{1}{c} = 0.</math>
Hence, it will be invariant under inversion if and only if c = 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the (n − 1)-spheres with equation
- <math>x_1^2 + \cdots + x_n^2 + 2a_1x + \cdots + 2a_nx + 1 = 0,</math>
which are invariant under inversion, orthogonal to the unit sphere, and have centers outside of the sphere. These together with the subspace hyperplanes separating hemispheres are the hypersurfaces of the Poincaré disc model of hyperbolic geometry.
Since inversion in the unit sphere leaves the spheres orthogonal to it invariant, the inversion maps the points inside the unit sphere to the outside and vice-versa. This is therefore true in general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice-versa; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere. These reflections generate the group of isometries of themodel, which tells us that the isometries are conformal. Hence, the angle between two curves in the model is the same as the angle between two curves in the hyperbolic space.
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