, inversive geometry
is the study of a type of transformations
of the Euclidean plane
, called inversions
. These transformations preserve angles and map
into generalized circles, where a generalized circle
means either a circle or a line
(a circle with infinite radius
). Many difficult problems in geometry become much more tractable when an inversion is applied.
The concept of an inversion can be generalized to higher dimensional spaces.
Inverse of a point
In the plane, the inverse of a point P with 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,
This circle with respect to which inversion is performed will be called the reference circle.
The transformation of the plane which takes each point P to its inverse P' is called the inversion relative to the given point. Note that this inversion takes P' back to P, so the transformation obtained applying the same inversion twice is simply the identity transformation.
It follows from the definition 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 the point at infinity, which is transformed back to the center of the circle. In summary, the closer a point is to the center, the further away its transformation is, and vise-versa. This inversive relationship between points P and P' is the reasoning for this transformation's name.
One may invert a set of points with respect to a circle by inverting each of the points which make it up. The following properties are what make 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.
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.
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.
Inversions in three dimensions
Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point in 3D with respect to a reference sphere centered at a point with radius is a point such that and the points and are on the same ray going from .
As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center of the reference sphere, then it inverts to a plane. Any plane not passing through , inverts to a sphere touching at .
Stereographic projection is a special case of sphere inversion. Consider a sphere of radius 1 and a plane touching at the South Pole of . Then is the stereographic projection of with respect to the North Pole of . Consider a sphere of radius 2 centered at . The inversion with respect to transforms into its stereographic projection .
According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus
in 1831. Since then this mapping has become an avenue to higher mathematics. Through some steps of application of the circle inversion map, a student of transformation geometry
soon appreciates the significance of Felix Klein
’s Erlangen program
, an outgrowth of certain models of hyperbolic geometry
The combination of two inversions in concentric circles results in a similarity
, homothetic transformation
, or dilation
characterized by the ratio of the circle radii.
When a point in the plane is interpreted as a complex number z
), with complex conjugate z
* = x
, then the reciprocal
of z is z
. Consequently the algebraic form of the inversion in a unit circle is
- w = 1/z* = (1/z)* .
Reciprocation is key in transformation theory as a generator of the Mobius group
. The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Mobius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane). However, inversive geometry is the larger study since it includes the raw inversion in a circle (not yet made, with conjugation, into reciprocation). Inversive geometry also includes the conjugation
mapping. Neither conjugation nor inversion-in-a-circle are in the Mobius group since they are non-conformal (see below). Mobius group elements are analytic functions
of the whole plane and so are necessarily conformal
As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0 . In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line
, often called the Riemann sphere. It was
subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Arthur Cayley
, Felix Klein
, and Henri Poincaré
. Thus inversive geometry includes the ideas originated by Lobachevsky
in their plane geometry. Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program
, in 1872. Since then many mathematicians reserve the term geometry
for the group of mappings of some space characterized by a group invariant, a measure, like distance or angle.
For example, Smogorzhevsky develops several theorems of inversive geometry before beginning Lobachevskian geometry. First he defines two points as symmetric when they are images of each other under inversion in a circle. Then he writes of mutually orthogonal circles when the tangents to the circles, or their radii, at a point of intersection are perpendicular to each other.
- If a circle q passes through two distinct points A and A', symmetrical with respect to a circle k, then the circles k and q are mutually orthogonal.
- If the circles k and q are mutually orthogonal, then a straight line passing through the center O of k and intersecting q, does so at points symmetrical with respect to k.
- Given a triangle OAB in which O is the center of a circle k, and points A' and B' are symmetrical to A and B with respect to k, then
- A straight line no passing through the pole of inversion is transformed by inversion into a circle passing through the inversion pole.
- Inversion transforms a circle passing through the pole of inversion into a straight line not passing through the pole of inversion.
- Inversion transforms a circle not passing through the pole of inversion into a circle that likewise does not pass through the inversion pole.
- The points of intersection of two circles p and q orthogonal to a circle k, are symmetrical with respect to k.
- If M and M' are two points, symmetrical with respect to a circle k on two curves m and m', also symmetrical 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 angles unaltered.
Inversion in higher dimensions
In the spirit of generalization to higher dimensions, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversion in an n-sphere:
where r is the radius of the inversion.
In 2 dimensions, with r = 1, this is circle inversion with respect to the unit circle.
As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is simply a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.
A remarkable fact about higher-dimensional conformal maps is that they arise strictly from inversions in n-spheres or hyperplanes and Euclidean motions: see Liouville's theorem (conformal mappings).
Inversion of an algebraic curve
Anticonformal mapping property
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal
if it preserves oriented
angles) . Algebraically, a map is anticonformal if at every point the Jacobian
is a scalar times an orthogonal matrix
with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that if J
is the Jacobian, then
Computing the Jacobian in the case zi
, where ||x
+ ... + xn2
, with k
, and additionally det(J
) is negative; hence the inversive map is anticonformal.
In the complex plane, the most obvious circle inversion map (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking z to 1/z. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal.
Inversive geometry and hyperbolic geometry
The (n − 1)-sphere with equation
will have a positive radius so long as a12 + ... + an2 is greater than c, and on inversion gives the sphere
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
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 the
model, 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.