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). 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
O P ⋅ O P ′ = r 2 . {\displaystyle OP\cdot OP^{\prime }=r^{2}.}
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 circle c with center M going through P. (Not labeled. It's the blue circle) 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 from O (center of circle Ø) through P'. (Not labeled, it's the horizontal line) Draw line s through P' perpendicular to r. (Not labeled. It's the vertical line) 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. 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
∠ O A B = ∠ O B ′ A ′ and ∠ O B A = ∠ O A ′ B ′ . {\displaystyle \angle OAB=\angle OB'A'\ {\text{ and }}\ \angle OBA=\angle OA'B'.}
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
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