Geometry

Inversive Geometry

f(z) = r²/(z̄ − c̄) — inversion in a circle of radius r centered at c maps circles+lines to circles+lines

Inversion in a circle of center O and radius r is the map sending each point P (other than O) to the point P' on ray OP with OP · OP' = r². In complex notation, f(z) = r²/(z̄ − c̄) + c. Key invariant: the family of circles and lines is preserved (lines through O map to themselves, lines not through O map to circles through O, circles through O map to lines, and other circles map to circles). Inversion is conformal (preserves angles) and is the unique anti-holomorphic generator of the Möbius group when combined with a reflection. Tools: Apollonius problem (find a circle tangent to three given circles) — solvable via clever inversions. Used in classical geometry to simplify problems with circles, in Steiner chains, and in the proof of Ptolemy's theorem.

  • DefinitionOP · OP' = r²
  • PreservesCircles + lines as a family
  • ConformalPreserves angles, not lengths
  • Apollonius problem8 solutions generically
  • OriginsApollonius ~200 BCE; Steiner 1820s
  • Möbius groupGenerated by inversions + reflections

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Why inversive geometry matters

Inversion in a circle was invented by Apollonius of Perga around 200 BCE as a tool for solving tangency problems, then quietly waited two thousand years before Steiner, Möbius, and Liouville turned it into a foundational technique. Today it is the cleanest manual lever for handling configurations of circles and the algebraic engine behind the entire conformal group of the plane.

  • Classical problem-solving. Many problems involving circles tangent to other circles or lines simplify dramatically when one of the circles is inverted to a line. Choosing the inversion center on the most awkward circle is a standard tactic in Olympiad and competition geometry.
  • Möbius and conformal geometry. The full conformal group of ℂℙ¹ is generated by inversions in circles. Every Möbius transformation factors as the product of two inversions, making inversion the elementary building block of conformal mappings.
  • Steiner chains and packing. Steiner's porism on chains of mutually tangent circles, the Pappus chain, the Descartes circle theorem on four mutually tangent circles, and Apollonian gaskets all yield to inversion in the right place.
  • Computer graphics texture mapping. Stereographic projection — used to map a sphere to a plane for environment-mapping and panorama projection — is itself an inversion. Inversive coordinates also drive certain conformal UV unwrappings of meshes.
  • Hyperbolic geometry. The isometries of the Poincaré disc model are exactly the Möbius transformations preserving the unit disc, generated by inversions in circles orthogonal to the unit circle. Hyperbolic reflections are inversions in geodesic circles.
  • Ptolemy's theorem. The classical identity for a cyclic quadrilateral falls out of a one-line inversion argument that converts the four cyclic points to four collinear points, where the relation is automatic.
  • Robotic locomotion and electrical engineering. Smith charts in RF design are inversions of the impedance plane, and certain robot trajectory planners use inversive coordinates so circular obstacles transform predictably under tool changes.
  • Apollonian gasket fractals. Iterating inversion in the gaps between mutually tangent circles produces an Apollonian gasket — a fractal whose Hausdorff dimension (~1.3057) and curvature spectrum are open research topics, including a recent positive resolution of the Apollonian local-global conjecture (2023).
  • Astronomy and stereographic charts. Star charts and historic celestial globes use stereographic projection — an inversive map — because angles between constellations are preserved exactly even under heavy distortion of distance.

Common misconceptions

  • "Inversion preserves distance." It does not — inversion is conformal but not isometric. A short formula gives the new distance: |P'Q'| = r²|PQ|/(|OP|·|OQ|), which can stretch or compress arbitrarily.
  • "Inversion is well-defined at the center." The center O has no finite image; it is sent to the "point at infinity" on the Riemann sphere. To make the map a bijection of ℂ ∪ {∞}, you have to add this point as part of the domain.
  • "Inversion changes orientation." Inversion in a real circle reverses orientation (it is anti-holomorphic in complex coordinates), but composing two inversions restores orientation and yields a Möbius transformation.
  • "Inversion only acts on circles." It acts on every point in the plane (except the center). Circles and lines are special because the family of circles+lines is preserved as a set; arbitrary curves transform to other (often complicated) curves.
  • "It's a rare or specialized technique." Inversion is a competition-geometry staple and shows up in undergraduate complex analysis, Möbius geometry, and Riemann surface theory; it is far from a niche tool.
  • "The radius and center don't matter." Different inversion circles produce different maps. Conjugating one inversion by a translation changes the center, and rescaling changes the radius — picking the right pair is the whole art of using inversion as a problem-solving tool.

Frequently asked questions

What is the formula for inversion in a circle?

Inversion in a circle of center O and radius r maps a point P (other than O) to the point P' on the ray OP satisfying OP · OP' = r². In Cartesian coordinates with center at the origin, P = (x, y) maps to P' = (r²x/(x² + y²), r²y/(x² + y²)). In complex notation with center c, the map is f(z) = c + r²/(z̄ − c̄), where the conjugate makes it anti-holomorphic. The center O itself has no image — it gets sent to "infinity" on the conformal Riemann sphere.

Why does it map circles to circles?

The general equation for a circle or line in the plane has the form A(x² + y²) + Bx + Cy + D = 0, where A = 0 is a line and A ≠ 0 is a circle. Substituting the inversion formulas shows that the equation transforms into A'(x² + y²) + B'x + C'y + D' = 0 — same family, with new coefficients. So the union {circles ∪ lines} is preserved. Lines through O map to themselves, lines not through O map to circles through O, circles through O map to lines, and circles not through O map to circles.

What is the Apollonius problem and how does inversion solve it?

The Apollonius problem (~200 BCE) asks for a circle tangent to three given circles. Generically there are eight solutions, distinguished by whether the new circle is internally or externally tangent to each given circle. Inversion centered at any point on one of the three circles converts that circle into a line, reducing the problem to finding a circle tangent to a line and two other circles — and then to finding a circle tangent to two lines and one circle, which is solvable by elementary methods. Inverting back recovers the eight solutions.

How does it help prove Ptolemy's theorem?

Ptolemy's theorem states that for four points A, B, C, D on a circle in cyclic order, |AC|·|BD| = |AB|·|CD| + |AD|·|BC|. Invert the configuration about A — the circle through A, B, C, D becomes a line containing the images B', C', D', collinear in the same order. The collinearity gives |B'D'| = |B'C'| + |C'D'|. Substituting the inversion distance formula |X'Y'| = r²|XY|/(|AX|·|AY|) and clearing denominators recovers Ptolemy's identity exactly.

What is a Steiner chain?

Given two non-intersecting circles, one inside the other, a Steiner chain is a sequence of circles each tangent to its two neighbors in the chain and to both starting circles. Steiner's porism (1826) says that whether the chain closes after n steps depends only on the two starting circles, not on where you begin — so if it closes once, it closes from every starting position, and you can rotate the entire chain like a ring of beads. Inversion centered at a point chosen to send the two starting circles to concentric circles makes this trivial: the chain becomes a regular ring of equal circles between two concentric ones.

How is it related to Möbius transformations?

On the Riemann sphere ℂℙ¹, Möbius transformations z ↦ (az + b)/(cz + d) are the orientation-preserving conformal automorphisms. Inversions are orientation-reversing conformal automorphisms — they generate the full conformal group of the sphere together with reflections. Composing two inversions in different circles produces a Möbius transformation; conversely every Möbius transformation factors as the composition of two inversions. So inversive geometry and Möbius geometry are the same theory viewed from two angles.