Inversion in a sphere

The results of Inversion in a circle are given in Wikipedia, but the results for inversion in a sphere are not proved there. This article is attempting to fill a gap.

This article is concerned with objects in three dimensions (3-D). As Prof Hinton found in 1980, very few people can, without training, visualise in 3-D. It is recommended that readers should consult Course 42, SIGGRAPH 2001

Generally

The inverse of a line is a circle through the centre of the reference sphere, and vice versa. The inverse of a plane is a sphere through the centre of the reference sphere, and vice versa. Otherwise the inverse of a circle is a circle; the inverse of a sphere is a sphere.

Proof

Let the reference sphere be Σ, with centre O and radius r denoted by {O, r}. All inverses, in this paper, are in the sphere Σ.

Definition

• Let P be a point at distance n > 0 from O.
• If P' be a point on OP, on the same direction as OP, such that OP.OP' = r², then P, and P' are inverse points
• If n > r, then OP' < r, so P' lies inside Σ, and vice-versa.
  
• Points on the surface of Σ are the only self-inverse points.

Construction

• As in inversion in a circle, the usual construction, for a point, P, outside the sphere, is to take a plane through OP, draw tangents, in the plane, from P to Σ, meeting it at S, T.
  
• The intersection of the chord ST with OP gives P'. (Triangles OPS, OSP' are similar.)
• For a point P inside Σ, take a plane through OP, draw a chord of the sphere in that plane, normal to OP at P, meeting Σ, at S, T.
  
• Draw tangents, in the plane, to meet at P', the inverse of P.
• In either case, The right angled triangles, OPT, OTP' are similar, so OP/OT = OT/OP'

(See fig 1)

Fig 1

Similar triangles

• Given two points A, B with inverses A', B'; OA'.OA = r², OB'.OB = r².
• So OA'/OB' = OB/OA.
• Since ∠AOB is ∠B'OA', the triangles AOB, B'OA' are similar.
• So ∠OAB = ∠OB'A', ∠OBA = ∠OA'B'.
• Similarly, since ∠AOB' is ∠BOA', AOB', BOA' are similar.
• So ∠OAB' = ∠OBA', ∠OB'A = ∠OA'B.

(See fig 2)

Fig 2

Inverse of a line

• If the line intersects Σ, then only the two points of intersection are self-inverse.
• If O lies on the line, then the line is self inverse;
• Else,

• Let P be the foot of the perpendicular from O to the line, with inverse P', and let X be any point on the line, with inverse X',
• By 'similar triangles', ∠OX'P' = ∠OPX = π/2.
• So X' lies on a circle through O, with OP' as diameter. (Angle in a semicircle is a right angle)
  

(See fig 3)

Fig 3

Note 4: Generally, the inverse of a line is a circle.

Inverse of a plane

• If the plane intersects Σ, then each point of the circle of intersection is self-inverse.
• If O lies on the plane, the inverse is the plane;
• Else:

• Let the foot of the perpendicular from O to the plane be P with inverse P'.
  
• Let X be any point on the plane with inverse X'.
  
• By 'similar triangles', ∠OX'P' = ∠OPX = π/2.
  
• X' lies on a sphere with diameter OP'.(angle in a semicircle is a rightangle)
  

Note 5: Generally, the inverse of a plane is a sphere.

Inverse of a Sphere

• Let the sphere be {A, a}, i.e. centre A and radius a > 0.
  
• If sphere{A, a} intersects Σ, the only self-inverse points are on the circle of intersection.
  
• If A is at O then the inverse of sphere{A, a} is a concentric sphere with radius r²/a;
  

(Trivially, if a = r, then every point on {A, a} is self-inverse.)

• Else
  

• if O lies on sphere{A, a},
  
• Then let OP be a diameter of sphere{A, a}, with P' the inverse of P.
  
• Let X be any point on sphere{A, a}, with X' as inverse.
  
• Then by 2.3, ∠OP'X' = ∠OXP = π/2.
  
• So X' lies on a plane through P' normal to OP'.
  

• Else,
  

• Let S, T be the intersections of OA and sphere{A, a}, with S', T' their inverses.
  
• ST is a diameter of {A, a}.
  
• Let OT = t > 0, OS = s, with t > |s| > 0.
  
• By 'similar triangles', for any point X on sphere{A, a}, with inverse X', ∠OXT = ∠OT'X', ∠OXS = ∠OS'X'.
  
• If T, S lie on the same side of O.
  
• Then t > s > 0.
  
• ∠OXT = ∠OXS = π/2.
  
• T'X'S' = ∠OT'X' -∠OS'X' = ∠OXT -∠OXS = π/2.
  
• So X' lies on a sphere, with T'S' as diameter (angle in a semicircle is a rightangle).
  

(See fig 4)

File:Inver in a sphere, Fig 4.pdf

Fig 4

• If T, S lie on opposite sides of O, then t > 0, and s < 0, with t > -s.
• ∠OXT + ∠OXS = π/2.
• π/2 = ∠OXT + ∠OXS = ∠OT'X' + ∠OS'X' = π - T'X'S'.
• So ∠T'X'S' = π/2, and X' lies on a sphere, with T'S' as diameter (angle in a semicircle is a rightangle).

(See fig 5)

Fig 5

Note 6: Generally the inverse of a sphere is a sphere
(The only exception is when the centre of the reference sphere lies on the sphere.)

Inverse of a circle

• Let the circle be c, with centre C and radius a, lying on a plane ψ .
• Let S, T be the nearest and furthest points of c, from O, (i.e. OT > OS), with T', S' their inverses,
  
• If C is at O then the inverse of c is a concentric circle with radius r²/a;
• Else
  

• if O lies on c,
• Then let OP be a diameter of c, with P' the inverse of P.
• Let X be any point of the circle, with inverse X'.
• By 'similar triangles', ∠OP'X' = ∠OXP = π/2.
• The inverse of points of the circle lie on a line in the plane of c, normal to OP';

• Else
  

• If O lies in the plane of c, then c is a great circle of sphere {C, a}, in a plane through O, S, T, so arguments that applied to inverse of a sphere also apply to the inverse of circle c, with similar results to all those of Section 6.

(Cf Figs 3, 4, 5)

• Else,
  

• in the general case, where O is not on ψ,the plane of c;
  
• Let A, B be two points on a line through C, perpendicular to ψ.
• Let Λ, Ω, be two spheres through c, with centres A, B, neither through O.
• Let a spheres, Λ', Ω', be the inverses of Λ, Ω (see Note 6).
• Every point of the inverse of c lies on both Λ' and Ω'.
• The intersection of the spheres Λ', Ω' is a circle c', say, the inverse of c.

Comment: Generally, the cone of projection on which c, and c' lie, is oblate and right elliptical.

• If c lies on sphere Σ, then every point of c is self-inverse;
• Else,
  

• If c intersects the sphere, the only self-inverse points are those two intersections.
  

Note 7: Generally the inverse of a circle is a circle (The only exception is when the centre of the reference sphere lies on the circle.)

Results of Inversion in a Sphere

1. A line through the centre of inversion is self-inverse.
   
2. Generally, the inverse of a line is a circle through the centre of inversion.
   
3. The inverse of a circle through the centre of inversion is a line.
   
4. Generally the inverse of a circle is a circle.
   
5. A plane through the centre of inversion is self-inverse.
   
6. Generally, the inverse of a plane is a sphere through the centre of inversion.
   
7. The inverse of a sphere through the centre of inversion is a plane.
   
8. Generally the inverse of a sphere is a sphere.