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.

Inversion in a sphere is a powerful transformation. One simple example is in map projection.
The usual projection of the North or South Pole is inversion from the Earth to a plane. If instead of making a pole the centre, we chose a city, then Inversion could produce a map where all the shortest routes (great circles) for flying from that city would appear as straight lines.

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.

The results in this article are dependent on three simple ideas:

1. Similar triangles: A scale model is the same shape as the original, i.e. all angles are kept.

2. The angle in a semicircle is a right angle. i.e. For any point on a semicircle, the diagonal makes a right angle (90^o).

3. The angles of a triangle add up to 180^o, so an external angle equals the sum of the other two internal angles.

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 any 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

Inversion of a pair of points

• 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'.

(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 'Inversion of a pair of points', ∠OX'P' = ∠OPX = 90^o.
• 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 through the centre of reference.

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 'Inversion of a pair of points', ∠OX'P' = ∠OPX = 90^o.
• 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 through the centre of reference.

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 P be a point diametrically opposite O on sphere{A, a}, with P' the inverse of P.
• Let X be any point on sphere{A, a}, with X' as inverse.
• Then by 'Inversion of a pair of points' ∠OP'X' = ∠OXP = 90^o (angle in a semicircle).
• This is true for all points on sphere{A, a}.
• 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 X be any point on sphere{A, a}, with inverse X'.
• ∠OXT = ∠OT'X', and ∠OXS = ∠OS'X'. (inverse of a pair of points)

• If T, S lie on the same side of O.

• ∠T'X'S' = ∠OX'T' -∠OX'S'
• = ∠OTX -∠OSX (Inversion of a pair of points).
• = ∠TXS (external angle equals sum of internal angles)
• = 90^o (angle in a semicircle is a right angle)
• So X' lies on a semicircle, with T'S' as diameter.
• This is true for every point on sphere {A, a}.
• So X' lies on a sphere, with T'S' as diameter.

(See fig 4)

Fig 4

• If T, S lie on opposite sides of O:

• ∠OXT + ∠OXS = 90^o (angle in a semi-circle is a rightangle).
• ∠T'X'S' = ∠OX'T' + ∠OX'S'
• = ∠OTX + ∠OSX (inverse of a pair of points).
• = 180^o - ∠TXS (angles in a triangle sum to 180^o)
• So ∠T'X'S' = 90^o, and X' lies on a semicircle, with T'S' as diameter (angle in a semicircle is a rightangle).
• As before:
• This is true for every point on sphere {A, a}.
• So X' lies on a sphere, with T'S' as diameter.

(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 .
• If c intersects the sphere, the only self-inverse points are those two intersections.
• 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 'Inversion of a pair of points', ∠OP'X' = ∠OXP = 90^o.
• 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.

• If O lis on the line AB, the cone of projection is right circular,

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

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.