Inversion in a sphere
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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.[1] It is recommended that readers should consult Course 42, SIGGRAPH 2001[2]
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 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)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Inver_in_a_sphere%2C_Fig_i.pdf/page1-450px-Inver_in_a_sphere%2C_Fig_i.pdf.jpg)
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)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Inverin_a_sphere%2C_Fig_2.pdf/page1-450px-Inverin_a_sphere%2C_Fig_2.pdf.jpg)
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)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Inver_in_a_sphere%2C_Fig_3h.pdf/page1-450px-Inver_in_a_sphere%2C_Fig_3h.pdf.jpg)
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)
- Let the foot of the perpendicular from O to the plane be P with inverse P'.
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'.
- if O lies on sphere{A, a},
- 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).
- Let S, T be the intersections of OA and sphere{A, a}, with S', T' their inverses.
(See 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)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Inver_in_a_sphere%2C_Fig_5.pdf/page1-450px-Inver_in_a_sphere%2C_Fig_5.pdf.jpg)
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.
- in the general case, where O is not on ψ,the plane 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.
- 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
- A line through the centre of inversion is self-inverse.
- Generally, the inverse of a line is a circle through the centre of inversion.
- The inverse of a circle through the centre of inversion is a line.
- Generally the inverse of a circle is a circle.
- A plane through the centre of inversion is self-inverse.
- Generally, the inverse of a plane is a sphere through the centre of inversion.
- The inverse of a sphere through the centre of inversion is a plane.
- Generally the inverse of a sphere is a sphere.