Spheres #

This file proves basic geometrical results about distances and angles in spheres in real inner product spaces and Euclidean affine spaces.

Main theorems #

mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi: Intersecting Chords Theorem (Freek No. 55).
mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero: Intersecting Secants Theorem.
mul_dist_add_mul_dist_eq_mul_dist_of_cospherical: Ptolemy’s Theorem (Freek No. 95).

TODO: The current statement of Ptolemy’s theorem works around the lack of a "cyclic polygon" concept in mathlib, which is what the theorem statement would naturally use (or two such concepts, since both a strict version, where all vertices must be distinct, and a weak version, where consecutive vertices may be equal, would be useful; Ptolemy's theorem should then use the weak one).

An API needs to be built around that concept, which would include:

strict cyclic implies weak cyclic,
weak cyclic and consecutive points distinct implies strict cyclic,
weak/strict cyclic implies weak/strict cyclic for any subsequence,
any three points on a sphere are weakly or strictly cyclic according to whether they are distinct,
any number of points on a sphere intersected with a two-dimensional affine subspace are cyclic in some order,
a list of points is cyclic if and only if its reversal is,
a list of points is cyclic if and only if any cyclic permutation is, while other permutations are not when the points are distinct,
a point P where the diagonals of a cyclic polygon cross exists (and is unique) with weak/strict betweenness depending on weak/strict cyclicity,
four points on a sphere with such a point P are cyclic in the appropriate order, and so on.

Geometrical results on spheres in real inner product spaces #

This section develops some results on spheres in real inner product spaces, which are used to deduce corresponding results for Euclidean affine spaces.

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theorem inner_product_geometry.mul_norm_eq_abs_sub_sq_norm {V : Type u_1} [inner_product_space ℝ V] {x y z : V} (h₁ : ∃ (k : ℝ), k ≠ 1 ∧ x + y = k • (x - y)) (h₂ : ∥z - y∥ = ∥z + y∥) :

∥x - y∥ * ∥x + y∥ = |∥z + y∥ ^ 2 - ∥z - x∥ ^ 2|

Geometrical results on spheres in Euclidean affine spaces #

This section develops some results on spheres in Euclidean affine spaces.

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theorem euclidean_geometry.mul_dist_eq_abs_sub_sq_dist {V : Type u_1} [inner_product_space ℝ V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b p q : P} (hp : ∃ (k : ℝ), k ≠ 1 ∧ b -ᵥ p = k • (a -ᵥ p)) (hq : has_dist.dist a q = has_dist.dist b q) :

has_dist.dist a p * has_dist.dist b p = |has_dist.dist b q ^ 2 - has_dist.dist p q ^ 2|

If P is a point on the line AB and Q is equidistant from A and B, then AP * BP = abs (BQ ^ 2 - PQ ^ 2).

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theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical {V : Type u_1} [inner_product_space ℝ V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b c d p : P} (h : euclidean_geometry.cospherical {a, b, c, d}) (hapb : ∃ (k₁ : ℝ), k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)) (hcpd : ∃ (k₂ : ℝ), k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)) :

has_dist.dist a p * has_dist.dist b p = has_dist.dist c p * has_dist.dist d p

If A, B, C, D are cospherical and P is on both lines AB and CD, then AP * BP = CP * DP.

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theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi {V : Type u_1} [inner_product_space ℝ V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b c d p : P} (h : euclidean_geometry.cospherical {a, b, c, d}) (hapb : euclidean_geometry.angle a p b = real.pi) (hcpd : euclidean_geometry.angle c p d = real.pi) :

has_dist.dist a p * has_dist.dist b p = has_dist.dist c p * has_dist.dist d p

Intersecting Chords Theorem.

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theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero {V : Type u_1} [inner_product_space ℝ V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b c d p : P} (h : euclidean_geometry.cospherical {a, b, c, d}) (hab : a ≠ b) (hcd : c ≠ d) (hapb : euclidean_geometry.angle a p b = 0) (hcpd : euclidean_geometry.angle c p d = 0) :

has_dist.dist a p * has_dist.dist b p = has_dist.dist c p * has_dist.dist d p

Intersecting Secants Theorem.

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theorem euclidean_geometry.mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {V : Type u_1} [inner_product_space ℝ V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b c d p : P} (h : euclidean_geometry.cospherical {a, b, c, d}) (hapc : euclidean_geometry.angle a p c = real.pi) (hbpd : euclidean_geometry.angle b p d = real.pi) :

has_dist.dist a b * has_dist.dist c d + has_dist.dist b c * has_dist.dist d a = has_dist.dist a c * has_dist.dist b d

Ptolemy’s Theorem.