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geometry.euclidean.sphere

Spheres #

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

Main theorems #

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.

theorem inner_product_geometry.mul_norm_eq_abs_sub_sq_norm {V : Type u_1} [inner_product_space V] {x y z : V} (h₁ : ∃ (k : ), k 1 x + y = k (x - y)) (h₂ : z - y = z + y) :
x - y * x + y = abs (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.

theorem euclidean_geometry.mul_dist_eq_abs_sub_sq_dist {V : Type u_1} [inner_product_space V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b p q : P} (hp : ∃ (k : ), k 1 b -ᵥ p = k (a -ᵥ p)) (hq : dist a q = dist b q) :
(dist a p) * dist b p = abs (dist b q ^ 2 - 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).

theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical {V : Type u_1} [inner_product_space 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 : ), k₁ 1 b -ᵥ p = k₁ (a -ᵥ p)) (hcpd : ∃ (k₂ : ), k₂ 1 d -ᵥ p = k₂ (c -ᵥ p)) :
(dist a p) * dist b p = (dist c p) * dist d p

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

theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi {V : Type u_1} [inner_product_space 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 : a p b = π) (hcpd : c p d = π) :
(dist a p) * dist b p = (dist c p) * dist d p

Intersecting Chords Theorem.

theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero {V : Type u_1} [inner_product_space 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 : a p b = 0) (hcpd : c p d = 0) :
(dist a p) * dist b p = (dist c p) * dist d p

Intersecting Secants Theorem.