Spheres

This file defines and proves basic results about spheres and cospherical sets of points in Euclidean affine spaces.

Main definitions

• EuclideanGeometry.Sphere bundles a center and a radius.

• EuclideanGeometry.Cospherical is the property of a set of points being equidistant from some point.

• EuclideanGeometry.Concyclic is the property of a set of points being cospherical and coplanar.

theorem EuclideanGeometry.Sphere.ext {P : Type u_2} : ∀ {inst : MetricSpace P} (x y : EuclideanGeometry.Sphere P), x.center = y.center → x.radius = y.radius → x = y

theorem EuclideanGeometry.Sphere.ext_iff {P : Type u_2} : ∀ {inst : MetricSpace P} (x y : EuclideanGeometry.Sphere P), x = y ↔ x.center = y.center ∧ x.radius = y.radius

structure EuclideanGeometry.Sphere (P : Type u_2) [MetricSpace P] : Type u_2

A Sphere P bundles a center and radius. This definition does not require the radius to be positive; that should be given as a hypothesis to lemmas that require it.

• center : P
• radius : ℝ

instance EuclideanGeometry.instNonemptySphere {P : Type u_2} [MetricSpace P] [Nonempty P] : Nonempty (EuclideanGeometry.Sphere P)

Equations

• (_ : Nonempty (EuclideanGeometry.Sphere P)) = (_ : Nonempty (EuclideanGeometry.Sphere P))

instance EuclideanGeometry.instCoeSphereSet {P : Type u_2} [MetricSpace P] : Coe (EuclideanGeometry.Sphere P) (Set P)

Equations

• EuclideanGeometry.instCoeSphereSet = { coe := fun (s : EuclideanGeometry.Sphere P) => Metric.sphere s.center s.radius }

instance EuclideanGeometry.instMembershipSphere {P : Type u_2} [MetricSpace P] : Membership P (EuclideanGeometry.Sphere P)

Equations

• EuclideanGeometry.instMembershipSphere = { mem := fun (p : P) (s : EuclideanGeometry.Sphere P) => p ∈ Metric.sphere s.center s.radius }

theorem EuclideanGeometry.Sphere.mk_center {P : Type u_2} [MetricSpace P] (c : P) (r : ℝ) : { center := c, radius := r }.center = c

theorem EuclideanGeometry.Sphere.mk_radius {P : Type u_2} [MetricSpace P] (c : P) (r : ℝ) : { center := c, radius := r }.radius = r

@[simp]

theorem EuclideanGeometry.Sphere.mk_center_radius {P : Type u_2} [MetricSpace P] (s : EuclideanGeometry.Sphere P) : { center := s.center, radius := s.radius } = s

@[simp]

theorem EuclideanGeometry.Sphere.coe_mk {P : Type u_2} [MetricSpace P] (c : P) (r : ℝ) : Metric.sphere { center := c, radius := r }.center { center := c, radius := r }.radius = Metric.sphere c r

theorem EuclideanGeometry.Sphere.mem_coe {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P} : p ∈ Metric.sphere s.center s.radius ↔ p ∈ s

@[simp]

theorem EuclideanGeometry.Sphere.mem_coe' {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P} : dist p s.center = s.radius ↔ p ∈ s

theorem EuclideanGeometry.mem_sphere {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P} : p ∈ s ↔ dist p s.center = s.radius

theorem EuclideanGeometry.mem_sphere' {P : Type u_2} [MetricSpace P] {p : P} {s : EuclideanGeometry.Sphere P} : p ∈ s ↔ dist s.center p = s.radius

theorem EuclideanGeometry.subset_sphere {P : Type u_2} [MetricSpace P] {ps : Set P} {s : EuclideanGeometry.Sphere P} : ps ⊆ Metric.sphere s.center s.radius ↔ ∀ p ∈ ps, p ∈ s

theorem EuclideanGeometry.dist_of_mem_subset_sphere {P : Type u_2} [MetricSpace P] {p : P} {ps : Set P} {s : EuclideanGeometry.Sphere P} (hp : p ∈ ps) (hps : ps ⊆ Metric.sphere s.center s.radius) : dist p s.center = s.radius

theorem EuclideanGeometry.dist_of_mem_subset_mk_sphere {P : Type u_2} [MetricSpace P] {p : P} {c : P} {ps : Set P} {r : ℝ} (hp : p ∈ ps) (hps : ps ⊆ Metric.sphere { center := c, radius := r }.center { center := c, radius := r }.radius) : dist p c = r

theorem EuclideanGeometry.Sphere.ne_iff {P : Type u_2} [MetricSpace P] {s₁ : EuclideanGeometry.Sphere P} {s₂ : EuclideanGeometry.Sphere P} : s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius

theorem EuclideanGeometry.Sphere.center_eq_iff_eq_of_mem {P : Type u_2} [MetricSpace P] {s₁ : EuclideanGeometry.Sphere P} {s₂ : EuclideanGeometry.Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂) : s₁.center = s₂.center ↔ s₁ = s₂

theorem EuclideanGeometry.Sphere.center_ne_iff_ne_of_mem {P : Type u_2} [MetricSpace P] {s₁ : EuclideanGeometry.Sphere P} {s₂ : EuclideanGeometry.Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂) : s₁.center ≠ s₂.center ↔ s₁ ≠ s₂

theorem EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere {P : Type u_2} [MetricSpace P] {p₁ : P} {p₂ : P} {s : EuclideanGeometry.Sphere P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : dist p₁ s.center = dist p₂ s.center

theorem EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere' {P : Type u_2} [MetricSpace P] {p₁ : P} {p₂ : P} {s : EuclideanGeometry.Sphere P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : dist s.center p₁ = dist s.center p₂

def EuclideanGeometry.Cospherical {P : Type u_2} [MetricSpace P] (ps : Set P) : Prop

A set of points is cospherical if they are equidistant from some point. In two dimensions, this is the same thing as being concyclic.

Equations

• EuclideanGeometry.Cospherical ps = ∃ (center : P) (radius : ℝ), ∀ p ∈ ps, dist p center = radius

theorem EuclideanGeometry.cospherical_def {P : Type u_2} [MetricSpace P] (ps : Set P) : EuclideanGeometry.Cospherical ps ↔ ∃ (center : P) (radius : ℝ), ∀ p ∈ ps, dist p center = radius

The definition of Cospherical.

theorem EuclideanGeometry.cospherical_iff_exists_sphere {P : Type u_2} [MetricSpace P] {ps : Set P} : EuclideanGeometry.Cospherical ps ↔ ∃ (s : EuclideanGeometry.Sphere P), ps ⊆ Metric.sphere s.center s.radius

A set of points is cospherical if and only if they lie in some sphere.

theorem EuclideanGeometry.Sphere.cospherical {P : Type u_2} [MetricSpace P] (s : EuclideanGeometry.Sphere P) : EuclideanGeometry.Cospherical (Metric.sphere s.center s.radius)

The set of points in a sphere is cospherical.

theorem EuclideanGeometry.Cospherical.subset {P : Type u_2} [MetricSpace P] {ps₁ : Set P} {ps₂ : Set P} (hs : ps₁ ⊆ ps₂) (hc : EuclideanGeometry.Cospherical ps₂) : EuclideanGeometry.Cospherical ps₁

A subset of a cospherical set is cospherical.

theorem EuclideanGeometry.cospherical_empty {P : Type u_2} [MetricSpace P] [Nonempty P] : EuclideanGeometry.Cospherical ∅

The empty set is cospherical.

theorem EuclideanGeometry.cospherical_singleton {P : Type u_2} [MetricSpace P] (p : P) : EuclideanGeometry.Cospherical {p}

A single point is cospherical.

theorem EuclideanGeometry.cospherical_pair {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ : P) (p₂ : P) : EuclideanGeometry.Cospherical {p₁, p₂}

Two points are cospherical.

structure EuclideanGeometry.Concyclic {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (ps : Set P) : Prop

A set of points is concyclic if it is cospherical and coplanar. (Most results are stated directly in terms of Cospherical instead of using Concyclic.)

• Cospherical : EuclideanGeometry.Cospherical ps
• Coplanar : Coplanar ℝ ps

theorem EuclideanGeometry.Concyclic.subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {ps₁ : Set P} {ps₂ : Set P} (hs : ps₁ ⊆ ps₂) (h : EuclideanGeometry.Concyclic ps₂) : EuclideanGeometry.Concyclic ps₁

A subset of a concyclic set is concyclic.

theorem EuclideanGeometry.concyclic_empty {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] : EuclideanGeometry.Concyclic ∅

The empty set is concyclic.

theorem EuclideanGeometry.concyclic_singleton {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p : P) : EuclideanGeometry.Concyclic {p}

A single point is concyclic.

theorem EuclideanGeometry.concyclic_pair {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ : P) (p₂ : P) : EuclideanGeometry.Concyclic {p₁, p₂}

Two points are concyclic.

theorem EuclideanGeometry.Cospherical.affineIndependent {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : Set P} (hs : EuclideanGeometry.Cospherical s) {p : Fin 3 → P} (hps : Set.range p ⊆ s) (hpi : Function.Injective p) : AffineIndependent ℝ p

Any three points in a cospherical set are affinely independent.

theorem EuclideanGeometry.Cospherical.affineIndependent_of_mem_of_ne {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : Set P} (hs : EuclideanGeometry.Cospherical s) {p₁ : P} {p₂ : P} {p₃ : P} (h₁ : p₁ ∈ s) (h₂ : p₂ ∈ s) (h₃ : p₃ ∈ s) (h₁₂ : p₁ ≠ p₂) (h₁₃ : p₁ ≠ p₃) (h₂₃ : p₂ ≠ p₃) : AffineIndependent ℝ ![p₁, p₂, p₃]

Any three points in a cospherical set are affinely independent.

theorem EuclideanGeometry.Cospherical.affineIndependent_of_ne {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ : P} {p₂ : P} {p₃ : P} (hs : EuclideanGeometry.Cospherical {p₁, p₂, p₃}) (h₁₂ : p₁ ≠ p₂) (h₁₃ : p₁ ≠ p₃) (h₂₃ : p₂ ≠ p₃) : AffineIndependent ℝ ![p₁, p₂, p₃]

The three points of a cospherical set are affinely independent.

theorem EuclideanGeometry.inner_vsub_vsub_of_mem_sphere_of_mem_sphere {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ : P} {p₂ : P} {s₁ : EuclideanGeometry.Sphere P} {s₂ : EuclideanGeometry.Sphere P} (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) : ⟪s₂.center -ᵥ s₁.center, p₂ -ᵥ p₁⟫_ℝ = 0

Suppose that p₁ and p₂ lie in spheres s₁ and s₂. Then the vector between the centers of those spheres is orthogonal to that between p₁ and p₂; this is a version of inner_vsub_vsub_of_dist_eq_of_dist_eq for bundled spheres. (In two dimensions, this says that the diagonals of a kite are orthogonal.)

theorem EuclideanGeometry.eq_of_mem_sphere_of_mem_sphere_of_mem_of_finrank_eq_two {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace ℝ P} [FiniteDimensional ℝ ↥(AffineSubspace.direction s)] (hd : FiniteDimensional.finrank ℝ ↥(AffineSubspace.direction s) = 2) {s₁ : EuclideanGeometry.Sphere P} {s₂ : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} {p : P} (hs₁ : s₁.center ∈ s) (hs₂ : s₂.center ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) (hs : s₁ ≠ s₂) (hp : p₁ ≠ p₂) (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hps₁ : p ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) (hps₂ : p ∈ s₂) : p = p₁ ∨ p = p₂

Two spheres intersect in at most two points in a two-dimensional subspace containing their centers; this is a version of eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two for bundled spheres.

theorem EuclideanGeometry.eq_of_mem_sphere_of_mem_sphere_of_finrank_eq_two {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [FiniteDimensional ℝ V] (hd : FiniteDimensional.finrank ℝ V = 2) {s₁ : EuclideanGeometry.Sphere P} {s₂ : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} {p : P} (hs : s₁ ≠ s₂) (hp : p₁ ≠ p₂) (hp₁s₁ : p₁ ∈ s₁) (hp₂s₁ : p₂ ∈ s₁) (hps₁ : p ∈ s₁) (hp₁s₂ : p₁ ∈ s₂) (hp₂s₂ : p₂ ∈ s₂) (hps₂ : p ∈ s₂) : p = p₁ ∨ p = p₂

Two spheres intersect in at most two points in two-dimensional space; this is a version of eq_of_dist_eq_of_dist_eq_of_finrank_eq_two for bundled spheres.

theorem EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center ≤ s.radius) : 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫_ℝ ∨ p₁ = p₂

Given a point on a sphere and a point not outside it, the inner product between the difference of those points and the radius vector is positive unless the points are equal.

theorem EuclideanGeometry.inner_nonneg_of_dist_le_radius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center ≤ s.radius) : 0 ≤ ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫_ℝ

Given a point on a sphere and a point not outside it, the inner product between the difference of those points and the radius vector is nonnegative.

theorem EuclideanGeometry.inner_pos_of_dist_lt_radius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center < s.radius) : 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫_ℝ

Given a point on a sphere and a point inside it, the inner product between the difference of those points and the radius vector is positive.

theorem EuclideanGeometry.wbtw_of_collinear_of_dist_center_le_radius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} {p₃ : P} (h : Collinear ℝ {p₁, p₂, p₃}) (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center ≤ s.radius) (hp₃ : p₃ ∈ s) (hp₁p₃ : p₁ ≠ p₃) : Wbtw ℝ p₁ p₂ p₃

Given three collinear points, two on a sphere and one not outside it, the one not outside it is weakly between the other two points.

theorem EuclideanGeometry.sbtw_of_collinear_of_dist_center_lt_radius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ : P} {p₂ : P} {p₃ : P} (h : Collinear ℝ {p₁, p₂, p₃}) (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center < s.radius) (hp₃ : p₃ ∈ s) (hp₁p₃ : p₁ ≠ p₃) : Sbtw ℝ p₁ p₂ p₃

Given three collinear points, two on a sphere and one inside it, the one inside it is strictly between the other two points.