bookshelf/Common/Real/Geometry/Basic.lean

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import Mathlib.Data.Real.Sqrt
import Common.Real.Basic
/-! # Common.Real.Geometry.Basic
A collection of useful definitions and theorems around geometry.
-/
namespace Real
/--
The undirected angle at `p₂` between the line segments to `p₁` and `p₃`. If
either of those points equals `p₂`, this is `π / 2`.
###### PORT
This should be replaced with the original Mathlib `geometry.euclidean.angle`
definition once ported.
-/
axiom angle (p₁ p₂ p₃ : ℝ²) :
noncomputable def port_geometry_euclidean_angle (p₁ p₂ p₃ : ℝ²) :=
if p₁ = p₂ p₂ = p₃ then π / 2 else angle p₁ p₂ p₃
notation "∠" => port_geometry_euclidean_angle
/--
Determine the distance between two points in `ℝ²`.
-/
noncomputable def dist (x y : ℝ²) :=
Real.sqrt ((abs (y.1 - x.1)) ^ 2 + (abs (y.2 - x.2)) ^ 2)
/--
Two sets `S` and `T` are `similar` **iff** there exists a one-to-one
correspondence between `S` and `T` such that the distance between any two points
`P, Q ∈ S` and corresponding points `P', Q' ∈ T` differ by some constant `α`. In
other words, `α|PQ| = |P'Q'|`.
-/
def similar (S T : Set ℝ²) : Prop :=
∃ f : ℝ² → ℝ², Function.Bijective f ∧
∃ s : , ∀ x y : ℝ², x ∈ S ∧ y ∈ T →
s * dist x y = dist (f x) (f y)
/--
Two sets are congruent if they are similar with a scaling factor of `1`.
-/
def congruent (S T : Set ( × )) : Prop :=
∃ f : ℝ² → ℝ², Function.Bijective f ∧
∀ x y : ℝ², x ∈ S ∧ y ∈ T →
dist x y = dist (f x) (f y)
/--
Any two `congruent` sets must be similar to one another.
-/
theorem congruent_similar {S T : Set ℝ²} : congruent S T → similar S T := by
intro hc
let ⟨f, ⟨hf, hs⟩⟩ := hc
conv at hs => intro x y hxy; arg 1; rw [← one_mul (dist x y)]
exact ⟨f, ⟨hf, ⟨1, hs⟩⟩⟩
end Real