62 lines
1.8 KiB
Plaintext
62 lines
1.8 KiB
Plaintext
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) : ℝ
|
||
|
||
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 : ℝ, ∀ 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 (ℝ × ℝ)) : 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 |