Redefine an `Orthogonal` rectangle as a subtype of a `Skew` one.
parent
c0f8895686
commit
68d46e1a7d
|
|
@ -1,5 +1,5 @@
|
|||
import Common.Geometry.Basic
|
||||
import Common.Geometry.Rectangle.Orthogonal
|
||||
import Common.Geometry.Rectangle.Skew
|
||||
import Common.Geometry.StepFunction
|
||||
|
||||
/-! # Common.Geometry.Area
|
||||
|
|
|
|||
|
|
@ -1,5 +1,4 @@
|
|||
import Mathlib.Data.Fin.Basic
|
||||
import Mathlib.Tactic.LibrarySearch
|
||||
|
||||
import Common.Geometry.Point
|
||||
import Common.Geometry.Segment
|
||||
|
|
@ -13,176 +12,27 @@ A characterization of an orthogonal rectangle.
|
|||
namespace Geometry.Rectangle
|
||||
|
||||
/--
|
||||
An `Orthogonal` rectangle is characterized by two points on opposite corners. It
|
||||
is assumed the edges of the rectangle are parallel to the coordinate axes.
|
||||
|
||||
A `Point` can alternatively be viewed as an `Orthogonal` rectangle in which the
|
||||
two points coincide. A horizontal or vertical `Segment` can alternatively be
|
||||
viewed as an `Orthogonal` rectangle with width or height (but not both) `0`.
|
||||
An `Orthogonal` rectangle is a `Skew` rectangle with edges parallel to the
|
||||
coordinate axes.
|
||||
-/
|
||||
structure Orthogonal where
|
||||
bl : Point -- bottom left
|
||||
tr : Point -- top right
|
||||
def Orthogonal := { r : Skew // r.bl.x = r.tl.x ∧ r.bl.y = r.br.y }
|
||||
|
||||
namespace Orthogonal
|
||||
|
||||
/--
|
||||
The width of the `Orthogonal` rectangle.
|
||||
-/
|
||||
def width (r : Orthogonal) := r.tr.x - r.bl.x
|
||||
|
||||
/--
|
||||
The height of the `Orthogonal` rectangle.
|
||||
-/
|
||||
def height (r : Orthogonal) := r.tr.y - r.bl.y
|
||||
|
||||
/--
|
||||
The top-left corner of the `Orthogonal` rectangle.
|
||||
-/
|
||||
def tl (r : Orthogonal) : Point := ⟨r.bl.x, r.bl.y + r.height⟩
|
||||
|
||||
/--
|
||||
The bottom-right corner of the `Orthogonal` rectangle.
|
||||
-/
|
||||
def br (r : Orthogonal) : Point := ⟨r.bl.x + r.width, r.bl.y⟩
|
||||
|
||||
/--
|
||||
The `Set` of `Point`s enclosed in the region determined by the edges of the
|
||||
`Orthogonal` rectangle. Edges of the rectangle are included in the result set.
|
||||
-/
|
||||
def toSet (r : Orthogonal) : Set Point :=
|
||||
{ p | r.bl.x ≤ p.x ∧ p.x ≤ r.br.x ∧ r.bl.y ≤ p.y ∧ p.y ≤ r.tl.y }
|
||||
{ p | r.val.bl.x ≤ p.x ∧ p.x ≤ r.val.br.x ∧
|
||||
r.val.bl.y ≤ p.y ∧ p.y ≤ r.val.tl.y }
|
||||
|
||||
/--
|
||||
An `Orthogonal` rectangle's top side is equal in length to its bottom side.
|
||||
Show the `toSet` definition of an `Orthogonal` rectangle is in agreement with
|
||||
its `Skew` counterpart.
|
||||
-/
|
||||
theorem dist_top_eq_dist_bottom (r : Orthogonal)
|
||||
: Point.dist r.tl r.tr = Point.dist r.bl r.br := by
|
||||
unfold tl br Point.dist width height
|
||||
norm_num
|
||||
|
||||
/--
|
||||
An `Orthogonal` rectangle's left side is equal in length to its right side.
|
||||
-/
|
||||
theorem dist_left_eq_dist_right (r : Orthogonal)
|
||||
: Point.dist r.tl r.bl = Point.dist r.tr r.br := by
|
||||
unfold tl br Point.dist width height
|
||||
norm_num
|
||||
|
||||
/--
|
||||
Convert an `Orthogonal` rectangle into a `Skew` one.
|
||||
-/
|
||||
def toSkew (r : Orthogonal) : Skew := ⟨r.tl, r.bl, r.br, sorry⟩
|
||||
|
||||
/--
|
||||
The set of `Orthogonal` rectangles are embedded in the set of `Skew` rectangles.
|
||||
-/
|
||||
def skewEmbedding : Orthogonal ↪ Skew :=
|
||||
have : Function.Injective toSkew := by
|
||||
unfold Function.Injective
|
||||
intro r₁ r₂ h
|
||||
unfold toSkew at h
|
||||
have ⟨⟨blx₁, bly₁⟩, ⟨trx₁, try₁⟩⟩ := r₁
|
||||
have ⟨⟨blx₂, bry₂⟩, ⟨trx₂, try₂⟩⟩ := r₂
|
||||
simp
|
||||
simp at h
|
||||
unfold tl br width height at h
|
||||
simp at h
|
||||
exact ⟨⟨h.left.left, h.right.left.right⟩, ⟨h.right.right.left, h.left.right⟩⟩
|
||||
⟨toSkew, this⟩
|
||||
|
||||
/-! ## Point -/
|
||||
|
||||
/--
|
||||
A `Point` is an `Orthogonal` rectangle in which all points coincide.
|
||||
-/
|
||||
abbrev AsPoint := Subtype (fun r : Orthogonal => r.bl = r.tr)
|
||||
|
||||
namespace AsPoint
|
||||
|
||||
/--
|
||||
The function mapping an `Orthogonal` rectangle with all points coinciding to a
|
||||
`Point`.
|
||||
-/
|
||||
def toPoint (p : AsPoint) : Point := p.val.tl
|
||||
|
||||
/--
|
||||
The function mapping a `Point` to an `Orthogonal` rectangle with all points
|
||||
coinciding.
|
||||
-/
|
||||
def fromPoint (p : Point) : AsPoint := ⟨Orthogonal.mk p p, by simp⟩
|
||||
|
||||
/--
|
||||
An isomorphism between an `Orthogonal` rectangle with all points coinciding and
|
||||
a `Point`.
|
||||
-/
|
||||
def isoPoint : AsPoint ≃ Point :=
|
||||
{
|
||||
toFun := toPoint,
|
||||
invFun := fromPoint,
|
||||
left_inv := by
|
||||
unfold Function.LeftInverse fromPoint toPoint
|
||||
intro ⟨r, hr⟩
|
||||
congr
|
||||
repeat {
|
||||
simp only
|
||||
unfold tl height
|
||||
rw [hr]
|
||||
simp
|
||||
}
|
||||
right_inv := by
|
||||
unfold Function.RightInverse Function.LeftInverse fromPoint toPoint
|
||||
intro ⟨r, hr⟩
|
||||
unfold tl height
|
||||
simp
|
||||
}
|
||||
|
||||
/--
|
||||
The width of an `AsPoint` is `0`.
|
||||
-/
|
||||
theorem width_eq_zero (p : AsPoint) : p.val.width = 0 := by
|
||||
unfold Orthogonal.width
|
||||
rw [p.property]
|
||||
simp
|
||||
|
||||
/--
|
||||
The height of an `AsPoint` is `0`.
|
||||
-/
|
||||
theorem height_eq_zero (p : AsPoint) : p.val.height = 0 := by
|
||||
unfold Orthogonal.height
|
||||
rw [p.property]
|
||||
simp
|
||||
|
||||
end AsPoint
|
||||
|
||||
/-! ## Segment -/
|
||||
|
||||
/--
|
||||
A `Segment` is an `Orthogonal` rectangle either width or height equal to `0`.
|
||||
-/
|
||||
abbrev AsSegment := Subtype (fun r : Orthogonal =>
|
||||
(r.bl.x = r.tr.x ∧ r.bl.y ≠ r.tr.y) ∨ (r.bl.x ≠ r.tr.x ∧ r.bl.y = r.tr.y))
|
||||
|
||||
namespace AsSegment
|
||||
|
||||
/--
|
||||
Either the width or height of an `AsSegment` is zero.
|
||||
-/
|
||||
theorem width_or_height_eq_zero (s : AsSegment)
|
||||
: s.val.width = 0 ∨ s.val.height = 0 := by
|
||||
apply Or.elim s.property
|
||||
· intro h
|
||||
refine Or.inl ?_
|
||||
unfold width
|
||||
rw [h.left]
|
||||
simp
|
||||
· intro h
|
||||
refine Or.inr ?_
|
||||
unfold height
|
||||
rw [h.right]
|
||||
simp
|
||||
|
||||
end AsSegment
|
||||
theorem orthogonal_set_eq_skew_set (r : Orthogonal)
|
||||
: r.toSet = r.val.toSet := sorry
|
||||
|
||||
end Orthogonal
|
||||
|
||||
|
|
|
|||
|
|
@ -1,5 +1,4 @@
|
|||
import Common.Finset
|
||||
import Common.Geometry.Point
|
||||
import Common.Geometry.Rectangle.Orthogonal
|
||||
import Common.List.Basic
|
||||
import Common.Set.Partition
|
||||
|
|
@ -34,7 +33,15 @@ def toSet (sf : StepFunction) : Set Point :=
|
|||
⋃ i ∈ Finset.finRange sf.p.ivls.length,
|
||||
let I := sf.p.ivls[i]
|
||||
Rectangle.Orthogonal.toSet
|
||||
⟨{ x := I.left, y := 0 }, { x := I.right, y := sf.toFun i }⟩
|
||||
⟨
|
||||
{
|
||||
tl := ⟨I.left, sf.toFun i⟩,
|
||||
bl := ⟨I.left, 0⟩,
|
||||
br := ⟨I.right, 0⟩,
|
||||
has_right_angle := sorry
|
||||
},
|
||||
by simp
|
||||
⟩
|
||||
|
||||
end StepFunction
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,6 @@
|
|||
import Mathlib.Algebra.BigOperators.Basic
|
||||
import Mathlib.Data.Real.Basic
|
||||
import Mathlib.Data.Finset.Basic
|
||||
import Mathlib.Tactic.LibrarySearch
|
||||
|
||||
/-! # Common.Real.Floor
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue