147 lines
3.5 KiB
Plaintext
147 lines
3.5 KiB
Plaintext
import Bookshelf.Real.Geometry.Basic
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/-! # Bookshelf.Real.Geometry.Rectangle
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A characterization of a rectangle. This follows the definition as outlined in
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[^1]. Note that a `Point` and a `LineSegment` are both considered rectangles,
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with one or both dimensions equal to `0` respectively.
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[^1]: Apostol, Tom M. Calculus, Vol. 1: One-Variable Calculus, with an
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Introduction to Linear Algebra. 2nd ed. Vol. 1. 2 vols. Wiley, 1991.
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-/
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namespace Real
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/--
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A `Rectangle` is characterized by three distinct points and the angle formed
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between line segments originating from the "bottom left" point.
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-/
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structure Rectangle where
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top_left : ℝ²
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bottom_left : ℝ²
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bottom_right : ℝ²
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forms_right_angle : ∠ top_left bottom_left bottom_right = π / 2
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namespace Rectangle
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/--
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The top-right corner of the rectangle, oriented with respect to the other
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vertices.
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-/
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def top_right (r : Rectangle) : ℝ² :=
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( r.top_left.fst + r.bottom_right.fst - r.bottom_left.fst
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, r.top_left.snd + r.bottom_right.snd - r.bottom_left.snd
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)
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/--
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A `Rectangle` is the locus of points bounded by its edges.
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-/
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def set_def (r : Rectangle) : Set ℝ² :=
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sorry
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/--
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A `Rectangle`'s top side is equal in length to its bottom side.
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-/
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theorem dist_top_eq_dist_bottom (r : Rectangle)
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: dist r.top_left r.top_right = dist r.bottom_left r.bottom_right := by
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unfold top_right dist
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repeat rw [add_comm, sub_right_comm, add_sub_cancel']
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/--
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A `Rectangle`'s left side is equal in length to its right side.
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-/
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theorem dist_left_eq_dist_right (r : Rectangle)
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: dist r.top_left r.bottom_left = dist r.top_right r.bottom_right := by
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unfold top_right dist
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repeat rw [
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sub_sub_eq_add_sub,
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add_comm,
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sub_add_eq_sub_sub,
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sub_right_comm,
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add_sub_cancel'
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]
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/--
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Computes the width of a `Rectangle`.
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-/
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noncomputable def width (r : Rectangle) : ℝ :=
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dist r.bottom_left r.bottom_right
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/--
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Computes the height of a `Rectangle`.
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-/
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noncomputable def height (r : Rectangle) : ℝ :=
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dist r.bottom_left r.top_left
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end Rectangle
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/--
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A `Point` is a `Rectangle` in which all points coincide.
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-/
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abbrev Point := Subtype (fun r : Rectangle =>
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r.top_left = r.bottom_left ∧ r.bottom_left = r.bottom_right)
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namespace Point
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/--
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A `Point` is the set consisting of just itself.
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-/
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def set_def (p : Point) : Set ℝ² := p.val.set_def
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/--
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The width of a `Point` is `0`.
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-/
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theorem width_eq_zero (p : Point) : p.val.width = 0 := by
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unfold Rectangle.width
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rw [p.property.right]
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unfold dist
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simp
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/--
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The height of a `Point` is `0`.
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-/
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theorem height_eq_zero (p : Point) : p.val.height = 0 := by
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unfold Rectangle.height
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rw [p.property.left]
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unfold dist
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simp
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end Point
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/--
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A `LineSegment` is a `Rectangle` in which two of the three points coincide.
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-/
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abbrev LineSegment := Subtype (fun r : Rectangle =>
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(r.top_left = r.bottom_left ∧ r.bottom_left ≠ r.bottom_right) ∨
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(r.top_left ≠ r.bottom_left ∧ r.bottom_left = r.bottom_right))
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namespace LineSegment
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/--
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A `LineSegment` `s` is the set of points corresponding to the shortest line
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segment joining the two distinct points of `s`.
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-/
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def set_def (s : LineSegment) : Set ℝ² := s.val.set_def
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/--
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Either the width or height of a `LineSegment` is zero.
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-/
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theorem width_or_height_eq_zero (s : LineSegment)
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: s.val.width = 0 ∨ s.val.height = 0 := by
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apply Or.elim s.property
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· intro h
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refine Or.inr ?_
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unfold Rectangle.height
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rw [h.left]
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unfold dist
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simp
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· intro h
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refine Or.inl ?_
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unfold Rectangle.width
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rw [h.right]
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unfold dist
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simp
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end LineSegment
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end Real |