feat(Geometry/Euclidean/Angle/Bisector): oriented angle bisection and equal distance #30477
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…Projection_of_le` Add a lemma that `orthogonalProjection s₁ (orthogonalProjection s₂ p) = orthogonalProjection s₁ p` for `s₁ ≤ s₂`. (A similar lemma for projection onto submodules rather than affine subspaces already exists.)
… and bisection lemmas
Add lemmas
```lean
lemma oangle_eq_neg_of_angle_eq_of_sign_eq_neg {w x y z : V}
(h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
(hs : (o.oangle w x).sign = -(o.oangle y z).sign) : o.oangle w x = -o.oangle y z := by
```
and
```lean
lemma angle_eq_iff_oangle_eq_neg_of_sign_eq_neg {w x y z : V} (hw : w ≠ 0) (hx : x ≠ 0)
(hy : y ≠ 0) (hz : z ≠ 0) (hs : (o.oangle w x).sign = -(o.oangle y z).sign) :
InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔
o.oangle w x = -o.oangle y z := by
```
and similar affine versions, corresponding to such lemmas that already
exist when the signs are equal rather than negations of each other.
Deduce lemmas relating oriented and unoriented versions of angle
bisection:
```lean
lemma angle_eq_iff_oangle_eq_or_sameRay {x y z : V} (hx : x ≠ 0) (hz : z ≠ 0) :
InnerProductGeometry.angle x y = InnerProductGeometry.angle y z ↔
o.oangle x y = o.oangle y z ∨ SameRay ℝ x z := by
```
and
```lean
lemma angle_eq_iff_oangle_eq_or_wbtw {p₁ p₂ p₃ p₄ : P} (hp₁ : p₁ ≠ p₂) (hp₄ : p₄ ≠ p₂) :
∠ p₁ p₂ p₃ = ∠ p₃ p₂ p₄ ↔ ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄ ∨ Wbtw ℝ p₂ p₁ p₄ ∨ Wbtw ℝ p₂ p₄ p₁ := by
```
… equal distance
Add a lemma
```lean
lemma dist_orthogonalProjection_eq_iff_oangle_eq {p p' : P} {s₁ s₂ : AffineSubspace ℝ P}
[s₁.direction.HasOrthogonalProjection] [s₂.direction.HasOrthogonalProjection]
(hp' : p' ∈ s₁ ⊓ s₂)
(hne : haveI : Nonempty s₁ := ⟨p', hp'.1⟩; haveI : Nonempty s₂ := ⟨p', hp'.2⟩;
(orthogonalProjection s₁ p : P) ≠ orthogonalProjection s₂ p)
(hp₁ : haveI : Nonempty s₁ := ⟨p', hp'.1⟩; orthogonalProjection s₁ p ≠ p')
(hp₂ : haveI : Nonempty s₂ := ⟨p', hp'.2⟩; orthogonalProjection s₂ p ≠ p') :
haveI : Nonempty s₁ := ⟨p', hp'.1⟩
haveI : Nonempty s₂ := ⟨p', hp'.2⟩
dist p (orthogonalProjection s₁ p) = dist p (orthogonalProjection s₂ p) ↔
∡ (orthogonalProjection s₁ p : P) p' p = ∡ p p' (orthogonalProjection s₂ p) :=
```
that is an oriented angle analogue of the existing
`dist_orthogonalProjection_eq_iff_angle_eq`. Because the minimal
nondegeneracy conditions required for the two directions of this lemma
are different (whereas the unoriented version doesn't need any
nondegeneracy conditions), those two directions are added as separate
lemmas, each with minimal nondegeneracy conditions, from which the
`iff` version is then deduced.
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PR summary ed54e09893Import changes exceeding 2%
|
| File | Base Count | Head Count | Change |
|---|---|---|---|
| Mathlib.Geometry.Euclidean.Angle.Bisector | 2114 | 2160 | +46 (+2.18%) |
Import changes for all files
| Files | Import difference |
|---|---|
Mathlib.Geometry.Euclidean.Angle.Bisector |
46 |
Declarations diff
+ dist_orthogonalProjection_eq_iff_oangle_eq
+ dist_orthogonalProjection_eq_of_oangle_eq
+ oangle_eq_of_dist_orthogonalProjection_eq
You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>
## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>The doc-module for script/declarations_diff.sh contains some details about this script.
No changes to technical debt.
You can run this locally as
./scripts/technical-debt-metrics.sh pr_summary
- The
relativevalue is the weighted sum of the differences with weight given by the inverse of the current value of the statistic. - The
absolutevalue is therelativevalue divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).
|
The increased imports are explicitly intended for this file and just a consequence of building up this file over multiple PRs rather than adding oriented and unoriented lemmas together, as noted in #29656 when I added |
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Oct 14, 2025
Add some lemmas that angles involving orthogonal projections are right angles. The unoriented versions abstract something used multiple times in `Mathlib.Geometry.Euclidean.Angle.Bisector` (which is thus refactored to use the new lemmas); I expect the oriented versions may be of use in future in further oriented angle bisection lemmas building on top of leanprover-community#30477.
Add a lemma
```lean
lemma orthogonalProjection_eq_iff_mem {s : AffineSubspace ℝ P} [Nonempty s]
[s.direction.HasOrthogonalProjection] {p q : P} :
orthogonalProjection s p = q ↔ q ∈ s ∧ p -ᵥ q ∈ s.directionᗮ := by
```
that gives the characteristic property of the orthogonal projection in
a more convenient form to use than the existing
`inter_eq_singleton_orthogonalProjection` (from which it is derived).
…ion_orthogonalProjection_of_le
…isector_dist_oangle
…ion_orthogonalProjection_of_le
…isector_dist_oangle
…ion_orthogonalProjection_of_le
…isector_dist_oangle
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
…ion_orthogonalProjection_of_le
…isector_dist_oangle
Co-authored-by: Jovan Gerbscheid <56355248+JovanGerb@users.noreply.github.com>
mathlib-bors bot
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…30522) Add some lemmas that angles involving orthogonal projections are right angles. The unoriented versions abstract something used multiple times in `Mathlib.Geometry.Euclidean.Angle.Bisector` (which is thus refactored to use the new lemmas); I expect the oriented versions may be of use in future in further oriented angle bisection lemmas building on top of #30477.
jsm28
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Oct 27, 2025
Add lemmas relating equal distance to two lines to bisecting the angle between them mod π (expressed as usual as equality of twice angles), building on the previous lemmas (leanprover-community#30477) dealing with bisection mod 2π.
This was referenced Oct 27, 2025
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Add a lemma
that is an oriented angle analogue of the existing
dist_orthogonalProjection_eq_iff_angle_eq. Because the minimal nondegeneracy conditions required for the two directions of this lemma are different (whereas the unoriented version doesn't need any nondegeneracy conditions), those two directions are added as separate lemmas, each with minimal nondegeneracy conditions, from which theiffversion is then deduced.orthogonalProjection_orthogonalProjection_of_le#30474