feat(Geometry/Euclidean/Angle): the unoriented angle at the center in terms of the angle at the circumference

[Scarlett-le]

Adds the unoriented counterpart of Sphere.oangle_center_eq_two_zsmul_oangle ("the angle at the
center of a circle is twice the angle at the circumference") to
Mathlib/Geometry/Euclidean/Angle/Sphere.lean. Since the unoriented angle at the center lies in
[0, π] while twice the angle at the circumference ranges over [0, 2π], two lemmas are needed,
splitting at a right angle at the circumference:

• Sphere.angle_center_eq_two_mul_angle_of_two_mul_angle_le_pi: when twice the angle at the
  circumference is at most π, the angle at the center equals twice it;
• Sphere.angle_center_eq_two_pi_sub_two_mul_angle_of_pi_le_two_mul_angle: when it is at least π,
  the angle at the center equals 2 * π minus twice it.

Dependencies

• depends on: [Merged by Bors] - feat(Geometry/Euclidean/Sphere): add lemmas about points on a sphere #41143

[github-actions]

PR summary 6246b7289d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ angle_center_eq_two_mul_angle_of_two_mul_angle_le_pi
+ angle_center_eq_two_pi_sub_two_mul_angle_of_pi_le_two_mul_angle

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 6246b72).

• +2 new declarations
• −0 removed declarations

+EuclideanGeometry.Sphere.angle_center_eq_two_mul_angle_of_two_mul_angle_le_pi
+EuclideanGeometry.Sphere.angle_center_eq_two_pi_sub_two_mul_angle_of_pi_le_two_mul_angle

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 6246b7289d
Reference commit 88d006abbc

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[felixpernegger]

awaiting-author

[Scarlett-le]

-awaiting-author

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat(Geometry/Euclidean/Sphere): add lemmas about points on a sphere #41143
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[jsm28]

What are the type class hypotheses on the new theorems? It looks to me like they're in a section of the file with assumptions of both Fact (finrank ℝ V = 2) and Module.Oriented ℝ V (Fin 2), meaning the statements will end up depending implicitly on a choice of orientation even though that choice ought to be made only inside the proof without the statements depending on it. (The hypothesis of being two-dimensional is genuinely needed, and probably more useful for most applications than a more general version that just requires the three points to be coplanar with the center of the sphere.) So I think this should be moved to an unoriented section of the file and instead made to choose an orientation only internally inside the proofs, not as a hypothesis of the theorems (something that would be more convenient given #39696).