feat: Frénet moving frame and Frénet equation for plane curves

[michael-novak-math]

We define the curvature function, normal vector function and the Frénet moving frame for plane curve.
We also prove the Frénet equations for plane curves.

A separate PR (#37489) will prove the fundamental theorem of plane curves.

[github-actions]

Welcome new contributor!

Thank you for contributing to Mathlib! If you haven't done so already, please review our contribution guidelines, as well as the style guide and naming conventions. In particular, we kindly remind contributors that we have guidelines regarding the use of AI when making pull requests.

We use a review queue to manage reviews. If your PR does not appear there, it is probably because it is not successfully building (i.e., it doesn't have a green checkmark), has the awaiting-author tag, or another reason described in the Lifecycle of a PR. The review dashboard has a dedicated webpage which shows whether your PR is on the review queue, and (if not), why.

If you haven't already done so, please come to https://leanprover.zulipchat.com/, introduce yourself, and mention your new PR.

Thank you again for joining our community.

[github-actions]

PR summary b43655dfe2

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Geometry.PlaneCurves (new file)	2434

---

Declarations diff

+ _root_.ContDiffOn.normal_of_twice_contDiffOn
+ _root_.HasDerivAt.normal
+ deriv_normal_eq_minus_orientedCurvature_times_deriv
+ frameAt
+ hasDerivAt_deriv_of_contDiffOn
+ hasDerivAt_integral_of_continuousOn_open_interval
+ hasDerivWithinAt_of_continuousOn_interval
+ hasDerivWithinAt_pi_euclidean
+ inner_deriv_curve_eq_zero_of_const_norm_curve
+ inner_deriv_deriv_normal_eq_minus_orientedCurvature
+ inner_deriv_normal_eq_zero
+ inner_deriv_normal_normal_eq_zero_of_norm_deriv_eq_one
+ inner_second_deriv_deriv_eq_zero_of_const_norm_deriv
+ inners_sum_eq_zero_of_const_inner_on_open
+ norm_normal_eq_one_of_non_zero_deriv
+ norm_normal_eq_one_of_norm_deriv_eq_one
+ normal
+ normal_eq
+ normal_eq_of_norm_deriv_eq_one
+ of
+ orientedCurvature
+ orientedCurvature_eq
+ orientedCurvature_of_norm_deriv_eq_one
+ second_deriv_eq_orientedCurvature_times_normal

You can run this locally as follows

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

Increase in tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type
6754	1	backward.isDefEq.respectTransparency

Current commit 659e77789e
Reference commit b43655dfe2

You can run this locally as

./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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[grunweg]

Three brief comments.

[grunweg]

I'm not sure how useful this lemma will be in practice. (Right now, you're using it more because you're considering unit speed curves more than necessary.) If this proves useful, please mention this in a doc-string to normal_eq also.