Frenet–Serret Formula ✍️

It explains how a curve reveals its hidden geometry by tracking the way it bends and twists through space. Imagine tracing the path of a roller coaster, a winding river, or the spiral of a DNA strand. At every point along the path, the curve is constantly changing direction, and the Frenet–Serret formulas provide a precise way to describe that change.

They do this by attaching a moving frame of three special directions to each point on the curve. The first points forward along the path, showing where the curve is heading. The second points inward, toward the direction of bending. The third stands perpendicular to both, capturing how the curve twists out of its plane. Together, they form a local coordinate system that travels with the curve itself.

As you move along the curve, these three directions rotate and evolve. The formulas measure this evolution using two key quantities: curvature and torsion. Curvature tells how sharply the path bends, while torsion tells how strongly it twists into three dimensions. If curvature vanishes, the path becomes straight; if torsion vanishes, the curve lies flat in a plane.

Mathematicians and physicists use the Frenet–Serret formulas to study motion, design smooth paths in engineering, understand particle trajectories, and analyze natural shapes. They transform a simple line into a rich geometric story, revealing exactly how space is being navigated at every step.

#FrenetSerretFormula #DifferentialGeometry #Geometry #Mathematics #Math #PureMathematics #AppliedMathematics #MathematicalPhysics #Physics #STEM #ScienceEducation #MathEducation #Curvature #Torsion #SpaceCurves #VectorCalculus #Calculus #LinearAlgebra #GeometricAnalysis

A pictorial introduction to differential geometry (2017)

https://arxiv.org/abs/1709.08492

#HackerNews #differentialgeometry #pictorialguide #mathematics #arxiv #geometry #education

Here's a differential geometry puzzle (inspired by one found in Weinberg's book on General Relativity).

On a fictional planet (the #ForgottenRealms), there are three cities: Candlekeep, Muran, and Ristavin.

The geodesic distance:
- from Candlekeep to Muran is 330 miles
- from Ristavin to Muran is 426 miles
- from Candlekeep to Ristavin is 636 miles.
(The margin of error for these measurements is ±3 miles.)

Is the planet positively curved (a sphere), negatively curved (a saddle), or zero curvature ('flat', i.e., a plane)? If it is a sphere, what is its radius?

#Mathematics #math #differentialgeometry #geography #cartography #puzzle

Today my paper "Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds" got published in the Journal of Geometry and Physics!

The paper defines: A globally hyperbolic spacetime $(X,g)$ is observer-refocusing if there exists a point $p$ and a timelike curve $\gamma$ in $X$ so that all lightrays emitted from $p$ intersect $\gamma$. The paper proves that spacetimes $(X,g)$ with $\dim(X)\geq 3$ which are observer-refocusing with respect to a compact timelike curve have compact Cauchy surfaces with finite fundamental group. This extends known results on strongly refocusing spacetimes, which are spacetimes with points $p,q$ so that all lightrays through $p$ go through $q$. Further, observer-refocusing spacetimes of dimension at least $3$ with an analytic metric are strongly refocusing.

These results lead to immediate corollaries in Riemannian geometry: Let $(M,h)$ be a connected, complete Riemannian manifold and let $x \in M$. We call $(M,h)$ a $Z^x$ manifold if all geodesics starting at $x$ return to $x$. We show that if $\dim(M) \geq 2$ and if the return time of unit-speed geodesics starting at $x$ is uniformly bounded, then $M$ is compact with finite fundamental group. Further, if the metric of a $Z^x$ manifold is analytic, then all unit-speed geodesics starting at $x$ return to $x$ at a common time. This resolves the question "Are all $Z^x$ manifolds $Y^x_l$ manifolds for some $l>0$?" posed in Besse's book "Manifolds all of whose geodesics are closed" affirmatively for analytic manifolds.

https://doi.org/10.1016/j.geomphys.2026.105834
https://arxiv.org/abs/2503.23565

#DifferentialGeometry #MathematicalPhysics #GeneralRelativity

[ Lumo Kaŭstikaĵo ]

Matematika diferenciala geometrio priskribanta la ebenan koverton de kurboj spuritaj de radioj disvastiĝantaj tra manifoldo. 🤓 #nerd

~briletanta~

\eZ

#miksang #dailypic #aphotoaday
#Esperanto #photography #photo
#physics #optics #mathematics #maths
#caustics #differentialgeometry
#manifold #manifolds
#shimmering

https://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1571-9.htm

Manifolds from Concrete Examples
by Atsushi Fujioka

#Math #DifferentialGeometry #MathBook

Change my mind: pseudogroups are the "wrong way" to formalize differential geometry.

What's wrong with formalizing charts and atlases, then a manifold is a set equipped with a maximal atlas?

We could formalize, e.g., complex manifolds using G-structures.

What is wrong with this approach?

#Mathematics #proofassistant #differentialgeometry