Differential Geometry of Curves in \(\mathbb{R}^3\)

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This page contains links to lecture material for the first section of the Differential Geometry course, where we will focus on curves in \(\mathbb{R}^3\). The goal is to include some animations and interactive pictures using Geogebra which will help you to understand some of the concepts.

Smooth paths

Regular and singular points

Reparametrising regular paths

Reparametrising by arclength

More examples of arclength parametrisations

Curvature

The Frenet Frame

Torsion and the osculating plane

Curves with zero torsion are planar

Orientation and signed curvature

A formula for curvature and torsion in any parametrisation

The Frenet Formulas

The Fundamental Theorem of Space Curves


Exercises

The following links will take you to interactive pictures explaining some of the curves that appear in the exercise sheets about curves in \(\mathbb{R}^3\).

A picture of the epicycloid from Exercise Sheet 1, Question 1

A picture of a pair of curves forming a ladder from Exercise Sheet 2, Question 10

Differential Geometry of Surfaces in \(\mathbb{R}^3\)

The links below will take you to the interactive pictures used in the lectures. They will not be complete lecture notes as for the curves section of the course.

Week 5, Lecture 1 (The picture of the hyperboloid as a level surface)

Week 8, Lecture 2 (Two basic examples of the Gauss map)

Week 8, Lecture 3 (An example of the shape operator)

Week 9, Lecture 4 (An example of a normal section to a surface)

Week 10, Lecture 5 (Classifying points on the torus)

Problem Class 2

The following links will take you to the interactive pictures used in Problem Class 2.

The Mercator projection

The Transverse Mercator projection

Stereographic Projection

The torus as a level set

Problem Class 3

Visualising the helicoid-catenoid transformation

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