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10 11 12</HEAD> 13<BODY> 14 15<h1 align="center">Differential Geometry of Curves in \(\mathbb{R}^3\)</h1> 16 17<p align="right"><a href="../index.html">Back to homepage</a></p> 18 19 20This 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 <a href="http://geogebra.org">Geogebra</a> which will help you to understand some of the concepts.<br /> 21 22<h3 align="left"><a href="Lecture_1_Smooth_Paths.html">Smooth paths</a></h3> 23 24<h3 align="left"><a href="Lecture_2_Regular_and_Singular_Points.html">Regular and singular points</a></h3> 25 26<h3 align="left"><a href="Lecture_3_Reparametrisations.html">Reparametrising regular paths</a></h3> 27 28<h3 align="left"><a href="Lecture_4_Arclength_Parametrisation.html">Reparametrising by arclength</a></h3> 29 30<h3 align="left"><a href="Lecture_5_More_Examples_Arclength_Parametrisation.html">More examples of arclength parametrisations</a></h3> 31 32<h3 align="left"><a href="Lecture_6_Curvature.html">Curvature</a></h3> 33 34<h3 align="left"><a href="Lecture_7_Frenet_Frame.html">The Frenet Frame</a></h3> 35 36<h3 align="left"><a href="Lecture_8_Torsion.html">Torsion and the osculating plane</a></h3> 37 38<h3 align="left"><a href="Lecture_9_Zero_torsion_planar.html">Curves with zero torsion are planar</a></h3> 39 40<h3 align="left"><a href="Lecture_10_Orientation_Signed_Curvature.html">Orientation and signed curvature</a></h3> 41 42<h3 align="left"><a href="Lecture_11_Curvature_Torsion_any_parametrisation.html">A formula for curvature and torsion in any parametrisation</a></h3> 43 44<h3 align="left"><a href="Lecture_12_Frenet_formulas.html">The Frenet Formulas</a></h3> 45 46<h3 align="left"><a href="Lecture_13_Fundamental_Theorem.html">The Fundamental Theorem of Space Curves</a></h3> 47 48<br /> 49 50<h2>Exercises</h2> 51 52The following links will take you to interactive pictures explaining some of the curves that appear in the exercise sheets about curves in \(\mathbb{R}^3\).<br /> 53 54 55<h3 align="left">A picture of the epicycloid from <a href="Exercises/Ex_1_Epicycloid.html">Exercise Sheet 1, Question 1</a></h3> 56 57<h3 align="left">A picture of a pair of curves forming a ladder from <a href="Exercises/Ex_1_Q9.html">Exercise Sheet 2, Question 10</a></h3> 58 59 60 61 62<h1 align="center">Differential Geometry of Surfaces in \(\mathbb{R}^3\)</h1> 63 64The links below will take you to the interactive pictures used in the lectures. <font color="red">They will not be complete lecture notes as for the curves section of the course.</font> 65 66<h3 align="left"><a href="Lecture_14_Hyperboloid_Level_Surface.html">Week 5, Lecture 1 (The picture of the hyperboloid as a level surface)</a></h3> 67 68<h3 align="left"><a href="Gauss_Map_Examples.html">Week 8, Lecture 2 (Two basic examples of the Gauss map)</a></h3> 69 70<h3 align="left"><a href="Shape_Operator_Example.html">Week 8, Lecture 3 (An example of the shape operator)</a></h3> 71 72<h3 align="left"><a href="Normal_Curvature_Example.html">Week 9, Lecture 4 (An example of a normal section to a surface)</a></h3> 73 74<h3 align="left"><a href="Torus_Point_Classification.html">Week 10, Lecture 5 (Classifying points on the torus)</a></h3> 75 76<h2 align="left"><a name="Problem_Class_2">Problem Class 2</a></h2> 77 78The following links will take you to the interactive pictures used in Problem Class 2.<br /> 79 80<h3 align="left"><a href="Problem_Classes/PC_2_Mercator.html">The Mercator projection</a> 81 82<h3 align="left"><a href="Problem_Classes/PC_2_Transverse_Mercator.html">The Transverse Mercator projection</a> 83 84<h3 align="left"><a href="Problem_Classes/PC_2_Stereographic.html">Stereographic Projection</a> 85 86<h3 align="left"><a href="Problem_Classes/PC_2_Torus.html">The torus as a level set</a> 87 88<h2 align="left"><a name="Problem_Class_3">Problem Class 3</a></h2> 89 90<h3 align="left"><a href="Problem_Classes/PC_3_Helicoid-Catenoid.html">Visualising the helicoid-catenoid transformation</a> 91 92<p> 93<a href="http://www.mathjax.org"><img align="right" title="Powered by MathJax" src="http://cdn.mathjax.org/mathjax/badge/badge.gif" border="0" alt="Powered by MathJax" />
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