Computer Aided Geometric Design

Computer Aided Geometric Design Class Exercise #10 Differential Geometry of Surfaces – Part I 1

Q1. Calculate the length of the parametric curve: in two ways: 1. As a curve in . 2. As a curve on a sphere, using the First Fundamental Form. 2

Solution For part 1 we ignore the fact that the curve is spherical and treat it as a curve in : 3

Solution Now: 4

Solution Therefore: 5

Solution For part 2 we ignore the fact that the curve is in , and use the fact that it is in on the (parameterized) half-sphere: 6

Solution A curve on a parameterized surface is the restriction of the parameterization to a curve in the parameter space: 7

Solution Recall the First Fundamental Form defined on tangent vectors to the surface at a point : Where is the inner product on the tangent space. 8

Solution (Remark: In this case is inherited from , but this need not be the case in abstract manifolds) 9

Solution In lectures you saw that using the chain rule, the following coefficients are obtained: When: 10

Solution Therefore, the length of as a curve in the surface parameterized by is: When all functions involved are evaluated at , namely: , etc. 11

Solution The coefficients in our parameterization are: 12

Solution and: 15

Final Remarks In local coordinates, the First Fundamental Form is the quadratic form: Where: 16

Final Remarks Can you think of a point of view for the first solution that also uses: ? 17

Q2. Consider a surface parameterized by and denote: Prove that at a point , the length of the normal to at is given by . 18

Solution The normal is given by: and the length satisfies: 19

Solution Where is the angle between and . Now: 20

Solution Writing this using inner products: 21

Final Remarks This means that the first fundamental form enables us to compute the surface area, using only the parameterization via This fact remains for any abstract manifold in higher dimensions. is then called a Riemannian metric, and is called the Riemannian Volume form. 22

Q3. Use question 2 to compute the surface area of a torus with radius . 23

Solution First we need a parameterization: (Why?) 24

Solution The First Fundamental Form is computed by differentiation: 25

Solution Which gives: 26

Solution Now: 27

Solution 28

Final Remarks How is this related to the surface area of a cylinder? (Hint: how can one get a torus from a cylinder?) What is the geometric interpretation of ? 29

Computer Aided Geometric Design