6) Curvature Of Surfaces

Question 1

Derive second fundamental form

Answer 1

Question 2

Derive 2nd fundamental form of a plane

Answer 2

Question 3

Derive 2nd fundamental form of a surface of revolution

Answer 3

Question 4

Derive 2nd fundamental form of unit
Sphere (through surface of revolution)

Answer 4

Question 5

Define each component and relate

Answer 5

Curvature is a linear combination of N^- and N cross γ^-•
κ_n is normal curvature
κ_s is geodesic curvature

Question 6

Requirement for normal and geodesic curvatures

Answer 6

γ is a unit speed reparam

Question 7

κ_n ?

Answer 7

Normal curvature

Question 8

κ_g ?

Answer 8

Geodesic curvature

Question 9

Total curvature

Answer 9

Question 10

Most important single fact about normal curvature κ_n of a curve γ on a surface σ

Answer 10

Depends only on σ and on tangent vector to γ
(Not on higher derivatives of γ)

Question 11

Prove

Answer 11

Question 12

2 matrices for 1st and 2nd fundamental forms of a surface σ

Answer 12

Question 13

Derive matrix of first and second fundamental forms

Answer 13

Question 14

Define principal curvature of a surface

Answer 14

Question 15

Denote the principal vector corresponding to the principal curvature and it’s requirement

Answer 15

Question 16

Relate principal curvatures at a point P of a surface σ

Answer 16

Question 17

Prove

Answer 17

Question 18

Euler’s Theorem

Answer 18

Question 19

Prove Euler’s Theorem

Answer 19

Question 20

Relate normal curvatures of all unit speed curves to principal curvatures at a point on a surface and principal directions

Answer 20

Question 21

Prove

Answer 21