Exam Prep

Question 1

Process to find unit speed reparam

Answer 1

Question 2

Process to find curvature

Answer 2

Question 3

Process to Find right handed orthonormal basis

Answer 3

Question 4

Frenet-serret equations in R^3

Answer 4

Question 5

Nowhere vanishing curvature implies

Answer 5

Torsion is always defined

Question 6

Greene’s theorem

Answer 6

For all smooth functions f and g

Provided that γ is traversed anti clockwise

Question 7

Unit normal to σ

Answer 7

Question 8

Unit normal

Answer 8

Question 9

A surface patch is regular if

Answer 9

Question 10

Regular point ?

Answer 10

Component vectors of cross product of partial derivatives of surface patch must be linearly independent
I.e. cross product never equals zero

Question 11

Process to find torsion

Answer 11

Either of

Question 12

Conformal?

Answer 12

A conformal parameterization has E=G and F=0

Question 13

1st fundamental form of a plane (in standard coordinates)

Answer 13

du² + dv²

Question 14

Calculate surface area

Answer 14

Question 15

For a curve on a regular surface patch, find the first fundamental form

Answer 15

Question 16

2 surfaces are isometries if

Answer 16

Question 17

What is a tangent developable

Answer 17

The union of the tangent lines to a curve in R³

Question 18

Find the parameterization of a tangent developable

Answer 18

Question 19

Any tangent developable is isometric to

Answer 19

(Part of) a plane

Question 20

Calculate 2nd fundamental form

Answer 20

Where N^- is the principal unit normal

Question 21

Normal curvature

Answer 21

For γ a unit speed curve on σ

Question 22

Geodesic curvature

Answer 22

For a unit speed γ on σ

Question 23

Relate normal curvature to geodesic curvature

Answer 23

Question 24

Matrices for fundamental forms

Answer 24

Question 25

Show that 2 tangent vectors are linear combinations of partial derivatives of surface patch

Answer 25

Question 26

Principal curvatures

Answer 26

Question 27

Euler’s Theorem

Answer 27

Question 28

Gaussian curvature and mean curvature

Answer 28

Question 29

Find explicit formulas for Gaussian curvature and mean curvature using 1st and 2nd fundamental forms

Answer 29

Question 30

For σ^~ = λ σ Give F₁ and F₂

Answer 30

Question 31

Gaussian curvature for a sphere and for a cylinder

Answer 31

Sphere constant positive Cylinder constant zero

Question 32

Find principal vector corresponding to principal curvature

Answer 32

Question 33

If k1 doesn’t equal k2

Answer 33

Any 2 principal vectors t1 t2corresponding to k1 k2 are perpendicular

Question 34

If k1 = k2

Answer 34

Every tangent vector to σ at P is a principal vector