5) The First Fundamental Form

Question 1

General nomenclature for rest of module unless stated otherwise, surface means

Answer 1

Regular surface patch

Question 2

How to calculate first fundamental form of a surface (patch)

Answer 2

Question 3

The first fundamental form depends on? But not?

Answer 3

Depends on surface (and it’s parameterization) but not the curve γ

Question 4

Calculate first fundamental form of plane

Answer 4

Question 5

Calculate first fundamental form of sphere in latitude longitudinal coordinates

Answer 5

Question 6

Calculate first fundamental form of generalised cylinder

Answer 6

Question 7

Changing parameterization of surface changes? In first fundamental form

Answer 7

Coefficients E F G, but not ds^2

Question 8

A surface parameterization is conformal if

Answer 8

E=G , F=0

Question 9

Prove conformal parameterization

Answer 9

Question 10

Requirements for a surface to have a conformal parameterization

Answer 10

ALL SURFACE HAVE

Question 11

Define a diffeomorphism

Answer 11

A map between the images of 2 surfaces that is bijective and smoother

Question 12

When are 2 images of surfaces isometric?

Answer 12

Question 13

A tangent developable is

Answer 13

The union of tangent lines to a curve in R^3

Question 14

Prove that any tangent developable is isometric to (part of) a plane

Answer 14

Question 15

Find the surface area of part of the surface corresponding to a region

Answer 15

Question 16

Answer 16

Question 17

Prove

Answer 17

Question 18

=?

Answer 18

Question 19

Prove that dA_σ remains unchanged from re parameterization

Answer 19

Question 20

Prove archimedes theorem

Answer 20

Question 21

Prove

Answer 21