4) Surfaces In 3d

Question 1

Define a surface as a set of points

Answer 1

Question 2

Equation of a plane in R^3

Answer 2

Question 3

Define a Surface patch

Answer 3

Question 4

Define open disc

Answer 4

Question 5

Define closed disc

Answer 5

Question 6

Define a point on plane in vectors

Answer 6

P and q are vectors parallel to plane
a is a reference point vector
r is the desired point vector

Question 7

Parameterize unit sphere by latitude θ and longitude φ

Answer 7

-π/2<=Θ<=π/2
0<=φ<=2π
To ensure that the function σ is injective
(However this is not an open set)

Question 8

Reparameteization of a surface patch?

Answer 8

Question 9

Reparam of unit sphere

Answer 9

Question 10

Define a curve on a surface

Answer 10

Question 11

Define a tangent space of a surface patch

Answer 11

Question 12

Prove the tangent space of a surface patch

Answer 12

Question 13

A surface patch is regular if?
Define unit normal to σ

Answer 13

Question 14

State the form of a generalised cylinder as a surface?
How to derive?

Answer 14

Question 15

State the form of a generalised cone and how to derive

Answer 15

Question 16

State the form of a quadratic surface and derive

Answer 16

Question 17

When is a quadratics surface a single point

Answer 17

If A is identity matrix and b(vector) and c are zero

Question 18

Form of elipsoide

Answer 18

Question 19

Form of hyperboloid of 1 sheet

Answer 19

Question 20

Form of hyperboloid of 2 sheets

Answer 20

Question 21

Form of elliptic parabaloid

Answer 21

Question 22

Form of hyperbolic parabaloid

Answer 22

Question 23

Form of quadric cone

Answer 23

Question 24

Form of elliptic cylinder

Answer 24

Question 25

Form of hyperbolic cylinder

Answer 25

Question 26

Form of parabolic cylinder

Answer 26

Question 27

Parameterization of elipsoide

Answer 27

Question 28

Parameterization of hyperboloid of one sheet

Answer 28

Question 29

Surface of revolution is made by taking?
Rotating?

Answer 29

Rotating a plane curve (the profile curve) around a straight line in the plane (z axis here)

Question 30

Parallels? Meridians?

Answer 30

On a surface of revolution,
Parallels are curves that are taken by rotating a single point
Meridians are curves that run down the rotation axis

Question 31

What is a ruled surface?

Answer 31

Union of straight lines (rulings of surface)
γ is a curve that meets these rulings
δ(u) is a non zero vector in direction of line passing through γ(u)
A point on this line has position vector