4) Surfaces In 3d Flashcards

1
Q

Define a surface as a set of points

A
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2
Q

Equation of a plane in R^3

A
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3
Q

Define a Surface patch

A
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4
Q

Define open disc

A
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5
Q

Define closed disc

A
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6
Q

Define a point on plane in vectors

A

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

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7
Q

Parameterize unit sphere by latitude θ and longitude φ

A

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

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8
Q

Reparameteization of a surface patch?

A
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9
Q

Reparam of unit sphere

A
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10
Q

Define a curve on a surface

A
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11
Q

Define a tangent space of a surface patch

A
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12
Q

Prove the tangent space of a surface patch

A
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13
Q

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

A
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14
Q

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

A
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15
Q

State the form of a generalised cone and how to derive

A
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16
Q

State the form of a quadratic surface and derive

17
Q

When is a quadratics surface a single point

A

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

18
Q

Form of elipsoide

19
Q

Form of hyperboloid of 1 sheet

20
Q

Form of hyperboloid of 2 sheets

21
Q

Form of elliptic parabaloid

22
Q

Form of hyperbolic parabaloid

23
Q

Form of quadric cone

24
Q

Form of elliptic cylinder

25
Form of hyperbolic cylinder
26
Form of parabolic cylinder
27
Parameterization of elipsoide
28
Parameterization of hyperboloid of one sheet
29
Surface of revolution is made by taking? Rotating?
Rotating a plane curve (the profile curve) around a straight line in the plane (z axis here)
30
Parallels? Meridians?
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
31
What is a ruled surface?
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