Exam Prep Flashcards

1
Q

Process to find unit speed reparam

A
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2
Q

Process to find curvature

A
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3
Q

Process to Find right handed orthonormal basis

A
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4
Q

Frenet-serret equations in R^3

A
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5
Q

Nowhere vanishing curvature implies

A

Torsion is always defined

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6
Q

Greene’s theorem

A

For all smooth functions f and g

Provided that γ is traversed anti clockwise

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7
Q

Unit normal to σ

A
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8
Q

Unit normal

A
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9
Q

A surface patch is regular if

A
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10
Q

Regular point ?

A

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

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11
Q

Process to find torsion

A

Either of

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12
Q

Conformal?

A

A conformal parameterization has E=G and F=0

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13
Q

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

A

du2 + dv2

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14
Q

Calculate surface area

A
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15
Q

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

A
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16
Q

2 surfaces are isometries if

A
17
Q

What is a tangent developable

A

The union of the tangent lines to a curve in R3

18
Q

Find the parameterization of a tangent developable

A
19
Q

Any tangent developable is isometric to

A

(Part of) a plane

20
Q

Calculate 2nd fundamental form

A

Where N- is the principal unit normal

21
Q

Normal curvature

A

For γ a unit speed curve on σ

22
Q

Geodesic curvature

A

For a unit speed γ on σ

23
Q

Relate normal curvature to geodesic curvature

A
24
Q

Matrices for fundamental forms

A
25
Q

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

A
26
Q

Principal curvatures

A
27
Q

Euler’s Theorem

A
28
Q

Gaussian curvature and mean curvature

A
29
Q

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

A
30
Q

For σ~ = λ σ
Give F1 and F2

A
31
Q

Gaussian curvature for a sphere and for a cylinder

A

Sphere constant positive
Cylinder constant zero

32
Q

Find principal vector corresponding to principal curvature

A
33
Q

If k1 doesn’t equal k2

A

Any 2 principal vectors t1 t2corresponding to k1 k2 are perpendicular

34
Q

If k1 = k2

A

Every tangent vector to σ at P is a principal vector