Exam Prep Flashcards
Process to find unit speed reparam
Process to find curvature
Process to Find right handed orthonormal basis
Frenet-serret equations in R^3
Nowhere vanishing curvature implies
Torsion is always defined
Greene’s theorem
For all smooth functions f and g
Provided that γ is traversed anti clockwise
Unit normal to σ
Unit normal
A surface patch is regular if
Regular point ?
Component vectors of cross product of partial derivatives of surface patch must be linearly independent
I.e. cross product never equals zero
Process to find torsion
Either of
Conformal?
A conformal parameterization has E=G and F=0
1st fundamental form of a plane (in standard coordinates)
du2 + dv2
Calculate surface area
For a curve on a regular surface patch, find the first fundamental form
2 surfaces are isometries if
What is a tangent developable
The union of the tangent lines to a curve in R3
Find the parameterization of a tangent developable
Any tangent developable is isometric to
(Part of) a plane
Calculate 2nd fundamental form
Where N- is the principal unit normal
Normal curvature
For γ a unit speed curve on σ
Geodesic curvature
For a unit speed γ on σ
Relate normal curvature to geodesic curvature
Matrices for fundamental forms
Show that 2 tangent vectors are linear combinations of partial derivatives of surface patch
Principal curvatures
Euler’s Theorem
Gaussian curvature and mean curvature
Find explicit formulas for Gaussian curvature and mean curvature using 1st and 2nd fundamental forms
For σ~ = λ σ
Give F1 and F2
Gaussian curvature for a sphere and for a cylinder
Sphere constant positive
Cylinder constant zero
Find principal vector corresponding to principal curvature
If k1 doesn’t equal k2
Any 2 principal vectors t1 t2corresponding to k1 k2 are perpendicular
If k1 = k2
Every tangent vector to σ at P is a principal vector
