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Curvature for arbitrary regular parametrizations
function in vector form
Curvature at a point on the graph y = f(x)
function in standard form
When is a parametrization r(t) regular?
r’(t) is not equal to 0
When r(t) is a regular parametrization, what is T(t) (the unit tangent vector) defined as?
derivative of vector parametrization divided by the magnitude of the derivative of the vector parametrization
Curvature in terms of the s function
K = ||T’(s)||
s (the arc length function) is equal to
integrate
How to find t in terms of s
isolate for t with the original function s(t)
N(t) (unit normal vector) =
T’(t)/||T’(t)||
Binormal Vector
B(t) =
B(t) = N(t) x T(t)
How to find center of osculating circle?
r(t0) + 1/Kp * N
Equation of a plane
(a,b,c) * (x-x0, y-y0, z - z0)
How to find equation of osculating plane
Dot product of Binomial vector + point on graph
T’(t)
T’(t) = k(t) * v(t) * N(t)
aT =
(remember the 3 main formulas, mainly the first two)
3 main formulas
aTT =
help
aN =
2 main formulas
in terms of curvature and sqrt version
aN =
cross product formula
Not in the textbook