How to compute curvature?
May 3, 2010 3:44 AM Subscribe
MathFilter - Help me understand this equation.
I'm using Matlab, btw.
I have some (m by 3) data that represents a 3D curve. What I'm supposed to do is parameterize by arc length and get r(s) = [x(s), y(s), z(s)]. I've done that. Then I'm supposed to fit a polynomial to it, which I believe I've done correctly, and I have three polynomials for x, y and z.
I'm now supposed to compute the curvature, at evenly spaced points along the curve, and the curvature is given by equation 27 on this page
Curvature
but I want to spell it out here...
k = |dr/dt x d^2r/dt^2| / |dr/dt|^3
Wolfram uses t as the variable for parameterization, even though they refer to s higher up in the page as arc length.
I guess my first question is what's the difference here between s and t and what pitfalls do I risk hitting?
As I have the coefficients of the polynomials, I can compute the derivatives no problem, but I'm just confused by the notation or something. At each point, aren't dr/dt and d^2r/dt^2 scalars? Is that a cross product notation or is there a convention to just not put those derivates next to one another? What do the magnitude bars mean here?
Please feel free to ask any followup questions if what I've written is unclear.
Thank you.
I'm using Matlab, btw.
I have some (m by 3) data that represents a 3D curve. What I'm supposed to do is parameterize by arc length and get r(s) = [x(s), y(s), z(s)]. I've done that. Then I'm supposed to fit a polynomial to it, which I believe I've done correctly, and I have three polynomials for x, y and z.
I'm now supposed to compute the curvature, at evenly spaced points along the curve, and the curvature is given by equation 27 on this page
Curvature
but I want to spell it out here...
k = |dr/dt x d^2r/dt^2| / |dr/dt|^3
Wolfram uses t as the variable for parameterization, even though they refer to s higher up in the page as arc length.
I guess my first question is what's the difference here between s and t and what pitfalls do I risk hitting?
As I have the coefficients of the polynomials, I can compute the derivatives no problem, but I'm just confused by the notation or something. At each point, aren't dr/dt and d^2r/dt^2 scalars? Is that a cross product notation or is there a convention to just not put those derivates next to one another? What do the magnitude bars mean here?
Please feel free to ask any followup questions if what I've written is unclear.
Thank you.
Yeah, in that link, "t" is an arbitrary parameter that the curve is originally defined with, whereas "s" is the actual arc length along the curve. Wolfram starts with "t" because a curve isn't necessarily parameterized with arc length to begin with, so that derivation is more useful in general.
And as Dr. Dracator says, "r(t)" is a vector, and so are its derivatives, thus the need for a cross product. And the magnitude bars are there because the curvature is a scalar, so you'll need to take the magnitude of the vectors to get the curvature.
posted by Salvor Hardin at 4:24 AM on May 3, 2010
And as Dr. Dracator says, "r(t)" is a vector, and so are its derivatives, thus the need for a cross product. And the magnitude bars are there because the curvature is a scalar, so you'll need to take the magnitude of the vectors to get the curvature.
posted by Salvor Hardin at 4:24 AM on May 3, 2010
This thread is closed to new comments.
Changing the curve parametrization doesn't change the curvature, the extra ds/dt, d2s,/d2t terms due to re-parametrization cancel out. Of course, you have to keep track of which s-values correspond to which t-values: curvature as a function of t will be different from curvature as a function of s but the values at corresponding points will be equal.
All this can be written out in beautiful, precise notation that AskMe doesn't really support, so I'm not even going to try giving the equations, but it's standard fare you will find in most textbooks.
[This is the essence of differential geometry btw, that things are independent of their representation. ]
posted by Dr Dracator at 3:59 AM on May 3, 2010