Members - Reputation: 101
Posted 19 September 2012 - 02:34 AM
I got a problem and I don't know how to figure it out.
In short I have an arc r(t)=[x(t), y(t), z(t)]
i'm gonna calculate the radius of this arc. As we know the curvature is the inverse of radius. If we consider the following formula:
K(t)=||r'(t) x r''(t)|/||r'(t)||^3
when ' means the first derivative to time and '' means the second derivative to the time and the the X is cross product.
So by having this equation and considering one simple example: A circle at the origin of a 2D space:
x(t)= R (1-t^2)/(1+t^2)
when tE[0,1] and R is the radius. (EQ1)
By the formula of curvature K(t) is expected to be 1/R but when I compute it strangely for t=0 I get 4*R and for the t=1 I get R and for the time between two margins the K(t) is a non-linear distribution from 4R to R.
I have no idea why it happens. It's very simple example and I am wondering whether I do a silly mistake in my calculations or it's just a wrong assumption in my calculation;.
I created one Maple file to do so. And the result is flawless in sense of numerical and parametric calculation I guess.
In addition, by selecting another form of a circle equation by sinusoidal ones. I get better consistency but it's not even the answer. So if I select
The comuted curvature for the whole range of t is R!!! it's wrong because we expect 1/R not R.
So I looked for any possible example. Finally I found one in a literature. They change the parameters and the new circle equation is:
And the result is correct we get the curvature by this equation 1/R!!
So I'm really confused why three different answers I get by the same formula when the resulting geometrical shape is the same. Please help me out thank you in advance.
all the best
Members - Reputation: 1366
Posted 19 September 2012 - 03:09 AM