Three part Multivariable Calculus: Parametric Curves?
Please show me how to do this step by step, as clear and concise as possible.
i)Computer the Curvature
ii)Understand and explain the result independently of the curvature computation, for example, by graphing the parametric curve, recognizing it geometrically, and finding the curvature from the geometric information.
iii) Verify the geometric assertion in ii) by direct algebraic manipulation
http://imageshack.us/a/img846/700/img2012100504230...
Thanks
Comments
1) x = (1 - t^2)/(1 + t^2), y = 2t/(1 + t^2).
This is a parameterization of the unit circle, which can be recognized by noting that x^2 + y^2 = 1.
Not too surprisingly, this has constant curvature 1.
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2) r(t) = <(2 - t^2)/(1 + t^2), (2t^2 - 2)/(1 + t^2), (3t^2 - 2)/(1 + t^2)>
This is a parameterization of a straight line; see link:
http://www.wolframalpha.com/input/?i=parametric+pl...
Not too surprisingly, this has constant curvature 0.
Algebraic justification:
Rewrite as
r(t) = <-1 + 3/(1 + t^2), 2 - 4/(1 + t^2), 3 - 5/(1 + t^2)>
Letting t' = 1/(1+t^2), we can reparameterize this as
r(t') = <-1 + 3t', 2 - 4t', 3 - 5t'>, an equation for a line.
I hope this helps!