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Curvature of Ellipse

by soe236
Tags: curvature, ellipse
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soe236
#1
Dec10-08, 10:20 AM
P: 24
Find Curvature of Ellipse given x=3*cos(t) and y=4*sin(t) at the points (3,0) and (0,4)

Relevant equations: curvature at r(s) is k(s)=||dT/ds|| when r(s) is arc length parametrization and T is the unit tangent vector
I usually use the formula k(t)= (||r'(t) x r''(t)||)/||r'(t)||^3

So, r(t)=<3cost,4sint> and r'(t)= <-3sint,4cost> and r''(t)=<-3cost,-4sint>
do I just plug them in for k(t)? And I'm clueless about how to use the given points. Someone help please. Thank you
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Dick
#2
Dec10-08, 10:29 AM
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Yes, you can just plug them into k(t). You'll need to do a cross product, so take the z component of the vectors to be zero. When you get k as a function of t, then you just need to go back and set (3*cos(t),4*sin(t))=(3,0) and figure out what t is for the first point. Ditto for the second.
soe236
#3
Dec10-08, 11:12 AM
P: 24
Thanks, but I'm not sure I completely understand

This is what I got so far, please correct me if I'm wrong:
r'(t) x r''(t)= 12sint^2+12cost^2 = 12
||r'(t) x r''(t)||= sqrt(144) =12
||r'(t)||^3 = (9sint^2+16cost^2)^(3/2)

k(t)= 12/(9sint^2+16cost^2)^(3/2)

from what you've said, (3*cos(t),4*sin(t))=(3,0), so 3cost=3 => t= 0 or 2pi and 4sint=0 => t=0 or 2pi
similarly (3*cos(t),4*sin(t))=(0,4), so 3cost=0 => t=pi/2 or 3pi/2 and 4sint=4 => t=pi/2

Sry if this is a stupid question, but how do I apply that to k(t)?

Dick
#4
Dec10-08, 01:27 PM
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P: 25,228
Curvature of Ellipse

Quote Quote by soe236 View Post
Thanks, but I'm not sure I completely understand

This is what I got so far, please correct me if I'm wrong:
r'(t) x r''(t)= 12sint^2+12cost^2 = 12
||r'(t) x r''(t)||= sqrt(144) =12
||r'(t)||^3 = (9sint^2+16cost^2)^(3/2)

k(t)= 12/(9sint^2+16cost^2)^(3/2)

from what you've said, (3*cos(t),4*sin(t))=(3,0), so 3cost=3 => t= 0 or 2pi and 4sint=0 => t=0 or 2pi
similarly (3*cos(t),4*sin(t))=(0,4), so 3cost=0 => t=pi/2 or 3pi/2 and 4sint=4 => t=pi/2

Sry if this is a stupid question, but how do I apply that to k(t)?
That looks ok to me. So the question is just asking you what is k(0) and k(pi/2), right? Those are the curvatures of the ellipse at the two given points.
soe236
#5
Dec10-08, 02:52 PM
P: 24
Oh I see.. okay, thank you very much!


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