[deity47]

Please help me with this intrinsic co-ordinates question. There not too bad so far but for this one, I have worked through to get an answer, albeit incorrect....

10. Find the radius of curvature at the point t=T on the curve with equations
x = 3cost - cos3t
y = 3sint - sin3t
where t is a parameter.

I have simply worked out dx/dt, dy/dt and the second derivatives, and put them into the equation for radius of curvature. After some cos^2 + sin^2 = 1 cancelling, and using the cos(A-B) equation I have got an answer of 1/12sinT.
However, in the back of the book it states 3sinT.

Please help! I cannot see where my working has gone wrong.
Thanks in advance

[RichE]

Originally Posted by deity47

Please help me with this intrinsic co-ordinates question. There not too bad so far but for this one, I have worked through to get an answer, albeit incorrect....

10. Find the radius of curvature at the point t=T on the curve with equations
x = 3cost - cos3t
y = 3sint - sin3t
where t is a parameter.

x' = -3sint + 3sin3t
x'' = -3cost + 9cos3t

y' = 3cost - 3 cos3t
y'' = -3sint + 9sin3t

rho = (x'^2+y'^2)^(3/2)/|x'y''-y'x''|

x'^2 + y'^2
= 9[(sin3t-sint)^2 +(cos3t-cost)^2]
= 9[2 - 2sintsin3t - 2cos3tcost]
= 18[1- cos(3t-t)]
= 18[1 - cos2t]
= 36 sin^2t

x'y''-y'x''
= (-3sint + 3sin3t)(-3sint + 9sin3t) - (3cost - 3 cos3t)(-3cost + 9cos3t)
= 9[(sin^2t -sin3tsint + 3sin^2(3t) - 3sintsin3t) + (cos^2t - cos3tcost - 3cost cos3t + 3cos^2(3t))]
= 9[4 - 4cos3tcost -4sin3tsint]
= 72sin^2t <arguing as previously>

Hence answer equals

rho = (36 sin^2t)^(3/2)/(72 sin^2t)

= 216 sin^3t /(72 sin^2t) = 3sint

[deity47]

Oh my! I miss read the textbook, power of 1/2 instead of 3/2. The answer fell out from there!

Cheers

PS Liverpool WIN!!!