# Differential Geometry Help Needed!

• May 13th 2008, 01:35 PM
il_bello1234
Differential Geometry Help Needed!
Hey guys... Have a Diff Geometry exam coming up and looking at the past papers this keeps coming up but I cant get my head around it! Can anyone help......

Sample Question
Q: The plane through a point on a curve г c R^3 perpendicular to the tangent line is called the normal plane to the curve at the point.

(a) Show that a curve lies in a sphere if the intersection of all normal planes is non-empty.

(b) Hence, or otherwise show that the curve parametrized by
P(ө) = (cos 2ө, -2cosө, sin2ө), ө is an element of [0, 2*Pi] lies in a sphere. Find the centre and radius of the sphere!

AGH! (Crying)

From a Confused Student...
• Jun 7th 2009, 09:26 PM
Rebesques
a) Let the curve be parametrized by arclength, $\displaystyle x=x(s)$ and let $\displaystyle \{t,\eta,b\}$ be the Frenet-Serret frame. Then the plane $\displaystyle \{\eta,b\}$ always passes through a point $\displaystyle P$. Express $\displaystyle P=\lambda(s)\eta(s)+\mu(s)b(s)$ and differentiate to obtain $\displaystyle \lambda, \mu=0$. So $\displaystyle \{\eta,b\}$ always crosses the origin, which means $\displaystyle x\in\{\eta,b\}$ or $\displaystyle x(s)=\Lambda(s)\eta(s)+M(s)b(s)$. Use the Frenet-Serret equations to show that $\displaystyle \Lambda, M$ are constants.

(Actually, the curvature of $\displaystyle x$ turns out to be constant and the torsion zero, so $\displaystyle x$ is a circle.)

b) Let the coordinates be x,y,z. We easily see that this circle is the intersection of the circular cylinder $\displaystyle x^2+z^2=1$ with the parabolic cylinder $\displaystyle y^2=2(1+x)$.
(...unless i messed up my calcs again)