Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

who can prove an easy and beautiful observation on a sheet of paper in a few lines? I have used a computer algebra system to verify (the possible input is contained in the "proof") the following.

Let $C\subset\mathbb R^3$ be a cone (quadric) with half-angle $0<\varphi<\tfrac{\pi}{2}$, $P\in C$ a point in distance $\overline{oP}>0$ to the apex $o$ of $C$ and let $E$ be a plane containing $P$ but not the line $oP$. Then the orthogonal projection of the conic section $C\cap E$ onto a plane $F$ orthogonal to $oP$ has curvature $\cot(\varphi)/\overline{oP}$ at the intersection of $F$ and $oP$.

Proof: Introducing an orthogonal coordinate system $(x,y,z)$ and appying an isometry, $C$ is the locus of $$ x^2 + (y\cos(\varphi) + z\sin(\varphi))^2 = \tan(\varphi)^2(z\cos(\varphi) - y\sin(\varphi))^2, $$ with the apex $o$ equal to the origin and $P:=(0,0,-d)$ lying on the $z$-axis for some $d>0$. The plane $E$ is the locus of $$ x\eta+y\xi-(z+d)=0 $$ for some $\eta,\xi\in\mathbb R$. The conic section $C\cap E$ is locally around $P$ parametrized by the $x$-coordinate and we have $\tfrac{\partial y}{\partial x}(0)=0$ and $\tfrac{\partial^2 y}{\partial x^2}(0)=\cot(\varphi)/d$, using a computer algebra system. $\square$

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.