# References for the basic theory of surfaces of revolution, cylinders and cones

I'm looking for references to books were the following types of problems about finding the equation defining a surface of revolution, a cylinder or a cone are treated. These are problems that are usually presented every semester in the University of Costa Rica's multivariable calculus course for engineers.

When I was in Costa Rica I asked the younger professors about references and nobody seemed to know, I have not seen such problems in calculus textbooks, and everybody seemed to have some old notes they had borrowed from somebody else when they faced the same situation of having to teach multivariable calculus and had to present these topics in class.

The types of problems I'm talking about are of the following sort.

1. Find the equation of the cylinder whose directrix is the curve $$\begin{eqnarray} x^2 + y^2 + 2z^2 &= 8\\ x - y + 2z &= 0 \end{eqnarray}$$ and whose generatrices are parallel to the line $(x, y, z) = (-3, 1, 5) + t(2, 1, -4), \quad t \in \mathbb{R}$.

2. Calculate the equation of the surface of revolution that results from rotating the line $$\begin{eqnarray} x + y + z &= 0\\ y - z &= 0 \end{eqnarray}$$ around the axis that is the intersection of the planes $x + y = 1$ and $z = 0$.

I actually know how to solve such problems by forming a system of equations and eliminating variables until one ends up with an equation involving only $x, y, z$ say, but I would like to have some references where the general theory of surfaces of revolution, cones and cylinders is treated.

Thank you very much for any help.

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I don't know about refs, but if presented with a problem of constructing a surface of revolution, I'd start by constructing parametric equations first for the curve to be rotated, and then worry about eliminating parameters later. –  Ｊ. Ｍ. Apr 18 '11 at 5:16
@J. M. Thanks for the edit. I guess I would have never noticed such a typo in the tags =) –  Adrián Barquero Apr 18 '11 at 5:22
Have a look at The differential geometry of curves and surfaces by M.P. do Carmo, I think I learned it from that book. Google doesn't let me check, however... –  t.b. Apr 18 '11 at 8:16
@Theo Thank you very much. I'll take a look at the book. –  Adrián Barquero Apr 18 '11 at 14:30

You could try Delaunay's "Sur la surface de revolution dont la courbure moyenne est constante", but it's only for surfaces of revolution of constant mean curvature. However, that being said a lot of interesting things are mentioned in here, one of which is the result from the calculus of variations.

If I attempt to solve the problem of finding a surface of revolution whose area is a minimum for constant volume, I get a problem in optimisation with a contraint. In fact the euler lagrange equations will result in a differential equation of the form

$\frac{2}{\sqrt{1+y'^2}} + \lambda y = C$, where $\lambda$ is some constant (The lagrange multiplier) and $C$ is some other constant.

Different values of $C$ give rise to different surfaces of revolution, $C=0$ gives rise to a circle rotated about the $x-axis$ or rather a sphere.

I'm not sure if that helps. Ben

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As an additional note: the CMC surfaces by Delaunay are surfaces whose meridian curves are roulettes of conics (curves traced by the focus of a conic rolling on a line). The catenoid is one such Delaunay surface (since the focus of a parabola rolling on a line traces a catenary, the meridian of a catenoid). The surfaces corresponding to ellipses and hyperbolas have parametric equations involving elliptic integrals. –  Ｊ. Ｍ. Apr 18 '11 at 6:44