DIFFERENTIAL GEOMETRY COURSE

14. *TORUS AND HELICOID SURFACE*

*Explanation 121 :* In this section we learn some interesting new surfaces. Thereafter we get some problems for repetition and enrichment about surfaces in general.

*Definition 122 :* Let C be the circle in the x,y-plane with radius r and center (0,0,0). The torus is the geometric locus of the circles with radius R (smaller than r) and center on C,
lying in a plane through the z-axis.

*Explanation 123 :* The torus is a sort of life buoy like the ones we use when learning to swim.

*Problem 124 :* Show we can parametrize the torus by (x,y,z) = (r cos θ + R cos θ cos φ, r sin θ + R sin θ cos φ, R sin φ).

What do the parameter lines look like?

Check by calculation that the first fundamental form is: (ds)^{2} = R^{2} (dφ)^{2} +
(r^{2} + R^{2}cos^{2}φ + 2rR cos φ) (dθ)^{2}.

Do the parameter lines form an orthogonal net?

*Definition 125 :* A helicoid surface (whose axis is the z-axis) has a parametrization in the form (x,y,z) = (u cos v, u sin v, f(u) + hv) (h constant).

If h=0 we have a surface of revolution.

*Explanation 126 :* We get a helicoid surface by "screwing" a planar curve (u, 0, f(u)) around the z-axis.

The u-lines are congruent to this curve, the v-lines are circular helices.

*Problem 127 :* Show that the line element of a helicoid surface is given by (ds)^{2} = (1 + (f ' )^{2}) (du)^{2} + 2 h f ' du dv +
(u^{2} + h^{2}) (dv)^{2}.

In which cases do the parameter lines form an orthogonal net?

*Problem 128 :* In 125, take f(u) = 0. Determine the curves on this right helicoid surface that make an angle of 45 degrees with the v-lines.

*Problem 129 :* Determine the line element of (a(u+v), b(u-v), uv) and the geometric locus of the points where the parameter lines are perpendicular to each other.

*Problem 130 :* Prove that the curves that in every point make equal angles with the parameter lines obey the following differential equation: a_{1 1} (du)^{2} =
a_{2 2} (dv)^{2}. This net is called the bissectrix net.

*Problem 131 :* Prove that the net (du)^{2} = (dv)^{2} on (a (cos u + cos v), a (sin u + sin v), b(u + v)) is orthogonal.