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 = R2 (dφ)2 + (r2 + R2cos2φ + 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 +
(u2 + h2) (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: a1 1 (du)2 = a2 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.