QUESTIONS AND PROBLEMS 395(xg, J/9), the ninth point of intersection of two cubic curves through the other eight
points, for which the rank will remain at 8.]
12.8. Find the Gaussian curvature of the hyperbolic paraboloid æ = (÷^2 - y^2 )/a
at each point using ÷ and y as parameters.
12.9. Find the Gaussian curvature of the pseudosphere obtained by revolving a
tractrix about the x-axis. Its parameterization can be taken as
r(u, v) = — atanh ^—^,asech ^—j cos(w),asech ^—j sin(u)^.Observe that the elements of area on both the pseudosphere and its map to the
sphere vanish when u = 0. (In terms of the first and second fundamental forms,
Å = 0 = g when u = 0.) Hence curvature is undefined along the circle that is the
image of that portion of the parameter space. Explain why the pseudosphere can
be thought of as "a sphere of imaginary radius." Notice that it has a cusp along
the circle in which it intersects the plane ÷ = 0.
12.10. Prove that the Euler relation V—£, + F = 2fora closed polyhedron is
equivalent to the statement that the sum of the angles at all the vertices is (2V—4)ð,
where V is the number of vertices. [Hint: Assume that the polygon has F faces,
and that the numbers of edges on the faces are ei,.. .,ep. Then the number of
edges in the polyhedron is Å = (e\ + V ep)/2, since each edge belongs to two
faces. Observe that a point traversing a polygon changes direction by an amount
equal to the exterior angle at each vertex. Since the point returns to its starting
point after making a complete circuit, the sum of the exterior angles of a polygon
is 2ð. Since the interior angles are the supplements of the exterior angles, we see
that their sum is å^ð — 2ð = (e* — 2)ð. The sum of all the interior angles of the
polyhedron is therefore (2E — 2F)7r.]
12.11. Give an informal proof of the Euler relation V - Å + F — 2 for closed
polyhedra, assuming that every vertex is joined by a sequence of edges to every
other vertex. [Hint: Imagine the polyedron inflated to become a sphere. That
stretching will not change V, E, or F. Start drawing the edges on a sphere with a
single vertex, so that V = 1 = F and Å = 0. Show that adding a new vertex by
distinguishing an interior point of an edge as a new vertex, or by distinguishing an
interior point of a face as a new vertex and joining it to an existing vertex, increases
both V and Å by 1 and leaves F unchanged, while drawing a diagonal of a face
increases Å and F by 1 and leaves V unchanged. Show that the entire polyhedron
can be constructed by a sequence of such operations.]