- Which parametrization (patch) did you like the best?
- Monge patch x
*(u,v) = (u, v, f(u,v))* - geographical coordinates with 2 angles and a radius from a center
like
x
*(u,v) = (R cos u cos v, R sin u cos v, R sin v)* - surface of revolution
x
*(u,v) = (g(u), h(u) cos v, h(u) sin v)*from a planar curve*alpha(u) = (g(u), h(u), 0)* - ruled surface
x
*(u,v) = beta(u) + v delta(u)*, where beta and delta are curves and x*(u,v)*is lines emanating from the directrix beta going in the direction of delta - other

Next write down examples of surfaces for each type of parametrization.

- Monge patch x
- What is the equation of a geodesic that an
arbitrary point
*y(theta,r)*satisfies, where*d*and*beta*are defined as in the hw and following picture:*theta=r**r=d sec (theta-beta)**r=d cos (theta-beta)**d=r sec (theta-beta)**arctan(s/d) = n(alpha/2)*

- In general on a cone of small enough cone angle,
a geodesic will self-intersect...

- they will generally not intersect
- at points vertically removed from each other
- at points horizontally removed from each other
- each time its lift crosses the seam of the covering
- infinitely many times

1

a: paraboloid

b: sphere

c: catenoid from catenary y=cosh(x)

e: helicoid, cone, cylinder

2. b

3. d