Hyperboloid

g := (x,y) -> [sinh(x)*cos(y), sinh(x)*sin(y), 3*cosh(x)]:
a1:=0: a2:=1.2: b1:=0: b2:=2*Pi:
c1 := .95: c2 := 1.05:
Point := 1:
f1:= (t) -> t:
f2:= (t) -> 1:

Cone

Make a 90 degree and 180 degree cone
Cone parameterization
g := (x,y) -> [(1-x)*cos(y), (1-x)*sin(y), x]:
a1:=0: a2:=3: b1:=0: b2:=1:
c1 := 0: c2 := 1:
Point := 1/2:
f1:= (t) -> 1/2:
f2:= (t) -> t:
latitude circle - discuss why it is not a geodesic using intrinsic arguments, including the lack of half-turn symmetry and the fact that it unfolds to circle.
Play with a's b's and c's to convince yourself that it is a double-cone.
I just changed b -2..2

g := (x,y) -> [(1-x)*cos(y), (1-x)*sin(y), x]:
a1:=0: a2:=3: b1:=-2: b2:=2:
c1 := 0: c2 := 1:
Point := 1/2:
f1:= (t) -> t:
f2:= (t) -> 1/2:

What is the cone angle? Looks like it will be 2*45 = 90 degrees

Finding other geodesics and wrap around geodesics? This is hard! We will come back to this problem later when we have some equations to work with.

Helicoid

g := (x,y) -> [x*cos(2*Pi*y), x*sin(2*Pi*y), y]:
a1:=0: a2:=3: b1:=0: b2:=1:
c1 := 0: c2 := 1:
Point := 1/2:
f1:= (t) -> t:
f2:= (t) -> 1/2:
Note the changes in a's, etc. Where did the curvature vector go?

g := (x,y) -> [x*cos(2*Pi*y), x*sin(2*Pi*y), y]:
a1:=0: a2:=3: b1:=0: b2:=1:
c1 := 0: c2 := 1:
Point := 1/2:
f1:= (t) -> 1/2:
f2:= (t) -> t:
Where is the curvature vector (pink)? Take off Geovector. Take off , linestyle=DASH from ExtVector command too.
Covering: Twist and unfold. Discuss what these lines are when they are flat.
Helicoid in Cartesian coordinates
Bubbles