### 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