Volcano: What function y(x) minimizes the distance between two points on a cone of half-angle α? Find the shortest path between the points (x,y,z) = (0,-1,0) and (0,1,0) on a cone with the base of radius r=1 and half angle α = π/4. What is the length of that path?

x

y

z

..

x = r cos�

y = r sin�

z = h� r cot↵

ds

2= dx

2+ dy

2+ dz

2= dr

2+ r

2d�

2+ cot

2↵ dr

2

ds2 = (1 + cot

2 ↵)dr2 + r2d�2= (1 + cot

2 ↵)(dr2 + r2d�2)

� ⌘ (1 + cot

2 ↵)�1�

Solution is a straight line!

for our endpoints :

Shortest connecting length:

r cos(� � �0) = r0General solution: Straight line in polar coordinates: