Notice “r” here does not mean the usual radial distance but rather the radial distance along the curve itself. I don’t see any obvious barrier to such a thing and I expect it exists, but that it actually does isn’t obvious to me; the resulting differential equation seems to have a problem at h=0, and not knowing much about differential equations I have no idea if it’s removable or not (since getting an explicit solution is not so easy).
You piqued my curiosity, so I sat to play a little with the equation. If R is the usual radial coordinate, I got:
dh/dR = c y^(1/3) / [(1 - c^2 y^(2/3))^(1/2)]
with y = 3h/(2c), using the definition of c in my previous comment. (I got this by switching from dh/dr to dh/dR with the relation between sin and tan, and replaciong r by r(h). Feel free to check my math, I might have made a mistake.) This was easily integrated by Mathematica, giving a result that is too long to write here, but has no particular problem at h = 0, other than dR/Dh being infinite there. That is expected, it just means dh/dR = 0 so the peak of the dome is a smooth maximum.
Notice “r” here does not mean the usual radial distance but rather the radial distance along the curve itself. I don’t see any obvious barrier to such a thing and I expect it exists, but that it actually does isn’t obvious to me; the resulting differential equation seems to have a problem at h=0, and not knowing much about differential equations I have no idea if it’s removable or not (since getting an explicit solution is not so easy).
You piqued my curiosity, so I sat to play a little with the equation. If R is the usual radial coordinate, I got:
dh/dR = c y^(1/3) / [(1 - c^2 y^(2/3))^(1/2)]
with y = 3h/(2c), using the definition of c in my previous comment. (I got this by switching from dh/dr to dh/dR with the relation between sin and tan, and replaciong r by r(h). Feel free to check my math, I might have made a mistake.) This was easily integrated by Mathematica, giving a result that is too long to write here, but has no particular problem at h = 0, other than dR/Dh being infinite there. That is expected, it just means dh/dR = 0 so the peak of the dome is a smooth maximum.