Theory
For any conformal map X, from C to R3, (the laplacian)X=2(lambda
squared)H(vector N) where (lambda)=< X (derivative with respect to x),
X(derivative with respect to x)> , H is the mean curvature, and N is the
normal vector. We now define U as the mean curvature half density
of the surface, equal to (lambda) * H. We can obtain a conformal
map by taking two arbitrary holomorphic functions of x and y, (psi1) and
(psi2), and plugging them into the formula:
X1 = (integral)(x0,y0,x,y) i/2 * ((psi2 bar)^2 + (psi1)^2) * dz  ((psi2)^2
+ (psi1 bar)^2)) * d(z bar)
X2 = (integral) (x0,y0,x,y)1/2 * ((psi2 bar)^2  (psi1)^2) * dz + (psi2)^2
 (psi1 bar)^2) * d(z bar)
X3 = (integral) (x0,y0,x,y)(psi2 bar)*(psi1)*dz + (psi2)*(psi1 bar)
d(z bar).
We now assume that (psi1) and (psi2) satisfy the differential equations:
(d/dz)(psi2)U*(psi1) = 0
(1)
(d/d(z bar))(psi1)U*(psi2) = 0
(2)
If we assume (psi1) and (psi2) are of the form:
(psi1) = h1(x)*e^iy
(3)
(psi2) = h2(x)*e^iy
(4)
Using (3) and (4), (1) and (2) can be rewritten in terms of h1 and h2
as follows:
h1' = 2U*h2 + h1
(5)
h2' = 2U*h1  h2
(6)
Using (1)(6), we obtain a surface of revolution as follows:
X1 = Sin(2y0) * (integral, x0,y0,x,y0)(h2^2h1^2)*
dx + (h1^2+h2^2) * Sin(2y0)/4 + (h1^2 + h2^2)/4 * Sin 2y
X2 = Cos(2y0) * (integral, x0,y0,x,y0)(h2^2h1^2)*
dx + (h1^2+h2^2) * Cos(2y0)/4 + (h1^2 + h2^2)/4 * Cos 2y
X3 = (integral) 2*h1*h2
We have the differential equations (5) and (6) satisfied by h1 and h2:
Sin2y * (integral, x0,y0,x,y0)(h2^2h1^2)* dx =
(h1^2+h2^2) * Sin(2y0)/4
So:
(h1^2 + h2^2)/4 *
Sin 2y
X={ (h1^2 + h2^2)/4 * Cos 2y
(integral) 2*h1*h2
As a result, the parametrization is a surface of revolution.
Thus, given the mean curvature half density of a surface, U, we can
solve the differential equations and obtain a parametrization for a surface
of revolution.
We have explored some of the constant mean curvature half density surfaces,
click
here to see them.
We have also made a program where the user can produce these kinds of
surfaces, to see the program click here.
