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)(h2^2-h1^2)* dx + (h1^2+h2^2) * Sin(2y0)/4 + (h1^2 + h2^2)/4 * Sin 2y
    X2 = Cos(2y0) * (integral, x0,y0,x,y0)(h2^2-h1^2)* 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)(h2^2-h1^2)* dx = -(h1^2+h2^2) * Sin(2y0)/4

(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.