A minimal surface obeys everywhere the condition of zero mean curvature. In the nonparametric problem [4], the curvature constraint applies directly to a height function
,
, (1)
where
are first and second partial derivatives with respect to the Cartesian coordinates. A formal solution can be represented by a series:
, (2)
which should converge on a limited region around
. Constraint equation (1) introduces dependencies into the set of
, but does not entirely determine the shape of the surface. The remaining degrees of freedom depend uniquely on a boundary condition,
,
with
a
-periodic parameter around the boundary
, a closed curve in
.
Equations (1)–(3) locally determine a surface of zero mean curvature. A minimal surface in all
exists whenever the boundary
admits a valid tiling. Valid tilings obey a few additional constraints: every segment of
must coincide with a segment of exactly one adjacent
, the interiors of
and
must be mutually disjoint, and the surface must obey equation (1) across all adjacencies.
The Schwarz D surface is an simple example in which the boundary extends along the six edges of a cube with vertices
:
,
without visiting
or
. Taking the surface normal along the axis
, the projection
becomes a regular hexagon, while the height
alternates with a period of
. To solve for this boundary, we rewrite equation (2) as
, (3)
,
.
After satisfying equation (1), we obtain the height function displayed above. The remaining degrees of freedom
determine the shape along the boundary. Settling on a numerical method, we insert decimal values and apply variations until the approximate boundary most nearly matches the exact boundary. The procedure determines one- and two-parameter approximations:
,
,
,
,
.
The "boundary convergence" plot shows the difference between approximate and exact boundaries. To make the tiny difference more visible, we magnify by a factor of 10 and by a factor of 100. In principle, the series method extends to arbitrary precision, where the approximate boundary continues to converge to the exact boundary.
As a test of approximation validity, we directly integrate the surface area
,
where the definite integral goes over one hexagonal domain. The integral equals
(1 parameter),
(2 parameters).
The approximate calculation reaches excellent precision, differing by less than
percent from the exact value from [3],
,
where
is the complete elliptic integral of the first kind.
[3] P. J. F. Gandy, D. Cvijović, A. L. Mackay and J. Klinowski, "Exact Computation of the Triply Periodic D ('Diamond') Minimal Surface,"
Chemical Physics Letters,
314(5–6), 1999 pp. 543–551.
doi:10.1016/S0009-2614(99)01000-3.
[4] J. C. C. Nitsche,
Lectures on Minimal Surfaces, Vol. 1. (J. M. Feinberg, trans., A. Schmidt, ed.), New York: Cambridge University Press, 1989.
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