Simple 3-D Visualization of Some Common
Mathematical Minimal Surfaces using MATLAB
Nasiha Muna
1
and Albert E. Patterson
2
1
Department of Physics, Chemistry, and Mathematics, Alabama Agricultural and Mechanical (A&M)
University, Normal, Alabama, 35762, USA. Email: nasiha.muna@aamu.edu
2
Department of Industrial and Enterprise Systems Engineering, University of Illinois at
Urbana-Champaign, Urbana, Illinois, 61801, USA. Email: pttrsnv2@illinois.edu
The MATLAB code for this work is available at: https://github.com/pttrsnv2/Minimal_
Surfaces_Visualization_Code. This report and its code are published under a CC-BY 4.0
International license, so it can be used and modified with proper attribution to the authors.
Abstract
This report presents a simple approach for visualizing some common mathematical mini-
mal surfaces using MATLAB tools. The studied minimal surfaces were the gyroid, lidinoid,
Schwarz P surface, Schwarz D surface, Scherk tower, helicoid surface, catenoid surface, and
Mobius strip. For each, the mathematic definition is given along with a simple matlab code
for it and an example 3-D surface plot generated from the given code.
1 Introduction
The selected minimal surfaces to be discussed in this report were the gyroid, lidinoid, Schwarz
P surface, Schwarz D surface, Scherk tower, helicoid surface, catenoid surface, and Mobius strip.
These were selected, as they represent a suitable subset of the available minimal surfaces and
serve as good cases to demonstrate the simple approach proposed in this report. Some very
complex surfaces, such as Costa surfaces, are not covered here and will be discussed in future
works. Most minimal surfaces are sometimes relatively difficult to plot in three dimensions, as they
are all typically defined by implicit or parametric functions (sometimes in the complex domain).
Fortunately, appropriate built-in MATLAB functions can be used to plot each of the surfaces
relative to its defining equation.
The gyroid and lidinoid functions are triply periodic minimal surfaces of genus 3 embedded in R
3
that possess no planar symmetry curves nor straight lines. They are the distinctive embedded
surfaces of the associate families of the Schwarz P and H surfaces [1–3]. The gyroid function is
typically defined as
f
g y
= sin(x) cos(y) + sin(y) cos(z) + sin(z) cos(x) = 0 (1)
while the lidinoid function is most often expressed as:
f
lid
=
1
2
(sin(2x) cos(y) sin(z) + sin(2y) cos(z) sin(x) + sin(2z) cos(x) sin(y))
−
1
2
(cos(2x) cos(2y) + cos(2y) cos(2z) + cos(2z)cos(2x)) + 0.15 = 0
(2)
The Schwarz minimal surfaces span many different kinds of forms, but the two most commonly-
seen are the Schwarz Primitive (“Schwarz P”) and Schwarz Diamond (“Schwarz D”) surfaces [3–5].
The Schwarz P surface is the triply periodic minimal surface built from a skew hexagon inscribed
in the edges of a regular octahedron. The Schwarz P surface encompasses sixfold junctions and is
typically expressed as:
1
- 1
- 2
- 3
- 4
- 5
- 6
前往页