The document summarizes research on the surface areas of several minimal surfaces. Tangent vectors, normal vectors, and surface area formulas are defined. Seven surfaces are analyzed - helicoid, catenoid, Catalan's surface, Henneberg surface, Enneper surface, Scherk surface, and pseudosphere. For each surface, the parameterization, plot, vectors, and bounds are documented. Surface areas could not be computed for some due to complexity.

1. Banks Osborne, Christain Braden
Dr. Russell Herman
MAT 365-001
15 November 2016
Minimal Surfaces Project
In this project, we studied the surface areas of several minimal surfaces and the effects
parameters have upon the overall shape. Throughout this report, we will refer to the following
formulae in order to make the same computation for different surfaces. To find the tangent vector
with respect to u, Tu, we will take the partial derivative of each component of x with respect to u
and similarly for finding the tangent vector with respect to v, Tv. Computing the normal vector N
to the the surface is done by taking the cross product of u and v as follows:
N = Tu x Tv .
Similarly, the magnitude of N is computed by the equation
|N| = (Tu x Tv)1/2 .
Finally, in order to actually compute each surface area in Maple, we used the formula
A(S) = ∫ ∫ |𝐍|dudv.
All work, handwritten and Maple, and may be found in the Appendix at the conclusion of this
study, with the corresponding figures of the surfaces distributed as needed in each section of the
study’s main body. Additionally, for certain results of various computations found to be tedious
and long, such as certain normal vectors and surface areas, results not be expressly given in the
main discussion of this study; so the Maple syntax will be referred to in their place.
Section 1. Helicoid.
Our first surface was a helicoid. The helicoid is a surface in the form of

2. x = ( 𝑢cos( 𝑣), 𝑢sin( 𝑣), 𝑣).
To graph the surface in Maple, we set the bounds as follows: 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 2𝜋. The
helicoid plot may be seen in Figure 1 below.
Figure 1. Helicoid.
We found the generalized tangent vectors, as well. The tangent vector with respect to u is
𝐓u = (cos(v), sin(v), 0) ,
and the tangent vector with respect to v is
Tv = (-usin(v), ucos(v), 1) .
The Normal vector to the helicoid is:
𝐍 = (sin( 𝑣), − cos( 𝑣), 𝑢) .

3. After finding the previously mentioned vectors, we found the surface area to be
S(A) = π√2 + ln(√2 + 1).
One interesting fact about this surface is that tangent vectors exist at every point. This is seen by
how both Tu and Tv are defined at all points. Some interesting observations about the surface are
that when the u boundary is changed the helicoid expands outward; and when the v boundary is
changed, more spiral are added to reflect the number of rotations. The helicoid was the simplest
of our surfaces.
Section 2. Catenoid.
The catenoid is a surface that looks similar to the stereotypical picture of a wormhole. It
is in the form of
x = (-cosh(u)cos(v), -cosh(u)sin(v), u) ,
and our bounds for this surface were -2 ≤ 𝑢 ≤ 2 and -2π ≤ 𝑣 ≤ 2𝜋. The bounds had to resemble
each other, otherwise the catenoid would have an extremely narrow middle section in relation to
the ends. The tangent vectors were simple in comparison to our later plots:
Tu = (-sinh(u)cos(v), -cosh(u)sin(v), 1) , and
Tv = (cosh(u)sin(v), -cosh(u)cos(v), 0) .
Furthermore, tangent vectors exist at all points on the surface. Figure 2 shows the plot of the
catenoid.

4. Figure 2. Catenoid.
Surface 3. Catalan’s Surface.
Our third surface is known as Catalan’s Surface. It exists in the form
x = (u – cosh(v)sin(u), 1 – cosh(v)cos(u), 4sin(u/2)sinh(v/2)) .
Our bounds for this surface were strange in comparison to the others. Whereas other surfaces’
bounds followed noticeable patterns, like both u and v matching in constants, the Catalan’s
Surface did not. Its bounds were 0 ≤ 𝑢 ≤ 13 and 0 ≤ 𝑣 ≤ 2𝜋. The tangent vector with respect to
u was
Tu = (1-cosh(v)cos(u), cosh(v)sin(u), 2cos((1/2)u)sinh((1/2)v)) ,
and the tangent vector with respect to v was
Tv = (-sinh(v) sin(u), -sinh(v)*cos(u), 2*sin((1/2)*u)*cosh((1/2)v).

5. Unfortunately, however, we were unable to compute the exact surface area because the
computation proved to be tedious by hand, and even Maple would not provide an exact output.
The most precise number given was
S(A) = 4.693772714*10^5+6.517063134*10^5*I .
An image of our surface in Maple is given below.
Figure 3. Catalan’s Surface.
Section 4. Henneberg’s Surface.
This surface is very similar to the Helicoid in terms of overall shape. There seems to be
two pinches on the surface that causes the overlap.

6. We can see that the hyperbolic functions included in the components may cause the pinching of
the surface at certain points. The tangent and normal vectors are included in the appendix due to
their large outputs.
Figure 4. Henneberg’s Surface.
The parameters of the surface were 0 ≤ 𝑢 ≤ 1 and 0 ≤ 𝑣 ≤ 2𝜋. The surface area could not be
computed under a normal run time through Maple, and calculation by hand proved to be tedious.
We assume our syntax included a mistake or we missed a command that would have made
Maple’s computation easier.
Section 5. Enneper Surface.
Again this surface continues the look of pinching and overlapping of the surface onto
itself. Enneper’s Surface, the graph of which is found at the end of Section 5, is in the form of
𝑥⃗ =

7. and its tangent vectors are as follows:
Tu = (-u2 + v2 + 1, 2vu, 2u) and
Tv = (2vu, u2 – v2 +1, -2v) .
The normal vector that results from the cross product of the two tangent vectors is
N = (4uv – 2u + 2uv2 – 2u3, 2v – 2u2v + 2v3 – 4uv, -u4 + 2u2v2 – v4 + 1 – 4uv) .
What is interesting is how there are no trigonometric functions within this parameterization, so
this observation made us wonder why there were so many intersections. The overlap could be
due to the polynomials causing intersections along the surface. However, we had to make sure
our bounds for this surface matched, lest the surface appear even more convoluted than it was
supposed to. The bounds we used were -5 ≤ 𝑢 ≤ 5 and -5 ≤ 𝑣 ≤ 5. At the end of our analysis of
Enneper’s Surface, we found the surface area to be
S(A) =
380900
9
.
Figure 5. Enneper’s Surface.

8. Section 6. Scherk’s Surface.
This surface is definitely different from the rest. The parameterized form of Scherk’s Surface is
𝒙⃗⃗⃗ =(u, v, log(cos(u)sec(v))) .
We can see from Figure 6 below that there is spacing amongst the surfaces, causing us to assume
there are not tangent and normal vectors for each point on the surface. Furthermore, the tangent
vectors’ components lead us to believe there are asymptotes where the surfaces are undefined.
The tangents are
Tu = (1, 0, -tan(u)), and
Tv = (0, 1, tan(v)) .
The uniqueness of this parameterization is most likely caused by the inclusion of the log
function. Nonetheless, the cross product still yields the following vector:
N = (tan(u), tan(v), 1) .
The range for this surface is −2𝜋 ≤ 𝑢 ≤ 6𝜋 and − 2𝜋 ≤ 𝑣 ≤ 2𝜋. Adjusting the u and v values
causes expansion of the number of surfaces present and the length of each surface. However,
despite the simplicity of the vectors and bounds, the overall surface area of this vector over these
bounds was another computation Maple was unable to perform. This is most likely due to a
similar reason mentioned previously in Section 4. Scherk’s Surface may be seen in Figure 6.
Figure 6. Scherk’s Surface.

9. Section 7. Seashell 1
This surface is self-explanatory in terms of physical observation – it is a seashell. This surface
uses exponentials in its parameterization. The exponents may help determine the shape and
dissipation of the shell as it spirals, which may be seen in Figure 7.
Figure 7. Seashell 1.
The range of values used were 0 ≤ 𝑢 ≤ 6𝜋 and 0 ≤ 𝑣 ≤ 2𝜋. Adjusting u affected how much of
the shell was shown, and adjusting the v value revealed more spirals iterated.
Section 8. Seashell 2
This seashell used a different parameterization than the preceeding seashell. It did not include
exponentials but instead used varying coefficients 𝜋 in the parameters. The difference is seen in
the smoothness and fill of the seashell when not using exponentials.
Figure 8. Seashell 2.

10. The range of values used were 0 ≤ 𝑢 ≤ 2𝜋 and − 2𝜋 ≤ 𝑣 ≤ 2𝜋. This also deviates from the
first seashell, where changing the u values shows a horizontal slicing of the shell in each level
and the v values gives iterates of the spiral.
Section 9. Pseudosphere.
This surface is similar to a spinning top. It relies heavily on hyperbolic functions to represent the
surface. This can show how sensitive the values are for the range when mapping the surface. The
pseudosphere’s vector in component form can be represented as follows:
𝑥⃗ = (sech( 𝑢)cos( 𝑣),sech( 𝑢)sin( 𝑣), 𝑢 − tanh( 𝑢))
Figure 9. Pseudosphere.

11. The range values used were−8 ≤ 𝑢 ≤ 8 and 0 ≤ 𝑣 ≤ 2𝜋. Changing the u values affects the
visual of the top or bottom part of the pseudosphere. While the v values change the revolution of
each half.
Section 10. Cornucopia
This surface resembles a horn or the representation of a Thanksgiving basket. This also uses
exponential in its parameterization. There are coefficients in the exponential part of the
parameters that affects the shape of the surface.
Figure 10. Cornucopia.
The range values used were −6 ≤ 𝑢 ≤ 6 𝑎𝑛𝑑 − 6𝜋 ≤ 𝑣 ≤ 6𝜋. These values are almost fixed to
give the correct representation of the surface. Where the bounds do not match in distance the
surface is skewed or missing patches.
Surface 11. Astroidal Ellipsoid.
This surface is similar to a sandspur. It is dependent on coefficients added to the
parameterization.
𝑥⃗ = [cos(u)^3cos(v)^3,sin(u)^3cos(v)^3,sin(v)^3]
Figure 11.

12. The range values used were −
𝜋
2
≤ 𝑢 ≤
𝜋
2
and − 𝜋 ≤ 𝑣 ≤ 𝜋. Like a majority of these surfaces
the dependence on matching bounds is pertinent to the correct mapping of the surface. The
changing of coefficients in the parameterization elicits the same results as changing the u and v
values in terms of affecting the overall shape.

13. Appendix
Helicoid
> with(plots):with(linalg):
> Surface:=[u*cos(v),u*sin(v),v]:
> Range:=u=0..1,v=0..2*Pi:
> plot3d(Surface,Range);
> T_u:=diff(Surface,u);
T_v:=diff(Surface,v);
> N:=crossprod(T_u,T_v); arg:=simplify(dotprod(N,N,'orthogonal'));
> int(int(sqrt(arg),u=0..1),v=0..2*Pi);

14. Catenoid Surface
>
>
>
>
>
>
>
>
>

15. Catalan Surface
>
>
>
>
>
>
>
>

16. Henneberg Surface
>
>
>
>
>
>
>
>
Warning, computation interrupted

17. Enneper Surface
>
>
>
>
>
>
>
>
>

18. Scherk Surface
Plot of the Surface: Scherk
> Surface:=[u,v,log(cos(u)*sec(v))]:
> Range:=u=-6..6,v=-6*Pi..6*Pi:
> plot3d(Surface,Range);
>
>
>
>
Warning, computation interrupted

19. Seashell Surface 1
> with(plots):with(linalg):
Plot of the Surface: Seashell1
> Surface:=[2*(1-exp(u/(6*Pi)))*cos(u)*cos(v/2)^2,2*(-1+exp(u/(6*Pi)))*sin(u)*cos(v/2)^2,1-
exp(u/(3*Pi))-sin(v)+exp(u/(6*Pi))*sin(v)]:
> Range:=u=0..6*Pi,v=0..2*Pi:
> plot3d(Surface,Range);
>
>
>
>
Warning, computation interrupted

20. Seashell Surface 2
> with(plots):with(linalg):
Plot of the Surface: Seashell2
> Surface:=[(5/4)*(1-(v/(2*Pi)))*cos(2*v)*(1+cos(u))+cos(2*v),(5/4)*(1-
(v/(2*Pi)))*sin(2*v)(1+cos(u))+sin(2*v),(10*v/(2*Pi))+(5/4)*(1-(v/(2*Pi)))*sin(u)+15]:
> Range:=u=0..2*Pi,v=-2*Pi..2*Pi:
> plot3d(Surface,Range);
>
>
>
>
Warning, computation interrupted
>

21. Pseudosphere
> with(plots):with(linalg):
Plot of the Surface: Pseudosphere
> Surface:=[sech(u)*cos(v),sech(u)*sin(v),u-tanh(u)]:
> Range:=u=-8..8,v=0..2*Pi:
> plot3d(Surface,Range);
>
>
>
>
Cornucopia
> with(plots):with(linalg):
Plot of the Surface: Cornucopia
>
Surface:=[exp(10*v)*cos(v)+exp(10*v)*cos(u)*cos(v),exp(10*v)*sin(v)+exp(10*v)*cos(u)*sin(
v),exp(10*v)*sin(u)]:
> Range:=u=-6..6,v=-6*Pi..6*Pi:
> plot3d(Surface,Range);
>

22. >
>
>

23. Astroidal Ellipsoid
> Surface:=[cos(u)^3*cos(v)^3,sin(u)^3*cos(v)^3,sin(v)^3]:
> Range:=u=-Pi/2..Pi/2,v=-Pi..Pi:
> plot3d(Surface,Range);
>
>
>
>