# Surface area of a truncated ellipsoid

• Murphy24
In summary, a biology PhD student is looking for help calculating the surface area of an ellipsoid truncated parallel to the long axis. The flat surface of the truncated part is relatively easy to calculate using an ellipse formula, but the other part requires a more complex integral. The student received help from others and found a python script to estimate the surface area. They also modified the equation for a spherical cap to incorporate the radii of the base of the truncated ellipsoid. This modification may not be exact, but it should be close enough for the student's purposes.
Murphy24
Hi,

I'm a biology PhD student looking for some help on how to calculate (or estimate) the surface area of an ellipsoid truncated parallel to the long axis. Any help would be greatly appreciated.

Thanks,

Murphy24

You would probably use a double integral to solve this type of problem.

Murphy24 said:
Hi,

I'm a biology PhD student looking for some help on how to calculate (or estimate) the surface area of an ellipsoid truncated parallel to the long axis. Any help would be greatly appreciated.

Thanks,

Murphy24

The flat surface of the truncated part is relatively easy. It is an ellipse with area A = pi*a * b where a and b are the two semi-diameters. (If a=b then you have a circle a=b=r and A=pi r^2).

The other part is a rather nasty integral, an elliptic integral without closed form solution (except to invent a new function whose value is that area).

There is a formula in terms of standard elliptic integrals and there is an approximation formula. Check out: http://en.wikipedia.org/wiki/Ellipsoid" .

EDIT: Ahhh... that only gives the total ellipsoidal area. To get the truncated area or the area of the piece to subtract out. Let me see what I can come up with. Possibly I can work out a way to numerically estimate via software.

Last edited by a moderator:
Below is a python script I wrote which will calculate the value you want. You can download the python interpreter from www.python.org if you don't already have it installed on your computer. (Open the file in IDLE and run.)

It's a slow brute force approach but it will work. The ellipsoid is assumed to be truncated at a certain height above the center which is given by z0. Note indention is critical in python as the interpreter uses it to group blocks of code. I get about 5 decimal place accuracy on the area of a sphere of radius 2. The equation for the ellipsoid is:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$$

$$z < z_0$$

As you change a,b, and c, and z0 be sure you enter them as decimals as otherwise python will do integer division truncating the fractions.

Code:
from math import *

z0 = 2.0  #Height at which ellipsoid is truncated.
a = 2.0
b = 2.0
c = 2.0

phi0 = acos(z0/c)
steps = 3000  #Gives about 5 decimal place accuracy in a few minutes.

def Jacob(phi,theta,A,B,C):
return abs(sin(phi))*sqrt( C*C*sin(phi)*sin(phi)*(B*B*cos(theta)*cos(theta)+ A*A*sin(theta)*sin(theta))+A*A*B*B*cos(phi)*cos(phi))

Int=0.0

dphi = (pi-phi0)/steps
dtheta = 2*pi/steps

for p in xrange(steps):
for t in xrange(steps):
phi = phi0 + p*dphi
theta = t*dtheta
Int += Jacob(phi,theta,a,b,c)*dtheta*dphi

print "The Area is:", Int

EDIT: BTW This code only gives the area of the ellipsoid surface. You'll need to add in the flat ellipse.
EDIT2: You can lower the number of steps for a quicker, less accurate answer.

Last edited:
Thanks for everyone's help.

I found this site yesterday and modified the equation for the surface area for a spherical cap to calculate the surface area of the ellipsoidal cap:

http://mathworld.wolfram.com/SphericalCap.html

So this site says that the surface area of a spherical cap = pi*(a^2+h^2). I suspect this is just an estimation, but its probably accurate enough for my purposes. I am aiming to estimate the surface area of a membrane that partially covers the top of reptile eggs, so it doesn't need to be a perfect estimate, just one that doesn't introduce too much error.

I reasoned that just as the area of an ellipse is pi*(a*b) whereas the area of a circle is pi*(r^2), that maybe I could modify the formula for the area of a spherical cap from pi*(a^2+h^2) to one that incorporates both radii of the base of the truncated ellipsoid: pi*(a*b+h^2).

First of all, is the site that I am referencing reliable at all? And secondly, is my modification of the equation for a spherical cap a valid one?

Thanks again,

Murphy24

Murphy24 said:
Thanks for everyone's help.

I found this site yesterday and modified the equation for the surface area for a spherical cap to calculate the surface area of the ellipsoidal cap:

http://mathworld.wolfram.com/SphericalCap.html

So this site says that the surface area of a spherical cap = pi*(a^2+h^2). I suspect this is just an estimation, but its probably accurate enough for my purposes. I am aiming to estimate the surface area of a membrane that partially covers the top of reptile eggs, so it doesn't need to be a perfect estimate, just one that doesn't introduce too much error.

I reasoned that just as the area of an ellipse is pi*(a*b) whereas the area of a circle is pi*(r^2), that maybe I could modify the formula for the area of a spherical cap from pi*(a^2+h^2) to one that incorporates both radii of the base of the truncated ellipsoid: pi*(a*b+h^2).

First of all, is the site that I am referencing reliable at all? And secondly, is my modification of the equation for a spherical cap a valid one?

Thanks again,

Murphy24

The formula they give is exact for spherical caps. It should be pretty darn close for your purposes (where a and b are relatively close to each other). Close enough that your worry is more that the egg isn't a perfect ellipsoid than that your formula isn't giving the perfectly exact area.

Good luck.

Thanks mate :)

## 1. What is the formula for calculating the surface area of a truncated ellipsoid?

The formula for calculating the surface area of a truncated ellipsoid is A = 2πb² + 2πa²c + (b² + a²)cot⁻¹(c/b), where a, b, and c are the semi-axes of the ellipsoid.

## 2. How is a truncated ellipsoid different from a regular ellipsoid?

A truncated ellipsoid is a modified version of a regular ellipsoid where a portion of its top or bottom has been cut off by a plane. This results in a flat surface instead of a curved one, changing the overall shape and surface area of the ellipsoid.

## 3. Can the surface area of a truncated ellipsoid be negative?

No, the surface area of a truncated ellipsoid cannot be negative. It is a measure of the total area of the ellipsoid's curved surface, so it will always be a positive value.

## 4. What units are typically used to measure the surface area of a truncated ellipsoid?

The surface area of a truncated ellipsoid is typically measured in square units, such as square centimeters (cm²) or square meters (m²).

## 5. How is the surface area of a truncated ellipsoid used in real-life applications?

The surface area of a truncated ellipsoid has various real-life applications, such as in engineering and architecture for designing structures with curved surfaces, in physics for calculating the surface area of objects in motion, and in biology for estimating the surface area of biological cells and organisms.

Replies
3
Views
1K
Replies
4
Views
2K
Replies
4
Views
5K
Replies
5
Views
4K
Replies
4
Views
1K
Replies
7
Views
5K
Replies
8
Views
1K
Replies
2
Views
6K
Replies
5
Views
5K
Replies
1
Views
1K