Solving Equation of Spheroid for Prolate & Oblate

  • Thread starter Amith2006
  • Start date
In summary, the conversation discusses the equations for prolate and oblate spheroids, which are obtained by rotating an ellipse around its major or minor axis. The equations for both spheroids appear to be identical, but by taking a different orientation for the major and minor axes, the equations can be distinguishable. The conversation also mentions the shapes of the prolate and oblate spheroids, which resemble a cigar and a cow-pie, respectively.
  • #1
Amith2006
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2

Homework Statement



I have a doubt on spheroid equations. A prolate spheroid is obtained by rotating the ellipse,
X^2/a^2 + Y^2/b^2 = 1 {Here a is major axis}
about the semi-major axis a(i.e. X axis). Its equation is,
X^2/a^2 + [Y^2+Z^2]/b^2 = 1
An oblate spheroid is obtained by rotating the ellipse,
X^2/a^2 + Y^2/b^2 = 1 {Here b is major axis}
about the semi-minor axis a(i.e. X axis).Its equation is,
X^2/a^2 + [Y^2+Z^2]/b^2 = 1
The problem is that both equations are identical. What I have done is that I have taken ‘a’ always along X axis and ‘b’ always along Y axis. Is it necessary that the equations be distinguishable?

Homework Equations



X^2/a^2 + Y^2/b^2 = 1

The Attempt at a Solution



In order distinguish between the two, I will have to take ‘a’ along Y axis for one of them. Suppose I take ‘a’ along the Y axis for oblate spheroid case, the equation of the oblate spheroid is got by rotating the ellipse,
X^2/b^2 + Y^2/a^2 = 1
about the semi-minor axis ‘b’(i.e. X axis).Its equation is,
X^2/b^2 + [Y^2+Z^2]/a^2 = 1
Another way is to rotate the ellipse,
X^2/a^2 + Y^2/b^2 = 1 {Here a is major axis}
first along X axis(i.e. ‘a’) for prolate spheroid in which case the equation becomes,
X^2/a^2 + [Y^2+Z^2]/b^2 = 1
And then along Y axis(i.e. ‘b’) for oblate spheroid in which case the equation becomes,
[X^2+Z^2]/a^2 + Y^2/b^2 = 1 {Here b is major axis}

Is there a better way to this? Please help.
 
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  • #2
Start with the general ellipsoid:
[tex] \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2 = 1[/tex]

if it is rotationally symmetric around anyone principle axis the the (a,b,c) coefficients for the other two must be equal. Example: if you have rotational symmetry about x then b=c. Further in this example if a is much bigger than b=c then the ellipsoid is cigar shaped. If a is much smaller than b=c then it is "cow-pie" shaped.

Regards,
J. Baugh
 
  • #3
jambaugh said:
Further in this example if a is much bigger than b=c then the ellipsoid is cigar shaped. If a is much smaller than b=c then it is "cow-pie" shaped.

Could you please say what are the shapes of Cigar and Cow-pie in this context?
 
  • #4
Amith2006 said:
Could you please say what are the shapes of Cigar and Cow-pie in this context?

Sure, take a highly eccentric ellipse. Rotate about the long axis and you have an ellipsoid that is long and cylindrical like a tapered cigar.

Take the same ellipse and rotate about the short axis and you have a tapered disk shaped ellipsoid, like the shape of a discus used in track-n-field events or like the pile of defecant a cow leaves behind.

Regards,
James Baugh
 
  • #5
So, the equation of prolate spheroid is,
X^2/a^2 + [Y^2+Z^2]/b^2 = 1
for a>b=c
and the equation of oblate spheroid is also,
X^2/a^2 + [Y^2+Z^2]/b^2 = 1
but here a<b=c. I that what u meant?
 

1. What is the equation for finding the volume of a prolate spheroid?

The equation for finding the volume of a prolate spheroid is (4/3)πa²c, where a is the semi-major axis and c is the semi-minor axis.

2. How do you solve for the surface area of an oblate spheroid?

To solve for the surface area of an oblate spheroid, you can use the equation 2πb² + 2πa√(b²+c²), where a is the semi-major axis and b and c are the two semi-minor axes.

3. Can the equation for a prolate spheroid be used for an oblate spheroid?

No, the equations for a prolate and oblate spheroid are different. The equation for a prolate spheroid uses the semi-major and semi-minor axes, while the equation for an oblate spheroid uses two semi-minor axes.

4. How do you convert an oblate spheroid equation to a prolate spheroid equation?

To convert an oblate spheroid equation to a prolate spheroid equation, you can use the geometric relationship a² = b² + c² and substitute it into the appropriate equation.

5. Are there any real-world applications for solving the equations of spheroids?

Yes, the equations for prolate and oblate spheroids are used in many fields, including astronomy, geology, and engineering, to calculate the volume and surface area of objects that have a spheroid shape, such as planets, asteroids, and ellipsoidal structures.

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