Solving Equation of Spheroid for Prolate & Oblate

[Amith2006]

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.

 

[jambaugh]

Start with the general ellipsoid:
\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2 = 1

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

 

[Amith2006]

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?

 

[jambaugh]

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

 

[Amith2006]

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?