Ellipsoids and Surfaces of Revolution

[IniquiTrance]

My textbook notes that if:

$$\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}}=1$$

and $$a \neq b \neq c$$

Then the ellipsoid is not a surface of revolution. It seems to me though that one can always find a curve in the plane, which when rotated around a line will produce the ellipsoid.

Why is this not true?

 

[LCKurtz]

My textbook notes that if:

$$\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}}=1$$

and $$a \neq b \neq c$$

Then the ellipsoid is not a surface of revolution. It seems to me though that one can always find a curve in the plane, which when rotated around a line will produce the ellipsoid.

Why is this not true?

Why is it not true that a rectangular block of wood is a surface of revolution? It just isn't.

A surface of revolution would require cross sections perpendicular to some axis that are circles. It is "apparent" that such cross sections of an ellipsoid don't exist unless two of a, b, and c are equal.

 

[IniquiTrance]

Ah, thanks for the explanation.