Classification of "polar manifolds"

Question

Is there a common name for the set of manifolds characterised by these examples:

• sphere
• ellipse,
• superquadric,
• donut
• 'a "sphere" with a wavy surface along the azimuth parameter'
• (polar roses? I'm uncertain about this one)

One defining factor is that none of them sort of 'twists'.

Another defining factor is that they can all be described like this (I think..):

    x = radius * cosinus ( theta ) * curve_X( phi );
    y = radius * sinus   ( theta );
    z = radius * cosinus ( theta ) * curve_Z( phi );

where

    curve_X ( phi ) = sin ( phi ) + something_X
    curve_Z ( phi ) = cos ( phi ) + something_Z

if you discard 'something', you get a perfect sphere. If you make 'something' into eg. a frequency-enchanced sinus (eg. sin( 10 x phi) for something_X and cos(10*phi) for something_Y), you get a "sphere" that is wavy around the azimuth parameter (on planet earth, it would be wavy along equator)

In this parametrisation, the y axis is sort of the 'backbone' around which the manifold is spun. If that makes sense..