Does there exist a circle, on the surface of a torus, that is not coplanar with the axis of revolution?

Question

I have got two closely related questions regarding Tori.

1.Does there exist a circle, on the surface of a torus, that is not co planar with the torus' axis of revolution?

2.Can the centre of a circle that is lying on the surface of a torus be the centre of a different circle also lying on the surface of the torus?

How does one proceed with proving (or disproving) them?

Answer 1

Both are true, so the easy way is to exhibit them:

Picture a standard torus with the axis of revolution along $z$ (i.e. rotated in the $x$-$y$ plane). Horizontal cuts are not coplanar with the $z$ axis (indeed, are perpendicular to it.) These are circles (and must be because they are parallel to the plane of rotation). At all but the extreme top and bottom, there are two them, sharing the same center: on the $z$ axis.

For 1, in addition, there are two subtler circles through any point: http://mathworld.wolfram.com/VillarceauCircles.html, which will also not be coplanar with the $z$ axis.

Answer 2

Provided that your torus bows to 2r=R, then a pair of those Villarceau circles can be chosen to conform with your nr. 2.

--- rk