The sphere

It projects as a circle of the radius of the sphere in both views, with the centre on the same vertical. Any plane section is a circle.

Every plane section is a circle, and its radius satisfies r² = R² - d², with d the distance from the centre to the plane:

Plane through the centre (d = 0): great circle, of radius R. It is seen in true size in the plan if the plane is horizontal.
Horizontal plane: in the plan it looks like a circle of radius r; in the elevation, like a segment.
Projecting plane: it looks like a segment in the edge-on view and like an ellipse in the other.

STEP 1 / 2

A horizontal plane cuts the sphere in a circle. In the plan it is seen as a circle (its real radius) and in the elevation as a segment.

If the distance from the centre to the line is greater than the radius there is no cut; if it is equal, the line is tangent (a single point).