One can construct rolling curves on the sphere in the same way as in the plane. A drawing stick is attached to the rolling circle and draws the curve. The point where the rolling circle touches the fixed circle (or: where the rolling wheel touches the street) is a momentary fixed point of the rolling motion of the sphere. This implies a tangent construction for the rolling curve: The tangent is orthogonal to the segment between the current curve point and the current fixed point ( or: orthogonal to the current “radius”)
Rolling curves drawn with sticks of varying length.
Rolling curves from varying stick length rendered as tubes.
Rolling curve with osculating circles. Note that these best approximating circles lie on the sphere. Their midpoints trace out a curve called evolute.
The normals of the rolling curves are drawn until they meet the evolute.