The literature dealing with curves of constant diameter is not very extensive, and the following notes may be of general interest.

If the distance between parallel tangents to any curve is constant, whatever their common direction, then the curve is a corve of constant diameter. Thus, if p (ψ) is the perpendicular distance from the origin to the tangent to the curve, where ψ is the angle the tangent makes with the positive x-axis, and D(ψ) is the diameter of the curve, that is, the distance between parallel tangents, then