Motivated by Gutkin curves arising in capillary floating and billiard ball problem, we
introduce a new projection function for convex plane curves, and study the curves
when the defined projection function is constant. These curves can be regarded as a
generalization of curves with constant width. We give a necessary and sufficient
condition for the existence of such curves, and provide an explicit expression for the
radius of curvature. Properties of our curves and their relation to Gutkin curves are
also discussed. Finally, we construct and exhibit some curves with constant projection
function.