The problem of interpolation by a convex curve to the vertices of a convex polygon is considered. A natural 1-parameter family of C∞ algebraic curves solving this problem is presented. This is extended to a solution, of a general Hermite-type problem, in, which the curve also interpolates to one or two prescribed tangents at any desired vertices of the polygon. The construction of these curves is a generalization of well known methods for generating conic sections. Several properties of this family of algebraic curves are discussed. In addition, the method is generalized to convex C∞ interpolation of strictly convex data sets in R3 by algebraic surfaces.