This paper presents a parametric family of curves that can smoothly morph from a circle (shape parameter s = 0) to any polygon circumscribed to that circle (s = 1). A fully closed solution r(φ; s, ψ) in polar coordinates is derived for the diamircle that seamlessly evolves between a diamond with interior angles ψ and 180°−ψ and its inscribed circle. Curvygons, closed curves with continuous derivative (class C1) that are intermediate to a circle of radius R and any given N-sided polygon circumscribed to that circle, are then immediately obtained by simply joining N scaled and suitably rotated diamircle segments. Conceptually related 3D curvyhedra, which smoothly morph between spheres and circumscribed polyhedra, are demonstrated. This work extends the Fernández-Guasti squircle (special case ψ = 90° of the diamircle) and the Fong sphube to much broader families of morphing 2D and 3D shapes. Application examples are showcased for computer animations, UI and industrial design, mechanical engineering, and heat diffusion physics.