On convolution surfaces in Euclidean 3-space

Abstract

In the present paper we study with the convolution surface $C=M\star N$ of a paraboloid $M\subset \mathbb{E}^{3}$ and a parametric surface $N\subset \mathbb{E}^{3}$. We take some spacial surfaces for $N$ such as, surface of revolution, Monge patch and ruled surface and calculate the Gaussian curvature of the convolution surface $C$. Further, we give necessary and sufficient conditions for a convolution surface $C$ to become flat.