Surface reconstruction is a problem that receives a lot of attention in Computer Graphics due to the importance and the number of its applications. A common solution is to represent these surfaces through triangular meshes. This work introduces an alternative structure of tetrahedrons with cubics defined in its interior, which are used to approximate the surface. Those cubics, called TetraQuads, are third-degree implicit surfaces defined as an interpolation of quadrics positioned at the tetrahedrons vertices. These elements are constructed for an efficient visualization by the graphics hardware. Objects defined in this manner contain more information from the differential geometry point of view than a triangle mesh, which entails a more complex modeling problem. This problem is discussed throughout the steps of surface reconstruction from point clouds through TetraQuads.