Geometry of skeletal structures and symmetry sets

In this thesis we study the geometry of symmetry sets and skeletal structures. The relationship between a symmetry point (skeletal point) and the associated midlocus point is studied and the impact of the singularity of the radius function on this relationship is investigated. Moreover, the concept of the centroid set associated to a smooth submanifold of Rn+1 is introduced and studied. Also, the relationship between the shape operator of a skeletal structure at a smooth point and the shape operator of its boundary at the associated point is studied

Some Geometric Characterizations of f-Curves Associated with a Plane Curve via Vector Fields

The differential geometry of plane curves has many applications in physics especially in mechanics. The curvature of a plane curve plays a role in the centripetal acceleration and the centripetal force of a particle traversing a curved path in a plane. In this paper, we introduce the concept of the f-curves associated with a plane curve which are more general than the well-known curves such as involute, evolute, parallel, symmetry set, and midlocus. In fact, we introduce the f-curves associated with a plane curve via its normal and tangent for both the cases, a Frenet curve and a Legendre curve. Moreover, the curvature of an f-curve has been obtained in several approaches

On geometry of the midlocus associated to a smooth curve in plane and space

The singularities of the midpoint map associated to a smooth plane curve, which is a map from the plane to the plane, are classified. The midlocus associated to a regular space curve is introduced. The geometric conditions for the midlocus of a space curve to have a crosscap or an S?1 singularities are investigated. A more general map, the ?-point map, associated to a space curve is introduced and many known surface singularities are realized as a special cases of this construction.</jats:p