Intersection Curves of Special Surfaces in Euclidean Space

In differential geometry of curves, there are many important properties and consequences. In the light of the existing studies, researchers always introduce new curves. Although, the differential geometry of curves in E^3 can be found in many textbooks and in the contemporary literature on geometric modeling, there is little literature in E^3 and hardly in E^4 and E^n in the study of differential geometry of intersection curves. Some intersection curves have been studied by many geometers and obtained some interesting results. Here, we work with one of the important classes of surfaces which are called the implicit surfaces in a four- dimensional Euclidean space E^4. And we present formulas for computing the differential geometry properties of the tangential intersection curve of three implicit surfaces in Euclidean 4-space E^4. These properties include (the tangent T, the principle normal N, the binormal vectors (B1;B2) and the curvatures (1; 2; 3) of the intersection curve). Furthermore, we give examples to explain our main results.

Lecturer of Computational-Differential Geometry, Mathematics Department, Faculty of Science, Sohag University, Egypt & Mathematics Department, Faculty of Science, Islamic University in Madina, KSA.Dr. H. S. Abdel Aziz, Dr. M. Khalifa Saad and Mr. Assem Abdel Salam (PhD student), are the group of researchers in differential geometry at mathematics department, faculty of science, Sohag University.