Introduction: It is known that if a space curve is given by its parametric equation, its geometric properties such as Frenet vectors and curvatures can be easily obtained. However, the computations of above curvatures of a surface curve become harder when the space curve is given by the intersection of two surfaces. In most cases, finding the parametric equation of the intersection curve of two surfaces is a difficult problem. That's why this is an interesting topic not only in differential geometry but also in Computer Aided Geometric Design. Since surfaces can be given by its implicit or parametric equations, the intersection of two surfaces can be of three types: implicitimplicit, implicitparametric, parametricparametric. Besides, when the normal vectors of the intersecting surfaces are linearly independent (linearly dependent) at an intersection point, the intersection is called transversal (tangential) intersection. The transversal intersection problem have been studied in various studies due to easy computation of the tangent vector of the intersection curve. On the other hand, we have less literature for the tangential intersection of two surfaces, since the computations become harder even for finding the tangential direction. Recently, a new method has been presented for the tangential intersection problem of two surfaces. In this recent study, by benefiting from the Rodrigues' rotation formula and the new defined operator, the authors obtain the curvature, tangent vector, principal normal vector and binormal vector of two intersecting surfaces. The computation of the torsion of the intersection curve was absent in this study. Purpose: Considering the Rodrigues' rotation formula and the new defined operator, the purpose of this research is to compute the torsion of the tangential intersection curve of two surfaces in Euclidean 3space. Scope: The scope of this research is restricted to the intersection problem of two parametric surfaces. Limitations: Since the intersection may be transversal or tangential at an intersection point, the intersection is limited to tangential intersection. Since all Frenet apparatus except the torsion of the intersection curve are obtained by using Rodrigues' rotation formula, this research is also limited to computation of the torsion of the tangential intersection curve. Method: The method used in this research is based on the Rodrigues' rotation formula and an operator D which has been defined by a vector product in 3space. When we have a nonzero vector x in 3space, D(x) is defined as a new vector by vector product of any vector y with x in which the vector y is chosen arbitrary such that it is linearly independent with x. This operator applied to intersection problem by using the common unit normal vector of two tangentially intersecting parametric surfaces. Applying D to the common normal vector yields a tangent vector which lies in the common tangent planes of the surfaces. This tangent vector is rotated about the normal direction of the surfaces at the tangential intersection point by using the Rodrigues' rotation formula. This rotation enables us to obtain the tangent vector of the intersection curve. Findings: We consider two parametric surfaces. We use the Darboux vector of the intersection curve to compute the torsion. To do this, we rotate the principal normal vector of the intersection curve about the axis defined by the unit Darboux vector. After this rotation, the principal normal vector coincides with the direction of the third order derivative vector of the intersection curve. We then obtain the torsion of the intersection curve. Conclusion: It has been concluded that the torsion of the tangential intersection curve can also be obtained by using the Rodrigues' rotation formula for the tangential intersection problem of two surfaces.
Anahtar Kelimeler: Intersection Curve, Tangential Intersection, Rodrigues' Rotation Formula
