Tangent vectors to a 3-D surface normal: A geometric tool to find orthogonal vectors based on the Householder transformation

• Authors: D. S. Lopes;M. T. Silva;J. A. AmbróSio
• Venue: Computer-Aided Design
• Year: 2013

An important geometric and linear algebraic problem denoted as vector orthogonalization, fundamental to handle contact detection and contact force descriptions in engineering applications, is here considered. The problem is to find a set of linearly independent vectors that span the entire R^3 Euclidean space given only one of the base vectors. This paper contains the explanation on how the Householder transformation, which is extensively used for matrix orthogonalization, provides an elegant analytical expression that solves the vector orthogonalization problem. Based on the QR matrix factorization method, the orthogonal vectors are produced using a Householder reflection that transforms the given vector into a multiple of the unit vector whose entries are all zero with the exception of the first. Based on efficiency, accuracy and numerical robustness criteria, the proposed technique is compared to other vector orthogonalization methods. The numerical results show that the Householder vector orthogonalization formula is the most efficient when it comes to outputting a set of orthonormal vectors, presenting speedups close to 1.017 times faster when compared to other efficient techniques. In addition, when dealing with C^n continuous implicit surfaces, with n1, the Householder vector orthogonalization formula reveals to be particularly useful for vector calculus since it provides a set of differential operators to calculate, not only the normal, but also the tangent and binormal surface vector fields which can be used to calculate surface curvatures. The major contribution of this paper is to explicitize how the Householder transformation holds an analytical expression that calculates the tangent and binormal vectors from a given normal at a surface point vector, which is computationally efficient and numerically robust for real-time computational geometry and computer graphics applications, namely, for contact mechanics applications with implicit surfaces of engineering problems with multiple contacts. Such a vector orthogonalization technique also has direct applications in several CAD/CAM processes, ranging from the elaboration of rough solid models to the precise manufacturing of a product.