The Position Relationship And Distance Computation Between Some Special Surfaces

The special surface is the surface which has some special geometric properties, such as sphere, ellipsoid, Bezier surface, torus, canal surface and so on. Because of their special geometric properties, they are widely used in the field of computer. Consequently, the research of distance computation and position relationship is very useful.A canal surface is the envelope of a one-parameter set of moving spheres. We present an accurate and efficient method for computing the distance between two canal surfaces using a set of cone-spheres as bounding volumes. For two canal surfaces, we use the distances between their bounding cone-spheres to approximate their distance; the accuracy of this approximation is improved progressively by subdividing the canal surfaces into more segments and using more cone-spheres to bound the segments, until a pre-specified threshold is reached. We present an efficient pruning technique that can eliminate most of the pairs of cone-spheres that do not contribute to the distance between the original canal surfaces. We use distance interval to instead the distance of bounding cone-sphere. It can ensure the correctness of the pruning technique. With the help of Bernstein polynomial, we present a method for computing tight bounding cone-spheres of a canal surface without solving any equations, which is an interesting problem in its own right. We also extend our method to distance computation of deformable canal surface. The movement and deformation of the surface is continuous. Based on this simple fact, we make our algorithm run efficiently. Experimental comparisons show that our new method is more efficient than Lee's method and our method in GMP2010. Our new method is especially efficient for computing the distance between two complex objects composed of many canal surfaces.In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional spaces about an axis coplanar with the circle. It is another special surface researched in this paper. We will study the position relationship between two tori. We reduce the order of torus by shrinking them and inversion transformation. Firstly, we change a torus which is a quartic surface into a circle which is a conic by shrinking. Then we change the circle into a beeline. By those methods, finally we change the problem of the position relationship between two tori into the problem of the roots of two quartic equations. By analyzing the roots of the two quartic equations, we can enumerate all kinds of position relationship between two tori and present the equivalent condition of every kind of position relationship. We present some results of our work about this in this paper.