Intersection Curves of Implicit and Parametric Surfaces in R3

We present algorithms for computing the differential geometry properties of Frenet apparatus and higher-order derivatives of intersection curves of implicit and parametric surfaces in 3 for transversal and tangential intersection. This work is considered as a continuation to Ye and Maekawa [1]. We obtain a classification of the singularities on the intersection curve. Some examples are given and plotted.   t,n,b,κ, τ 


Introduction
The intersection problem is a fundamental process needed in modeling complex shapes in CAD/CAM system.It is useful in the representation of the design of complex objects, in computer animation and in NC machining for trimming off the region bounded by the self-intersection curves of offset surfaces.It is also essential to Boolean operations necessary in the creation of boundary representation in solid modeling [1].The numerical marching method is the most widely used method for computing intersection curves in .The Marching method involves generation of sequences of points of an intersection curve in the direction prescribed by the local differential geometry [2,3].Willmore [4] described how to obtain the unit tangent, the unit principal normal, the unit binormal, the curvature and the torsion of the transversal intersection curve of two implicit surfaces [5].Kruppa [6] explained that the tangential direction of the intersection curve at a tangential intersection point corresponds to the direction from the intersection point towards the intersection of the Dupin indicatrices of the two surfaces.Hartmann [7] provided formulas for computing the curvature of the transversal intersection curves for all types of intersection problems in Euclidean 2-space.Kriezis et al. [8] determined the marching direction for tangential intersection curves based on the fact that the determinant of the Hessian matrix of the oriented distance function is zero.Luo et al. [9] presented a method to trace such tangential intersection curves for parametric-parametric surfaces employing the marching method.The marching direction is obtained by solving an undetermined system based on the equilibrium of the differentiation of the two normal vectors and the projection of the Taylor expansion of the two surfaces onto the normal vector at the intersection point.Ye and Maekawa [1] presented algorithms for computing all the differential geometry properties of both transversal and tangentially intersection curves of two parametric surfaces.They described how to obtain these properties for two implicit surfaces or parametric-implicit surfaces.They also gave algorithms to evaluate the higher-order derivative of the intersection curves.Aléssio [10] studied the differential geometry properties of intersection curves of three implicit surfaces in for transversal intersection, using the implicit function theorem.


In this study, we present algorithms for computing the deferential geometry properties of both transversal and tangentially intersection curves of implicit and Parametric surfaces in as an extension to the works of [1].
3  This paper is organized as follows: Section 2 briefly introduces some notations, definitions and reviews of differential geometry properties of curves and surfaces in .Section 3 derives the formulas to compute the properties for the transversal intersection.Section 4 derives the formulas to compute the properties for the tangential intersection.Some examples of transversal and tangentially intersection are given and plotted in Section 5. Finally, conclusion is given in Section 6.
The notation for the differentiation of the curve in relation to the arc length s is . Then from elementary differential geometry, we have where is the unit tangent vector field and t  α is the curvature vector.The factor is the curvature and is the unit principal normal vector.The unit binormal vector is defined as The vectors are called collectively the Frenet-Serret frame.The Frenet-Serret formulas along α are given by , , , where is the torsion which is given by τ provided that the curvature does not vanish.

Differential Geometry of the Parametric Surfaces in 3 
Assume that is a regular parametric surface.In other words where ) denote to partial derivatives of the surface .The unit sur-face normal vector field of the surface is given by The first fundamental form coefficients of the surface are given by R , ; , 1,2 The second fundamental form coefficients of the surface R are given by in the 1 2 u u -plane defines a curve on the surface which can be written as Then the three derivatives of the curve are given by α The projection of the curvature vector onto the unit normal vector field of the surface is given by

Differential Geometry of the Implicit Surfaces in  3
Assume that   , then the unit surface normal vector field of the surface f is given by be a curve on the surface f with constraint , , , , , , , , .
The projection of the curvature vector onto the unit normal vector field of the surface where

Transversal Intersection Curves
Consider the intersecting implicit and parametric surfaces and . Then the intersection curve of these surfaces can be viewed as a curve on both surfaces as s , s , s ; , , 0, where Then the surface Thus the intersection curve is given by

Tangential Direction
where , Since  α is the unit tangent vector field of the curve (3.2), then we have which can be written as (3.7) The unit tangent vector field of the intersection curve is given by substituting (3.7) into (2.12) as follows

Curvature and Curvature Vector
The curvature vector is given by differentiation (3.8) with respect to s as follows The unit principal normal vector field, the curvature and the unit binormal vector are given by using (2.3) (2.4) and (2.5) as follows (3.10)

h u h h h h h h h h h h h h h h h
2 and 2 into (2.14)we obtain the third-order derivative vector of the intersection curve.Hence the torsion can be obtained by (2.7).We can compute all higher-order derivatives of the intersection curve by a similar way.

Tangentially Intersection Curves
Assume that the surfaces and are intersecting tangentially at a point on the curve (3.2) then the unit surface normal vector field of both surfaces are parallel to each other.In other words which can be written as Then we can write

Tangential Direction
Projecting the curvature vector onto the two unit normal vectors of both surfaces yields Using (2.15) (2.21) and (4.4) we obtain where  R This can be written in a matrix form as follows where and is the Hessian matrix of the surface f Solving (4.6) for Then the unit tangent vector field of the intersection curve is given by From the previous formulas, it is easy to see that, there are four distinct cases for the solution of (4.6) depending upon the discriminant these cases are as the following [1]   where Since the curvature vector is perpendicular to the tangent vector, then we have ,    α α 0. Using (2.12) (2.13) and (4.9) we obtain  , and 2 into (2.14)we obtain the third-order derivative vector of the intersection curve.Then we can obtain the torsion by using (2.7). 2 , 9sin2 , 1,0,0 , 18cos 2 , 0, 36sin 2 , 1, 0, 0 , 3 0, cos , sin , 3 0,sinsin u ,cos , 3 0,coscosu , sin .

Conclusions
Algorithms for computing the differential geometry properties of intersection curves of implicit and parametric surfaces in are given for transversal and tangential intersection.This paper is an extension to the works of Ye and Maekawa [1].They gave algorithms to compute the differential geometry properties of intersection curves between two parametric surfaces then they applied it on a simple example for implicit and parametric surfaces intersection.This paper presented direct and simple formulas to compute all differential geometry properties, which may reduce the time it takes to calculate those properties.The types of singularities on the intersection curve are characterized.The questions of how to exploit and extend these algorithms to compute the differential geometry properties of intersection curves between three surfaces in , can be topics of future research.