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6.4.1 Tangential direction

The unit tangential vector $ {\bf t}$ of $ {\bf c}(s)$ at $ P$ must lie on the common tangent plane of $ A$ and $ B$ . Therefore, $ {\bf t}$ can be represented as a linear combination of $ {\bf r}^A_{u_A}$ and $ {\bf r}^A_{v_A}$ , as well as $ {\bf r}^B_{u_B}$ and $ {\bf r}^B_{v_B}$ , as in (6.17), i.e.
$\displaystyle {\bf t}= {\bf r}^A_{u_A}u'_A+ {\bf r}^A_{v_A}v'_A = {\bf r}^B_{u_B}u'_B+ {\bf r}^B_{v_B}v'_B\;.$     (6.56)

Equation (6.56) consists of two linear equations with four unknowns $ (u'_A,$ $ v'_A,$ $ u'_B$ , $ v'_B)$ , since the tangent vector is constrained in the tangent plane and does not have a normal component. Since $ {\bf N}^A = {\bf N}^B = {\bf N}$ at $ P$ , we find that $ \kappa_n^A = \kappa_n^B$ from (6.25). Thus, from (6.29) we have
$\displaystyle L^A(u'_A)^2 + 2 M^Au'_Av'_A + N^A(v'_A)^2 = L^B(u'_B)^2 + 2 M^Bu'_Bv'_B + N^B(v'_B)^2\;.$     (6.57)

This equation is a quadratic equation in $ (u'_A, v'_A, u'_B, v'_B)$ . Thus together with the unit length constraint of the tangent vector, (6.56) and (6.57) form a system of four nonlinear equations in four unknowns. This nonlinear system can be solved by representing $ u'_B$ and $ v'_B$ in terms of linear combinations of $ u'_A$ and $ v'_A$ from (6.56), and then substituting the results into (6.57). By taking the cross product of both sides of (6.56) with $ {\bf r}^B_{u_B}$ and $ {\bf r}^B_{v_B}$ , and projecting the resulting equations onto the common surface normal vector $ {\bf N}$ at $ P$ , $ u'_B$ and $ v'_B$ can be represented as the following linear combinations of $ u'_A$ and $ v'_A$
$\displaystyle u'_B = a_{11}u'_A + a_{12} v'_A\;,$     (6.58)
$\displaystyle v'_B = a_{21}u'_A + a_{22} v'_A\;,$     (6.59)

where
$\displaystyle a_{11} = \frac{({\bf r}^A_{u_A}\times {\bf r}^B_{v_B})\cdot{\bf N... ...ac{det({\bf r}^A_{u_A}, {\bf r}^B_{v_B}, {\bf N})}{\sqrt{E^B G^B - (F^B)^2}}\;,$     (6.60)
$\displaystyle a_{12} = \frac{({\bf r}^A_{v_A}\times {\bf r}^B_{v_B})\cdot{\bf N... ...ac{det({\bf r}^A_{v_A}, {\bf r}^B_{v_B}, {\bf N})}{\sqrt{E^B G^B - (F^B)^2}}\;,$     (6.61)
$\displaystyle a_{21} = \frac{({\bf r}^B_{u_B}\times {\bf r}^A_{u_A})\cdot{\bf N... ...ac{det({\bf r}^B_{u_B}, {\bf r}^A_{u_A}, {\bf N})}{\sqrt{E^B G^B - (F^B)^2}}\;,$     (6.62)
$\displaystyle a_{22} = \frac{({\bf r}^B_{u_B}\times {\bf r}^A_{v_A})\cdot{\bf N... ...ac{det({\bf r}^B_{u_B}, {\bf r}^A_{v_A}, {\bf N})}{\sqrt{E^B G^B - (F^B)^2}}\;.$     (6.63)

Substituting (6.58) and (6.59) into (6.57), we have

$\displaystyle b_{11} (u'_A)^2 + 2 b_{12} (u'_A)(v'_A) + b_{22} (v'_A)^2 = 0\;,$     (6.64)

where
$\displaystyle b_{11} = a_{11}^2 L^B + 2 a_{11} a_{21} M^B + a_{21}^2 N^B - L^A\;,$     (6.65)
$\displaystyle b_{12} = a_{11} a_{12} L^B + (a_{11} a_{22} + a_{21} a_{12}) M^B + a_{21} a_{22} N^B - M^A\;,$      
$\displaystyle b_{22} = a_{12}^2 L^B + 2 a_{12} a_{22} M^B + a_{22}^2 N^B - N^A\;.$      

If we denote $ \omega= \frac{u'_A}{v'_A}$ when $ b_{11} \ne 0$ or $ \mu= \frac{v'_A}{u'_A}$ when $ b_{11} = 0$ and $ b_{22}\neq 0$ , and solve (6.64) for $ \omega$ or $ \mu$ , then $ {\bf t}$ can be computed as
$\displaystyle {\bf t} = \frac{\omega {\bf r}^A_{u_A} + {\bf r}^A_{v_A}}{\vert\omega {\bf r}^A_{u_A} + {\bf r}^A_{v_A}\vert}\;,$     (6.66)

or
$\displaystyle {\bf t} = \frac{{\bf r}^A_{u_A} + \mu{\bf r}^A_{v_A}}{\vert {\bf r}^A_{u_A} + \mu {\bf r}^A_{v_A}\vert}\;.$     (6.67)

There are four distinct cases to the solution of (6.64) depending upon the discriminant $ b^2_{12} - b_{11}b_{22}$ :

  1. Isolated tangential contact point: If $ b_{12}^2 - b_{11} b_{22} < 0$ then (6.64) does not have any real solution. Thus, $ P$ is an isolated contact point of $ A$ and $ B$ .
  2. Tangential intersection curve: If $ b_{12}^2 - b_{11} b_{22} = 0$ and $ b_{11}^2+b_{12}^2+b_{22}^2\neq 0$ then (6.64) has a double root and $ {\bf t}$ is unique. Thus, $ A$ and $ B$ intersect at $ P$ and at its neighborhood.
  3. Branch Point: If $ b_{12}^2 - b_{11} b_{22} > 0$ then (6.64) has distinct roots. Thus, $ P$ is a branch point of the intersection curve $ {\bf c}(s)$ , i.e. there is another intersection branch crossing $ {\bf c}(s)$ at $ P$ .
  4. Higher order contact point: If $ b_{11}=b_{12}=b_{22}=0$ then (6.64) vanishes for any values of $ u'_A$ and $ v'_A$ . Thus, $ A$ and $ B$ has a contact of at least second order (i.e., curvature continuous) at $ P$ . In related work by Pegna and Wolter [305], they developed mathematical criteria for curvature continuity between two surfaces. Those criteria were later generalized to arbitrary higher order continuity (contact) in [161].
When $ P$ is a flat point of one of the surfaces, say $ {\bf r}^B$ , then $ L^B$ , $ M^B$ , $ N^B$ all vanish, however we can still evaluate (6.64). When $ P$ is a flat point of both surfaces, then the two surfaces have a contact of order at least 2 at $ P$ which is addressed under case 4.

There is a geometric interpretation to the tangent direction $ {\bf t}$ at $ P$ . Recall that the Dupin's indicatrix of a surface at point $ P$ is a conic section (see Sect. 3.6). Since $ A$ and $ B$ intersect tangentially at $ P$ , they have the same tangent-plane at $ P$ . Equation (6.57) indicates that along $ {\bf t}$ , the Dupin's indicatrices of $ A$ and $ B$ at $ P$ intersect. Conversely, $ {\bf t}$ is the vector(s) on the common tangent-plane at $ P$ along which the Dupin's indicatrices of $ A$ and $ B$ intersect. The two Dupin's indicatrices may not intersect at all (isolated tangential contact point), or intersect at two points tangentially, or intersect transversally at four points (branch point), or overlap (higher order contact point). In the case of overlap, they must be the same at $ P$ , and $ A$ and $ B$ are at least curvature continuous at $ P$ . Figure 6.3 shows the possible combinations of Dupin's indicatrices of two surfaces for four distinct cases. Although the coordinate system of the two indicatrices are chosen to be the same for simplicity, in general they may have different orientations. At hyperbolic points the Dupin's indicatrix is a set of conjugate hyperbolas depending on which side of the tangent plane the normal section is locally lying. However, for simplicity we have only illustrated the cases for one of the conjugate hyperbolas. The Dupin's indicatrices of case 2 upper right in Fig. 6.3 are parallel to each other and do not intersect. This is the case when two surfaces $ A$ and $ B$ intersect tangentially at a parabolic point $ P$ where they have the same principal directions. We assume without loss of generality that $ u-$ and $ v-$ parameter curves are in the directions of the principal directions, where $ u=constant$ being the principal direction with zero curvature. With these assumptions, we have $ M^A=M^B=N^A=N^B=0$ and $ a_{12}=a_{21}=0$ thus (6.64) reduces to

$\displaystyle (a_{11}^2L^B-L^A)(u_A')^2=0\;,$     (6.68)

Therefore it has a double root with unique direction ( $ u_A=constant$ ) for $ {\bf t}$ , provided that $ a_{11}^2L^B-L^A\ne 0$ . When $ a_{11}^2L^B-L^A = 0$ , the two surfaces are at least curvature continuous at $ P$ , and their Dupin's indicatrices overlap.

Figure 6.3: Dupin's indicatrices of two tangentially intersecting surfaces (adapted from [458])
\begin{figure}\centerline{ \psfig{file=fig/dupin_dupin.eps,height=6.0in} }\end{figure}

Implicit-implicit and parametric-implicit intersection cases can be handled in a similar way. For the implicit-implicit intersection case we first equate the normal curvatures of the two implicit surfaces $ A$ and $ B$ using (6.34), where the unknowns are the unit tangent vector $ (x', y', z')$ . We can eliminate one of the component, say $ z'$ , using (6.20) yielding the quadratic equation in $ x'$ and $ y'$ similar to (6.64).

For the parametric-implicit intersection case we equate the normal curvatures (6.29) and (6.34) of the parametric and implicit surfaces where the unknowns are $ x'$ , $ y'$ , $ z'$ , $ u'$ and $ v'$ . We can replace $ x'$ , $ y'$ , $ z'$ in terms of $ u'$ and $ v'$ using (6.17), which leads us to a quadratic equation similar to (6.64). Upon solving the quadratic equation and applying the unit length constraint, we obtain the unit tangent vector.

There are also four distinct cases for implicit-implicit and parametric-implicit intersections, depending on the discriminant of the quadratic equation.

next up previous contents index
Next: 6.4.2 Curvature and curvature Up: 6.4 Intersection curve at Previous: 6.4 Intersection curve at   Contents   Index
December 2009