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Next: 11.3.5 Self-intersection of offsets Up: 11.3 Offset surfaces Previous: 11.3.3 Self-intersection of offsets   Contents   Index


11.3.4 Self-intersection of offsets of explicit quadratic surfaces

Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. We have seen in Sect. 8.3, that any regular surface can be locally approximated in the neighborhood of a point $ P$ by an explicit quadratic surface of the form
$\displaystyle {\bf r}(x,y) = [x,y,\frac{1}{2}(\alpha x^2 + \beta y^2)]^T\;,$     (11.63)

to the second order where $ -\alpha$ and $ -\beta$ are the principal curvatures at point $ P$ . The minus signs are consistent with curvature sign convention (b). Therefore investigations of the self-intersection mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces.

In the sequel we assume $ d>0$ , $ \beta>0$ and $ \alpha\leq \beta$ without loss of generality. According to this assumption the surface is a hyperbolic paraboloid when $ \alpha < 0$ , an elliptic paraboloid when $ 0<\alpha < \beta$ , a paraboloid of revolution when $ 0<\alpha =\beta$ , and a parabolic cylinder when $ \alpha =0$ as illustrated in Fig. 8.9. The paraboloid of revolution and the parabolic cylinder can be considered as degenerate cases of the elliptic paraboloid. When $ \alpha=\beta$ , the principal direction is not defined and the point $ (0,0,0)$ will become an umbilic. If $ \alpha$ and $ \beta$ vanish at the same time, the surface is part of a plane, and we do not investigate such cases.

In the case for offsets of explicit quadratic surfaces, there are no self-intersections due to global distance function properties [26], thus if $ d>0$ the maximum absolute value of the negative minimum principal curvature determines the largest offset $ d$ without degeneracy. The largest magnitude of offset distance without degeneracy is called the maximum offset distance $ \vert d_{max}\vert$ . In Sect. 8.3 we discussed how to find the global minimum of the minimum principal curvature of explicit quadratic surfaces.

Due to Lemma 8.3.1 the global minimum of the minimum principal curvature of the explicit quadratic surface occurs at the origin, except for a parabolic cylinder ((11.63) with $ \alpha$ =0) which has minima along the $ x$ -axis with curvature value $ \kappa_{min} =-\beta$ , and hence the maximum offset distance is determined to be $ \vert d_{max}\vert= \frac{1}{\beta}$ . If the offset distance exceeds $ \frac{1}{\beta}$ , the offset starts to degenerate from the point $ (0,0,\frac{1}{\beta})$ on the offset surface except for a parabolic cylinder progenitor, where the offset starts to degenerate along the line $ (x,0,\frac{1}{\beta})$ .

Substitution of the expression of the offset of the explicit quadratic surface (11.63)

$\displaystyle \hat{\bf r}(x,y) = [x,y,\frac{1}{2}(\alpha x^2 + \beta y^2)]^T + \frac{d}{\sqrt{1 + \alpha^2x^2 + \beta^2y^2}}[-\alpha x, -\beta y, 1]^T\;,$     (11.64)

into (11.29) yields a vector equation for the self-intersection curve of such offset. The $ x$ , $ y$ and $ z$ components of this vector equation are given by
$\displaystyle \sigma - \frac{\alpha \sigma d}{\sqrt{1+\alpha^2 \sigma^2 + \beta^2 t^2}} = u - \frac{\alpha u d}{\sqrt{1+\alpha^2 u^2 + \beta^2 v^2}}\;,$     (11.65)
$\displaystyle t - \frac{\beta t d}{\sqrt{1+\alpha^2 \sigma^2 + \beta^2 t^2}} = v - \frac{\beta v d}{\sqrt{1+\alpha^2 u^2 + \beta^2 v^2}}\;,$      
$\displaystyle \frac{1}{2}(\alpha \sigma^2 + \beta t^2) + \frac{d}{\sqrt{1+\alph... ...}{2}(\alpha u^2 + \beta v^2) + \frac{d}{\sqrt{1+\alpha^2 u^2 + \beta^2 v^2}}\;,$      

The self-intersection curve of an offset can be considered as the locus of the center of a sphere, whose radius is the offset distance, rolling on the progenitor surface with two contact points. It is evident from (11.64) that the offsets of explicit quadratic surfaces are symmetric with respect to $ xz$ and $ yz$ planes. Because of the symmetry of the offsets of explicit quadratic surfaces, the center of the rolling sphere must move only on the planes of symmetry and hence the self-intersection curves are on the plane of symmetry. Therefore we can set $ \sigma =-u$ and $ t=v$ in (11.65). The $ y$ and $ z$ components will only result in identities, while the $ x$ component results in
$\displaystyle \sqrt{1+\alpha^2 \sigma^2 + \beta^2 t^2} = \alpha d\;,$     (11.66)

where the trivial solution $ \sigma=u=0$ is excluded. Similarly, we can set $ \sigma=u$ and $ t=-v$ in the (11.65). The $ x$ and $ z$ components will only result in identities, while the $ y$ component results in
$\displaystyle \sqrt{1+\alpha^2 \sigma^2 + \beta^2 t^2} = \beta d\;,$     (11.67)

where the trivial solution $ t=v=0$ is excluded. Therefore (11.65) have been reduced to two uncoupled equations in $ \sigma$ and $ t$ . Next we give a useful theorem for evaluating the self-intersection curves of offsets of explicit quadratic surfaces (11.63) and their corresponding planar curve in the $ xy$ -plane, i.e. pre-image of the self-intersection curve. In this theorem we assume that the $ x$ and $ y$ axes are taken as the directions of maximum and minimum principal curvatures.

Theorem 11.3.2. The self-intersection curves of offsets of the explicit quadratic surfaces $ {\bf r}(x,y)=[x,y,\frac{1}{2}(\alpha x^2 +\beta y^2)]^T$ and their pre-images in the $ xy$ -plane are as follows [248]:

  1. An offset of a hyperbolic paraboloid ( $ \alpha < 0 < \beta$ ) self-intersects only in the $ y$ -direction when $ \frac{1}{\beta}<d$ . The resulting self-intersection curve is a parabola given by
    $\displaystyle z = \frac{\alpha\beta}{2(\beta - \alpha)}x^2 + \frac{(\beta d)^2 + 1}{2\beta}, \;\;y=0\;,$     (11.68)
    $\displaystyle where \;\;\;-\frac{\beta-\alpha}{\alpha\beta}\sqrt{(\beta d)^2 - 1}\leq x \leq \frac{\beta-\alpha}{\alpha\beta}\sqrt{(\beta d)^2 - 1}\;,$      

    and its pre-image in the $ xy$ -plane is an ellipse when $ \vert\alpha\vert\neq\beta$ or a circle when $ \vert\alpha\vert=\beta$ , (see Fig. 11.21 (a)) given by
    $\displaystyle \frac{x^2}{\left(\frac{\sqrt{(\beta d)^2 - 1}}{\alpha}\right)^2} + \frac{y^2}{\left(\frac{\sqrt{(\beta d)^2 - 1}}{\beta}\right)^2} = 1\;.$     (11.69)

  2. An offset of an elliptic paraboloid ( $ 0<\alpha < \beta$ ) self-intersects only in the $ y$ -direction when $ \frac{1}{\beta}<d<\frac{1}{\alpha}$ and self-intersects in both $ x$ and $ y$ -directions when $ \frac{1}{\alpha}<d$ . The self-intersection curve which self-intersects in the $ y$ -direction is a parabola (see Fig. 11.20) given by (11.68) and its pre-image in the $ xy$ -plane is an ellipse (see Figs. 11.20, 11.21 (b)) given by (11.69). The self-intersection curve which self-intersects in the $ x$ -direction is also a parabola given by
    $\displaystyle z = \frac{\alpha\beta}{2(\alpha-\beta)}y^2 + \frac{(\alpha d)^2 + 1}{2\alpha}, \;\;x=0\;,$     (11.70)
    $\displaystyle where\;\;\;-\frac{\alpha-\beta}{\alpha\beta}\sqrt{(\alpha d)^2 - 1}\leq y \leq \frac{\alpha-\beta}{\alpha\beta}\sqrt{(\alpha d)^2 - 1}\;,$      

    and its pre-image in the $ xy$ -plane is an ellipse (see Figs. 11.21 (c), (d)) given by
    $\displaystyle \frac{x^2}{\left(\frac{\sqrt{(\alpha d)^2 - 1}}{\alpha}\right)^2} + \frac{y^2}{\left(\frac{\sqrt{(\alpha d)^2 - 1}}{\beta}\right)^2} = 1\;.$     (11.71)

  3. An offset of a paraboloid of revolution ( $ 0<\alpha =\beta$ ) self-intersects in all directions, when $ \frac{1}{\beta}=\frac{1}{\alpha}<d$ . The self-intersection curve is a point $ (0, 0, \frac{(\beta d)^2 -1}{2\beta})$ , and its pre-image in the $ xy$ -plane is a circle (see Fig. 11.21 (e)) given by
    $\displaystyle x^2 + y^2 = \left(\frac{\sqrt{(\beta d)^2 - 1}}{\beta}\right)^2\;.$     (11.72)

  4. An offset of a parabolic cylinder ( $ \alpha= 0 < \beta$ ) self-intersects only in the $ y$ -direction when $ \frac{1}{\beta}<d$ . The resulting self-intersection curve is a straight line in the $ xz$ -plane
    $\displaystyle z = \frac{(\beta d)^2 -1}{2\beta},\;\;y=0\;,$     (11.73)

    and its pre-image in the $ xy$ -plane (see Fig. 11.21 (f)) are two straight lines given by
    $\displaystyle y = \pm \frac{\sqrt{(\beta d)^2 - 1}}{\beta}\;.$     (11.74)

Proof:
Case (1): Since $ \alpha$ is negative for the hyperbolic paraboloid and we are assuming $ d>0$ and the left hand side of (11.66) is always positive, this equation cannot be used to derive the self-intersection curve. This implies that the offset of a hyperbolic paraboloid does not self-intersect in the $ x$ -direction (maximum principal direction). However, we can use (11.67) to derive the self-intersection curve in the $ y$ -direction (minimum principal direction). Upon squaring and replacing $ \sigma$ by $ x$ and $ t$ by $ y$ we obtain

$\displaystyle \alpha^2 x^2 + \beta^2 y^2 = (\beta d)^2 - 1\;.$     (11.75)

This equation describes an ellipse in the $ xy$ -plane and is equivalent to (11.69). Since the left hand side of (11.75) is always positive, the equation is only valid when $ \frac{1}{\beta}<d$ . This indicates that there is no self-intersection unless the offset distance exceeds the maximum offset distance $ \vert d_{max}\vert= \frac{1}{\beta}$ . Now we can obtain the self-intersection curve in the $ xz$ -plane by mapping the ellipse in $ xy$ -plane (see (11.75)) into the 3-D coordinates using (11.64), resulting in:
$\displaystyle \hat{\bf r}(x,y) = [\hat{x}(x,y),\; \hat{y}(x,y),\; \hat{z}(x,y)]... ...})x, \;0, \;\frac{1}{2\beta}\{\alpha(\beta-\alpha)x^2 + (\beta d)^2 + 1\}]^T\;,$     (11.76)

where $ -\frac{\sqrt{(\beta d)^2 - 1}}{\alpha}\leq x \leq \frac{\sqrt{(\beta d)^2 - 1}}{\alpha}$ . The range of parameter $ x$ comes from (11.75). If we eliminate the parameter $ x$ from (11.76) and replace $ \hat{x}$ by $ x$ and $ \hat z$ by $ z$ , we obtain the same result as (11.68).

Case (2): Since $ \alpha$ is positive for the elliptic paraboloid, both (11.66), (11.67) can be used to obtain the self-intersection curves in the $ xy$ -plane. This implies that the offset of an elliptic paraboloid may self-intersect in both principal directions. Since we have already derived the equation from (11.67), we derive another equation from (11.66). Upon squaring and replacing $ \sigma$ by $ x$ and $ t$ by $ y$ we obtain

$\displaystyle \alpha^2 x^2 + \beta^2 y^2 = (\alpha d)^2 - 1\;,$     (11.77)

which is equivalent to (11.71). Also this equation is only valid when $ \frac{1}{\alpha}<d$ . The self-intersection curve in 3-D coordinates can easily be obtained in a similar manner with Case (1).

Case (3): If we set $ \alpha=\beta$ in (11.75) and (11.77), both equations reduce to (11.72). Also if we set $ \alpha=\beta$ in (11.76), the parabola reduces to the point $ (0, 0, \frac{(\beta d)^2 +1}{2\beta})$ .

Case (4): Since $ \alpha$ is zero for the parabolic cylinder, (11.66) is not valid. Thus we set $ \alpha =0$ in (11.67) and replacing $ t$ by $ y$ we obtain $ \beta^2y^2= (\beta d)^2-1$ , which is equivalent to (11.74). The self-intersection curve in three dimensional coordinates can easily be obtained in a similar manner with Case (1).

Note that the self-intersection curve of the offset of an elliptic paraboloid (when $ \frac{1}{\beta}<d<\frac{1}{\alpha}$ ) has a positive quadratic term, while those of a hyperbolic paraboloid and an elliptic paraboloid (when $ \frac{1}{\alpha}<d$ ) have negative quadratic terms.

Example 11.3.4. Consider an elliptic paraboloid $ z= \frac{1}{2}(2x^2 + 4y^2)$ with offset distance $ d=0.3$ . Since $ \frac{1}{\beta}=\frac{1}{4}<d=0.3<\frac{1}{2}=\frac{1}{\alpha}$ , the offset surface self-intersects only in the $ y$ -direction. The self-intersection curve is $ z=2x^2+0.305$ (dashed line in Fig. 11.20) and its pre-image in the $ xy$ -plane is $ \frac{x^2}{0.11} + \frac{y^2}{0.0275}=1$ (solid line in Fig. 11.20). The dot dashed line in this figure illustrates the set of footpoints of the self-intersection curve on the progenitor surface. A pair of thin solid straight lines emanating from two distinct points on the surface $ {\bf r}(\sigma,t)$ , $ {\bf r}(u,v)$ and intersecting along the parabola are the pairs of vectors $ d{\bf N}(\sigma,t)$ and $ d{\bf N}(u,v)$ .

Figure 11.20: Self-intersection curves of an offset of elliptic paraboloid ($ \alpha =2$ ,$ \beta =4$ ) with $ d=0.3$ (adapted from [248])
\begin{figure}\centerline{ \hspace*{-30mm} \psfig{file=fig/off_mechanism.ps,height=4.0in}} \end{figure}

To illustrate Theorem 11.3.2, we plot pre-images of the self-intersection curves along with cuspidal edges in the $ xy$ -plane for the hyperbolic paraboloid ($ \alpha =-2$ , $ \beta =2$ , $ d=0.6$ ), the elliptic paraboloids ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.55$ ), ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.6$ ) and ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.65$ ), the paraboloid ( $ \alpha=b=2$ , $ d=0.6$ ) and the parabolic cylinder ($ \alpha =0$ , $ \beta =2$ , $ d=0.6$ ) as depicted in Figs. 11.21 (a) to (f).

\begin{figure*}\centerline{ \psfig{figure=fig/self_ridge_2D_hp60.ps,height=3.5in... ...centerline{ \psfig{figure=fig/self_ridge_2D_ep60.ps,height=3.5in}} \end{figure*}





Figure 11.21: Pre-images of self-intersection curves and cuspidal edges of offsets of explicit quadratic surfaces (adapted from [248]). The solid lines correspond to pre-images of the self-intersection curves for self-intersection in the $ y$ -direction. The dashed lines correspond to $ \kappa _{min}(x,y)=-\frac {1}{d}$ . The dot dashed lines correspond to pre-images of the self-intersection curves for self-intersection in the $ x$ -direction. The dot dot dashed lines correspond to $ \kappa _{max}(x,y)=-\frac {1}{d}$ . Symbols $ \times $ and $ *$ represent the locations of generic lemon type umbilics and non-generic umbilics, respectively. (a) Hyperbolic paraboloid ($ \alpha =-2$ , $ \beta =2$ , $ d=0.6$ ), (b) elliptic paraboloid ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.55$ ), (c) elliptic paraboloid ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.6$ ), (d) elliptic paraboloid ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.65$ ), (e) paraboloid of revolution ( $ \alpha =\beta =2$ , $ d=0.6$ ), (f) parabolic cylinder ($ \alpha =0$ , $ \beta =2$ , $ d=0.6$ )
\begin{figure}\centerline{ \psfig{figure=fig/self_ridge_2D_ep65.ps,height=3.5in}... ...enterline{ \psfig{figure=fig/self_ridge_2D_hc60.ps,height=3.5in} }\end{figure}

It is interesting to note that when the progenitor surface is a hyperbolic paraboloid or an elliptic paraboloid (see Fig. 11.21 (a) to (d)), the pre-images of the self-intersection curve of its offset which self-intersects in the $ y$ -direction and the cuspidal edge $ \kappa _{min}(x,y)=-\frac {1}{d}$ always intersect tangentially at $ y=0$ . The pre-image of the self-intersection curve of the offset of an elliptic paraboloid (see Fig. 11.21 (c)), which self-intersects in $ x$ -direction, and the cuspidal edge $ \kappa _{max}(x,y)=-\frac {1}{d}$ intersect tangentially at $ x=0$ , when the two cuspidal edges intersect with the $ y$ -axis within the two umbilics. Whereas when the two cuspidal edges intersect the $ y$ -axis outside the two umbilics (see Fig. 11.21 (d)), the pre-images of the self-intersection curve and the cuspidal edge $ \kappa _{min}(x,y)=-\frac {1}{d}$ intersect tangentially at $ x=0$ .

It is apparent from (11.21) that the direction of the normal vector of the offset surface is opposite to that of the progenitor surface inside the loop of $ \kappa _{min}(x,y)=-\frac {1}{d}$ (dashed line) in the absence of the loop of $ \kappa _{max}(x,y)=-\frac {1}{d}$ (see Figs. 11.21 (a), (b), (e)), and the regions between outside the loop of $ \kappa _{max}(x,y)=-\frac {1}{d}$ (dot dot dashed line) and inside the loop of $ \kappa _{min}(x,y)=-\frac {1}{d}$ (see Figs. 11.21 (c), (d)), while the direction is the same within the loop of $ \kappa _{max}(x,y)=-\frac {1}{d}$ (see Figs. 11.21 (c), (d)).

Figures 11.22, 11.23 and 11.24 show self-intersecting offset surfaces, self-intersection curves and cuspidal edges in 3-D space and the trimmed offset surface of a hyperbolic paraboloid ($ \alpha =-2$ , $ \beta =2$ , $ d=0.6$ ), an elliptic paraboloid ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.6$ ) and an elliptic paraboloid ( $ \alpha =1.75$ , $ \beta =2$ , $ d=0.65$ ), respectively.

Figure 11.22: Self-intersecting offset surface (top), region bounded by self-intersection curve (middle) and trimmed offset surface (bottom) of a hyperbolic paraboloid $ z=\frac {1}{2}(-2x^2 + 2y^2)$ with $ d=0.6$ (adapted from [248])
\begin{figure}\centerline{ \psfig{figure=fig/off_hp60.ps,height=3.0in}} \vspace*... ...line{ \hspace*{-20mm} \psfig{figure=fig/trim_hp60.ps,height=3in} }\end{figure}

Figure 11.23: Self-intersecting offset surface (top), region bounded by self-intersection curves (middle) and trimmed offset surface (bottom) of elliptic paraboloid $ z=\frac {1}{2}(1.75x^2 + 2y^2$ ) with $ d=0.6$ (adapted from [248])
\begin{figure}\centerline{ \hspace*{-20mm} \psfig{figure=fig/off_ellpar60.ps,hei... ...0mm} \psfig{figure=fig/trim_ep60.ps,height=3in} } \vspace*{-10mm} \end{figure}

Figure 11.24: Self-intersecting offset surface (top), region bounded by self-intersection curves (middle) and trimmed offset surface (bottom) of elliptic paraboloid $ z=\frac {1}{2}(1.75x^2 + 2y^2$ ) with $ d=0.65$ (adapted from [248])
\begin{figure}\centerline{ \hspace*{-20mm} \psfig{figure=fig/off_ellpar65.ps,hei... ...0mm} \psfig{figure=fig/trim_ep65.ps,height=3in} } \vspace*{-10mm} \end{figure}

Figure 11.25 illustrates the self-intersections of the offset of a bicubic Bézier surface patch. Figure 11.25 $ (a)$ shows the pre-images of the self-intersection curve in the parameter domain. The thick line represents the numerically traced self-intersection curve, while the thin line represents the ellipses of (11.69), (11.71), which are in quite good agreement. The same bullet symbols are mapped to the same locations on the offset surface. Figure 11.25 $ (b)$ shows the mapping of the self-intersection curves in the parameter domain onto the progenitor surface. Finally, Fig. 11.25 $ (c)$ shows the offset surface with its self-intersections.

Figure 11.25: Self-intersections curves of the offset of a bicubic surface patch when $ d$ =0.75 (adapted from [253]): (a) pre-images of the self-intersection curve in parameter domain, where the same bullet symbols are mapped to the same locations in the offset surface, (b) a set of footpoints of self-intersection curves on the progenitor surface, (c) the offset surface with self-intersections
\begin{figure}\centerline{ \psfig{file=fig/croof1_para_075.PS,height=3in}} \cent... ...0.5in} \centerline{ \psfig{file=fig/croof1_off_075.PS,height=3in}}\end{figure}

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Next: 11.3.5 Self-intersection of offsets Up: 11.3 Offset surfaces Previous: 11.3.3 Self-intersection of offsets   Contents   Index
December 2009