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9.5 Local extrema of principal curvatures at umbilics

In this section we discuss a criterion which assures the existence of local extrema of the principal curvature functions $ \kappa_{max}$ and $ \kappa_{min}$ at umbilical points of the surface [257]. The problem of detecting local extrema of principal curvature functions is motivated by engineering applications. When a ball-end mill cutter is used for NC machining, the cutter radius must be smaller than the smallest concave radius of curvature of the surface to be machined to avoid local overcut (gouging) (see Sect. 11.1.2). Gouging is the one of the most critical problems in NC machining of free-form surfaces. Therefore, we must determine the distribution of the principal curvatures of the surface, which are upper and lower bounds of the normal curvature at a given point, to select the cutter size. A natural approach to locate local extrema of the functions $ \kappa_{max}$ and $ \kappa_{min}$ would in principle be to search for zeros of the gradient vector field $ \nabla\kappa_{max}$ and $ \nabla\kappa_{min}$ and then use tools from differential calculus to decide if at those zeros the principal curvature functions attain extrema. The problem with this approach however is that the curvature functions $ \kappa_{max}$ and $ \kappa_{min}$ are generally not differentiable at the umbilics although those points may also be candidates for local principal curvature extrema. We will present a necessary and sufficient criterion, which always detects the existence of a local extremum of the principal curvature functions $ \kappa_{max}$ and $ \kappa_{min}$ at an umbilic, except in presence of rare well defined and easily computable conditions. Under such rare condition, the criterion will become only sufficient. This criterion is practical because it is almost always applicable and easily evaluated.

We discuss the local behavior of the functions $ \kappa_{max}$ and $ \kappa_{min}$ in the neighborhood of an umbilic. First let us consider a Taylor expansion around an umbilic $ (u_o,v_o)$ for the function defined in (9.25). We obtain


$\displaystyle W(u, v) = W(u_o,v_o) + \left[(u-u_o)W_u(u_o,v_o) + (v-v_o)W_v(u_o,v_o)\right]$     (9.53)
$\displaystyle + \frac{1}{2!}[(u-u_o)^2W_{uu}(u_o,v_o) + 2(u-u_o)(v-v_o)W_{uv}(u_o,v_o)$      
$\displaystyle + (v-v_o)^2W_{vv}(u_o,v_o)] + R(u-u_o, v-v_o)\vert(u-u_o, v-v_o)\vert^2\;,$      

with
$\displaystyle \lim_{u \rightarrow u_o, v \rightarrow v_o}R(u-u_o, v-v_o) = 0\;.$     (9.54)

Note that (9.53) describes the remainder term in case of a second order Taylor approximation of a $ C^2$ smooth function which is guaranteed if the surface is $ C^4$ . In the special case where all the second partial derivatives of $ W$ vanish, the condition $ W(u,v)\geq 0$ implies that the third order partial derivatives must also vanish. If we consider the total number of possibilities where we have non-vanishing partial derivatives up to third order, the case where all partial derivatives vanish is statistically very rare. Therefore, we focus our attention now on the generic case where at least one of the second order partial derivatives of $ W$ does not vanish. Using (9.26), we obtain $ W(u_o, v_o)=0$ and $ \nabla W(u_o, v_o) = {\bf0}$ at the umbilic, therefore (9.53) reduces to
$\displaystyle W(u,v) = W_Q(u,v) + R(u-u_o, v-v_o)\vert(u-u_o, v-v_o)\vert^2\;,$     (9.55)

where
$\displaystyle W_Q(u,v) = \frac{1}{2}(u-u_o, v-v_o)\begin{pmatrix}W_{uu} & W_{uv} \cr W_{uv} & W_{vv}\end{pmatrix}(u-u_o, v-v_o)^T$      
$\displaystyle = \frac{1}{2}\frac{(u-u_o, v-v_o)}{\vert(u-u_o, v-v_o)\vert}\begi... ...}\frac{(u-u_o, v-v_o)^T}{\vert(u-u_o, v-v_o)\vert}\vert(u-u_o, v-v_o)\vert^2\;.$     (9.56)

Now we can Taylor expand $ \sqrt{W(u,v)}$ up to first order 9.2
$\displaystyle \sqrt{W(u,v)} = \sqrt{W_Q} + \frac{R(u-u_o, v-v_o)}{2\sqrt{W_Q}}\vert(u-u_o, v-v_o)\vert^2$      
$\displaystyle = [C(u,v) + \frac{R(u-u_o,v-v_o)}{2C(u,v)}]\vert(u-u_o, v-v_o)\vert\;,$     (9.57)

where
$\displaystyle C(u,v) = \sqrt{\frac{1}{2}\frac{(u-u_o, v-v_o)}{\vert(u-u_o, v-v_... ...{uv} & W_{vv}\end{pmatrix}\frac{(u-u_o, v-v_o)^T}{\vert(u-u_o, v-v_o)\vert}}\;.$     (9.58)

Next we Taylor expand the mean curvature $ H(u,v)$ as follows:

$\displaystyle H(u,v) = H(u_o, v_o) + (H_L(u,v) + R(u-u_o, v-v_o))\vert(u-u_o, v-v_o)\vert\;,$     (9.59)

where
$\displaystyle H_L(u,v) = [H_u(u_o,v_o), H_v(u_o,v_o)]\frac{(u-u_o, v-v_o)^T}{\vert(u-u_o, v-v_o)\vert}\;.$     (9.60)

Although the function $ R(u-u_o, v-v_o)$ in the remainder terms are different in (9.55), (9.57), (9.59), we nonetheless use the same notation for simplicity, since we are essentially interested in the common property described in (9.54).

Consequently $ \kappa(u,v)$ in equation $ \kappa(u,v) = H(u,v) \pm \sqrt{H^2(u,v) - K(u,v)}$ can be expanded in the vicinity of an umbilic $ (u_o,v_o)$ as follows:

$\displaystyle \kappa(u, v) = H(u_o, v_o) + (H_L(u,v) \pm C(u,v) + R(u-u_o,v-v_o))\vert(u-u_o, v-v_o)\vert$      
$\displaystyle = H(u_o, v_o) + {\bar H}_L(u,v) \pm {\bar C}(u,v) +{\bar R}(u-u_o,v-v_o)\;,$     (9.61)

where
$\displaystyle {\bar H}_L(u,v) = H_L(u,v)\vert(u-u_o, v-v_o)\vert\;,$     (9.62)
$\displaystyle {\bar C}(u,v) = C(u,v)\vert(u-u_o, v-v_o)\vert\;,$     (9.63)
$\displaystyle {\bar R}(u-u_o, v-v_o) = R(u-u_o, v-v_o)\vert(u-u_o, v-v_o)\vert\;.$     (9.64)

Therefore $ \kappa(u,v)$ can be considered as sum of the constant term $ H(u_o, v_o)$ , the plane $ {\bar H}_L(u,v)$ , which is the tangent plane of $ H(u,v)$ at $ (u_o,v_o)$ , and the elliptic cone $ {\bar C}(u,v)$ whose axis of symmetry is perpendicular to $ uv$ -plane, since $ W(u_o, v_o)=0$ , $ \nabla W(u_o, v_o) = {\bf0}$ . First we assume that $ {\bar H}_L(u,v)=0$ , in other words the tangent plane of $ H(u,v)$ coincides with the $ uv$ -plane. In this case $ \kappa(u,v) - H(u_o, v_o)$ reduces to $ \pm {\bar C}(u,v)$ . Figure 9.4 shows a positive elliptic cone $ +{\bar C}(u,v)$ (maximum principal curvature) having a minimum at $ (u_o,v_o)$ . When the elliptic cone is negative, minimum principal curvature has a maximum at $ (u_o,v_o)$ . The condition $ {\bar H}_L(u,v)=0$ occurs when all the third order partial derivatives of the height function in the Monge form are zero. This can be proved as follows [257]:

Proof : The coefficients of first and second fundamental forms of the surface in Monge form are given in (3.63) and (3.65). Their first order partial derivatives with respect to $ x$ are readily evaluated:

$\displaystyle E_x = 2h_xh_{xx},\;\;\;F_x = h_{xx}h_y + h_xh_{xy},\;\;\;G_x = 2h_yh_{xy}\;,$      
$\displaystyle L_x = \frac{h_{xxx}\sqrt{(1+h_x^2+h_y^2)} + h_{xx}(1+h_x^2+h_y^2)^{-\frac{3}{2}}(h_xh_{xx} + h_yh_{xy})}{1 + h_x^2 + h_y^2}\;,$      
$\displaystyle M_x = \frac{h_{xxy}\sqrt{(1+h_x^2+h_y^2)} + h_{xx}(1+h_x^2+h_y^2)^{-\frac{3}{2}}(h_xh_{xx} + h_yh_{xy})}{1 + h_x^2 + h_y^2}\;,$      
$\displaystyle N_x = \frac{h_{xyy}\sqrt{(1+h_x^2+h_y^2)} + h_{xx}(1+h_x^2+h_y^2)^{-\frac{3}{2}}(h_xh_{xx} + h_yh_{xy})}{1 + h_x^2 + h_y^2}\;.$     (9.65)

Now we will differentiate (3.67) 9.3 with respect to $ x$
$\displaystyle H_x = \frac{(2F_xM + 2FM_x - E_xN - EN_x - G_xL - GL_x)(EG-F^2)}{2(EG- F^2)^2} - \frac{(2FM- EN - GL)(E_xG + EG_x - 2FF_x)}{2(EG- F^2)^2}\;.$     (9.66)

If the surface is in Monge form with an umbilic at the origin, we have $ h_x = h_y = 0$ , which leads to $ E_x = F_x = G_x = 0$ . Consequently if $ h_{xxx} = h_{xxy} = h_{xyy} = 0$ , then $ L_x = M_x = N_x = 0$ and hence $ H_x = 0$ . Similarly we can say that if $ h_{xxy} = h_{xyy} = h_{yyy} = 0$ , then $ H_y = 0$ . Since $ H_u$ and $ H_v$ can be written as
$\displaystyle H_u = H_xx_u + H_yy_u, \;\;\; H_v = H_xx_v + H_yy_v\;.$     (9.67)

We can conclude that if $ h_{xxx} = h_{xxy} = h_{xyy} =h_{yyy} = 0$ then $ H_u = H_v = 0$ .

Note that in case $ {\bar H}_L = 0$ the term $ {\bar R}(u-u_o,v-v_o)$ is negligible for local extremum properties of the function $ \kappa(u,v) - H(u_o, v_o)$ . Consequently when $ {\bar H}_L(u,v)=0$ , or alternatively when $ \nabla H(u_o,v_o) = {\bf0}$ , the function $ \kappa(u,v) - H(u_o, v_o)$ , hence $ \kappa(u,v)$ has a local extremum at $ (u_o,v_o)$ , or more precisely, $ \kappa_{max}$ has a local minimum and $ \kappa_{min}$ has a local maximum at an umbilical point $ (u_o,v_o)$ .

\begin{figure}\centerline{\psfig{figure=fig/cone.eps,height=3in}} \par \centerli... ...ne ${\bar H}_L(u,v)$ (adapted from \protect\cite{Maekawa96a})} } \end{figure}

Figure 9.5: (a) The plane intersects the cone, (b) the plane does not intersect the cone, (c) the plane and the cone are tangent to each other (adapted from [257])
\begin{figure}\centerline{\psfig{figure=fig/cone_plane_2D.eps,height=2.3in}}\end{figure}

It is also possible that $ \kappa(u,v)$ may have a local extremum at the umbilic when $ {\bar H}_L(u,v) \neq 0$ . This is the situation when the plane $ {\bar H}_L(u,v)$ is tilted against the uv-plane.9.4 There are three possible cases, the plane $ {\bar H}_L(u,v)$ intersects the cone $ {\bar C}(u,v)$ transversally (see Fig. 9.5 $ (a)$ ), the plane $ {\bar H}_L(u,v)$ does not intersect the cone $ {\bar C}(u,v)$ , apart from its apex, (see Fig. 9.5 $ (b)$ ) and the plane $ {\bar H}_L(u,v)$ and the cone $ {\bar C}(u,v)$ are tangent to each other (see Fig. 9.5 $ (c)$ ). Figure 9.5 $ (a)$ is the case when the plane intersects the cone in two straight lines. In this case for some directions the plane has a steeper slope than the cone, thus the sum $ {\bar H}_L(u,v) \pm {\bar C}(u,v)$ does not have an extremum at $ (u_o,v_o)$ , while in case $ (b)$ the plane intersects the cone only at $ (u_o,v_o)$ , and the cone always has a steeper slope than the plane, thus $ {\bar H}_L(u,v) \pm {\bar C}(u,v)$ has a local extremum at $ (u_o,v_o)$ . Consequently we need to examine the equation $ \pm {\bar C}(u,v) = -{\bar H}_L(u,v)$ which upon squaring and using (9.60) and (9.58) can be reduced to

$\displaystyle (W_{uu} - 2H_u^2)(u - u_o)^2 + 2(W_{uv} - 2H_uH_v)(u - u_o)(v - v_o) + (W_{vv} - 2H_v^2)(v - v_o)^2 = 0\;.$     (9.68)

We can rewrite (9.68) as
$\displaystyle A(u - u_o)^2 + 2B(u - u_o)(v - v_o) + C(v - v_o)^2 = 0\;,$     (9.69)

so that we can view (9.68) as a quadratic equation with unknown $ u-u_o$ or $ v-v_o$ . If $ B^2-AC>0$ there exist two distinct real roots, and thus there will be a real intersection between the plane and the cone made up of two straight lines. If $ B^2-AC=0$ there exist two identical real roots, and thus the cone and the plane are tangent to each other, and additional evaluation of higher order terms in the Taylor expansion is necessary to decide if we have an extremum at the umbilic. If $ B^2-AC<0$ there will be no real root, and thus there is no intersection between the cone and the plane. Consequently the criterion to have a local extremum of principal curvatures, when $ {\bar H}_L(u,v) \neq 0$ or $ \nabla H(u_o,v_o) \neq {\bf0}$ , is equivalent to the condition $ B^2-AC<0$ . Hence the condition is
$\displaystyle (W_{uv}-2H_uH_v)^2-(W_{uu}-2H_u^2)(W_{vv}-2H_v^2) < 0\;,$     (9.70)

or equivalently upon using $ W(u,v) = H^2(u,v) - K(u,v)$
$\displaystyle (2HH_{uv}-K_{uv})^2-(2HH_{uu} - K_{uu})(2HH_{vv} - K_{vv}) < 0\;.$     (9.71)

Finally we can state the criterion as follows [257]:

Theorem 9.5.1. (Criterion for extrema of principal curvature functions at umbilics): If we assume that $ W(u,v)$ is at least $ C^2$ smooth and at least one of the second order partial derivatives of $ W(u,v)$ does not vanish then:

  1. If $ \nabla H = {\bf0}$ at the umbilic, then $ \kappa_{max}$ has a local minimum and $ \kappa_{min}$ has a local maximum.
  2. If $ \nabla H \neq {\bf0}$ at the umbilic, then $ \kappa_{max}$ has a local minimum and $ \kappa_{min}$ has a local maximum if and only if $ D=(2HH_{uv}-K_{uv})^2-(2HH_{uu} - K_{uu})(2HH_{vv} - K_{vv}) < 0$ provided $ D\neq 0$ . In case $ D=0$ , additional evaluation of higher order terms in the Taylor expansion is necessary.

$ D=0$ occurs when the cone and plane are tangent to each other, which is very rare. The condition $ D=0$ forces the criterion to be only sufficient and not necessary. It is quite plausible that the plane-cone tangential case (Fig. 9.5 (c)) is the rare one, while cases plane and cone are intersecting (Fig. 9.5 (a)) or plane and cone are non-intersecting (Fig. 9.5 (b)) are the generic ones.

When all the second order partial derivatives of $ W(u,v)$ vanish, we need to Taylor expand up to fourth order in (9.53), since the condition $ W(u,v)\geq 0$ implies that the third order partial derivatives must vanish. Also we need to Taylor expand up to second order in (9.59). Consequently $ \kappa(u,v)$ can be expanded in the neighborhood of an umbilic $ (u_o,v_o)$ as sum of constant, linear and quadratic terms of mean curvature and the square root of fourth order terms of $ W(u,v)$ as

$\displaystyle \kappa(u,v) = H(u_o,v_o) + H_L(u,v)\vert(u-u_o, v-v_o)\vert + (H_Q(u,v) + Q_T(u,v) + R(u-u_o,v-v_o))\vert(u-u_o, v-v_o)\vert^2$      
$\displaystyle = H(u_o,v_o) + {\bar H}_L(u,v) + {\bar H}_Q(u,v) + {\bar Q}_T(u,v) + {\bar R}(u-u_o,v-v_o)\;,$     (9.72)

where $ {\bar H}_Q(u,v)$ and $ {\bar Q}_T(u,v)$ are the second order partial derivatives terms of the Taylor expansion of the mean curvature and the square root of fourth order partial derivatives terms of the Taylor expansion of $ W(u,v)$ . It is apparent that $ {\bar H}_Q(u,v)$ and $ {\bar Q}_T(u,v)$ have stationary point at $ (u_o,v_o)$ . It follows that the plane $ {\bar H}_L$ (linear term) will determine the local behavior of the function $ \kappa(u,v)$ . This implies now that in case $ {\bar H}_L\neq 0$ , $ \kappa(u,v)$ cannot have a local extremum at the umbilic due to strong monotonicity behavior of the linear function. Therefore, if the second order partial derivatives of $ W(u,v)$ vanish at the umbilic, then $ \kappa(u,v)$ can only have an extremum in case $ {\bar H}_L(u,v)=0$ . However $ {\bar H}_L(u,v)=0$ is not sufficient to guarantee a local extremum for $ \kappa(u,v)$ , since the point $ (u_o,v_o)$ can be a saddle point for the function $ {\bar H}_Q(u,v) + {\bar Q}_T(u,v)$ .


Footnotes

... order9.2
Note that here the Taylor expansion of the square root first yields an approximation instead of the equal sign in (9.57). However absorbing here the error term of this square root Taylor expansion in the remainder of (9.57) justifies the equality sign.
...eqn:surf_explicit_H) 9.3
Equation (3.67) is based on convention (a) of the normal curvature, while we are currently using convention (b) (see Fig. 3.7 (b) and Table 3.2).
... uv-plane.9.4
Note that we use the following observation illustrated by Fig. 9.5 $ (b)$ . The term $ {\bar R}(u-u_o,v-v_o)$ is negligible for investigating the local extrema properties of the function $ \kappa(u,v)$ at the umbilic $ (u_o,v_o)$ , provided the cone $ {\bar C}(u,v)$ and the plane $ {\bar H}_L = 0$ meet only at the point $ (u_o,v_o)$ . Namely in that case we have a positive number $ \alpha$ such that $ \vert{\bar C}(u,v)-{\bar H}_L(u,v)\vert \geq \alpha\vert(u-u_o, v-v_o)\vert$ . $ \alpha$ is related to the smallest possible slope between plane and cone. Hence clearly $ R(u-u_o, v-v_o)\vert(u-u_o, v-v_o)\vert$ is negligible to $ \alpha\vert(u-u_o, v-v_o)\vert$ .
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December 2009