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3.5.2 Implicit surfaces

Using (3.66), (3.67) for an explicit surface, we can derive equations for the Gaussian and mean curvature of an implicit surface $ f(x,y,z)=0$ . At a point where $ f_z \neq 0$ , $ z$ can be expressed as a function of $ x$ and $ y$ , say $ z=h(x,y)$ [166]. In such cases variables $ x$ and $ y$ are independent but $ z$ is a function of both $ x$ and $ y$ . Since $ f$ constantly satisfies the equation $ f(x,y,z)=0$ , the partial differentiation of $ f$ with respect to the independent variable $ x$ (by holding $ y$ fixed) must vanish [166]. Thus,
$\displaystyle \left(\frac{\partial f}{\partial x}\right)_y = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial x}=0\;,$     (3.68)

where $ \left(\frac{\partial f}{\partial x}\right)_y$ on the left-hand side is considered as $ f$ being expressed in terms of $ x$ and $ y$ only and $ y$ is held constant in the differentiation with respect to $ x$ , while $ \frac{\partial f}{\partial x}$ on the right-hand side is considered as $ f$ being expressed in terms of $ x$ , $ y$ , $ z$ and $ y$ , $ z$ are held constant in the $ x$ differentiation. Similarly we have
$\displaystyle \left(\frac{\partial f}{\partial y}\right)_x = \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial y}=0\;.$     (3.69)

Consequently we have
$\displaystyle h_x = -\frac{f_x}{f_z},\;\;\;\;\;\;h_y=-\frac{f_y}{f_z}\;.$     (3.70)

The second order partial derivatives $ h_{xx}$ , $ h_{xy}$ , $ h_{yy}$ are provided by differentiating (3.70). For example,

$\displaystyle h_{xx}=-\frac{\left(\frac{\partial f_x}{\partial x}\right)_yf_z -... ...}\right)_yf_x}{f_z^2}=\frac{2f_xf_zf_{xz} - f_x^2f_{zz} -f_z^2f_{xx}}{f_z^3}\;.$     (3.71)

Similarly, we have
$\displaystyle h_{xy}= \frac{f_xf_zf_{yz} + f_yf_zf_{xz} -f_xf_yf_{zz} - f_z^2f_{xy}}{f_z^3}\;,$     (3.72)
$\displaystyle h_{yy}= \frac{2f_yf_zf_{yz} - f_y^2f_{zz} -f_z^2f_{yy}}{f_z^3}\;.$     (3.73)

Equations (3.70) to (3.73) may be substituted into (3.63) and (3.65) to obtain the first and second fundamental form coefficients, and into (3.66) and (3.67) to compute the Gaussian and mean curvature of an implicit surface. If $ f_z=0$ , alternate formulae may be found by cyclic permutation of $ x$ , $ y$ , $ z$ .

Table 3.1: Classification of implicit quadrics

Implicit Quadrics		 $ \zeta $ $ \eta$ $ \xi$ $ \delta$

Ellipsoid		 1 1 1 1

Hyperboloid of One Sheet		 1 1 -1 1
  1 -1 1 1
  -1 1 1 1

Hyperboloid of Two Sheets		 1 -1 -1 1
  -1 1 -1 1
  -1 -1 1 1

Elliptic Cone		 1 1 -1 0
  1 -1 1 0
  -1 1 1 0

Elliptic Cylinder		 1 1 0 1
  1 0 1 1
  0 1 1 1

Hyperbolic Cylinder		 1 -1 0 1
  -1 1 0 1
  1 0 -1 1
  -1 0 1 1
  0 1 -1 1
  0 -1 1 1

For every quadric surface, it is possible to find a suitable 3-D rotation such that the cross terms $ dxy$ , $ eyz$ and $ fxz$ cancel out in (1.15). If a quadric surface has a center3.2, its axes can be translated to the center as origin so that the equation of the quadric surface does not have any first degree terms [79]. Therefore after these transformations the implicit quadrics, ellipsoids, hyperboloids of one and two sheets, elliptic cones, elliptic cylinders and hyperbolic cylinders can be expressed in a standard form

$\displaystyle f(x,y,z)=\zeta\frac{x^2}{a^2} + \eta\frac{y^2}{b^2} + \xi\frac{z^2}{c^2} - \delta = 0\;,$     (3.74)

where $ \zeta $ , $ \eta$ and $ \xi$ take values either -1, 0 or 1 and $ \delta$ takes values either 0 or 1, depending on the classification of quadrics (see Table 3.1).

By evaluating (3.70), (3.71), (3.72) and (3.73) for $ f$ given in (3.74), and substituting into (3.66) and (3.67), we obtain

$\displaystyle K(x,y,z) = \frac{\zeta\eta\xi\delta}{a^2b^2c^2(\zeta^2\frac{x^2}{a^4} + \eta^2\frac{y^2}{b^4} + \xi^2\frac{z^2}{c^4} )^2}\;,$     (3.75)
$\displaystyle H(x,y,z) = -\frac{\zeta^2b^2c^2(\xi b^2 + \eta c^2)x^2 + \eta^2a^... ...frac{x^2}{a^4} + \eta^2\frac{y^2}{b^4} + \xi^2\frac{z^2}{c^4} )^\frac{3}{2}}\;,$     (3.76)

where $ (x, y, z)$ satisfy $ f(x,y,z)=0$ . The principal curvatures can be obtained by substituting (3.75) and (3.77) into
$\displaystyle \kappa(x,y,z) = H \pm \sqrt{H^2 - K}\;,$     (3.77)

where we will not show the substituted expression because it is too cumbersome.

The curvatures of a hyperbolic cylinder ( $ \zeta=\delta=1$ , $ \eta=-1$ , $ \xi=0$ )

$\displaystyle f(x,y)=\frac{x^2}{a^2} -\frac{y^2}{b^2} -1 =0\;,$     (3.78)

can be obtained by evaluating (3.75), (3.76) and (3.77) resulting
$\displaystyle K=0,\;\;\;\;\;H=\frac{b^2x^2 - a^2 y^2}{2a^4b^4(\frac{x^2}{a^4}+\frac{y^2}{b^4})^\frac{3}{2}}\;,$     (3.79)
$\displaystyle \kappa_{max}=\frac{b^2x^2 - a^2 y^2}{a^4b^4(\frac{x^2}{a^4}+\frac{y^2}{b^4})^\frac{3}{2}},\;\;\;\;\;\kappa_{min}=0\;,$     (3.80)

where $ (x,y)\in f(x,y)=\frac{x^2}{a^2} -\frac{y^2}{b^2} -1 =0$ .

Similarly, the curvatures of an ellipsoid ( $ \zeta=\eta=\xi=\delta=1$ )

$\displaystyle f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}-1 =0\;,$     (3.81)

are evaluated as
$\displaystyle K = \frac{1}{a^2b^2c^2\left(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \... ...eft(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4} \right)^\frac{3}{2}}\;,$     (3.82)
$\displaystyle \kappa = \frac{x^2 + y^2 + z^2 - a^2 -b^2 -c^2} {2a^2b^2c^2 \left... ...eft(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4} \right)^\frac{3}{2}}\;,$     (3.83)

where $ (x,y,z)\in f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}-1 =0$ . Here we note that in the derivation of the mean curvature in (3.82), we used (3.81) to simplify the expression. For the case of a sphere of radius $ R$ , (3.81) simplifies to $ f(x,y,z) = \frac{1}{R^2}(x^2 + y^2 +z^2) -1 =0$ , and (3.82) and (3.83) simplify to $ K=\frac{1}{R^2}$ , $ H=\kappa = -\frac{1}{R}$ , which shows that a sphere is made of entirely nonflat umbilics (see Sects. 9.1 and 9.2). The negative sign comes from the sign convention of the curvature (see Fig. 3.7 and Table 3.2).

Finally, the curvatures of an elliptic cone ( $ \zeta = \eta = 1$ , $ \xi=-1$ and $ \delta = 0$ )

$\displaystyle f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0\;,$     (3.84)

excluding the apex (0,0,0) are given by
$\displaystyle K =0, \;\;\; H = -\frac{x^2 + y^2 + z^2 }{2a^2b^2c^2\left(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4} \right)^\frac{3}{2}}\;,$     (3.85)
$\displaystyle \kappa_{max} = 0, \kappa_{min}= -\frac{x^2 + y^2 + z^2 }{a^2b^2c^2\left(\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4} \right)^\frac{3}{2}},$     (3.86)

where $ (x,y,z)\in f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0$ . Here we also used (3.84) to simplify the expression of mean curvature in (3.85).


Footnotes

... center3.2
A center of a quadric surface is defined as a point bisecting every chord passing through it [79]. Here chord is a line which joins two points on a surface. Ellipsoids and hyperboloids have centers, while paraboloids do not have centers. The elliptic/hyperbolic cylinder is a limiting case of the ellipsoid/hyperboloid and the elliptic cone is asymptotic to hyperboloids of one and two sheets.
next up previous contents index
Next: 3.6 Euler's theorem and Up: 3.5 Gaussian and mean Previous: 3.5.1 Explicit surfaces   Contents   Index
December 2009