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curvature equation for implicit functions, level sets is usually given in two forms: One is the divergence of gradient of the unit normal: $\kappa = \bigtriangledown \cdot \frac{\bigtriangledown \phi}{|\bigtriangledown \phi|}$ The other is $\kappa = \frac{\phi_{xx}\phi_y^2 - 2\phi_x\phi_y\phi_{xy} + \phi_{yy}\phi_x^2}{(\phi_x^2+\phi_y^2)^{3/2}}$ How do we derive the second equation from the first? |
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Just expand in coordinates: $$\begin{align}\kappa &= \nabla \cdot \frac{\nabla \phi}{|\nabla \phi|} = \nabla \cdot \frac{(\phi_x,\phi_y)}{\sqrt{\phi_x^2+\phi_y^2}}\\ &=\left(\partial_x \frac{\phi_x}{\sqrt{\phi_x^2+\phi_y^2}}\right)+ \left(\partial_y \frac{\phi_y}{\sqrt{\phi_x^2+\phi_y^2}}\right) \\ &= \frac{\phi_{xx}}{\sqrt{\phi_x^2+\phi_y^2}} - \frac{\phi_x (\phi_x\phi_{xx}+\phi_y\phi_{xy})} {(\phi_x^2+\phi_y^2)^{3/2}} + \frac{\phi_{yy}}{\sqrt{\phi_x^2+\phi_y^2}} - \frac{\phi_y(\phi_x\phi_{xy}+\phi_y\phi_{yy})} {(\phi_x^2+\phi_y^2)^{3/2}} \\ &= \frac{\phi_{xx}(\phi_x^2+\phi_y^2) - \phi_x (\phi_x\phi_{xx}+\phi_y\phi_{xy}) +\phi_{yy}(\phi_x^2+\phi_y^2) - \phi_y(\phi_x\phi_{xy}+\phi_y\phi_{yy})}{(\phi_x^2+\phi_y^2)^{3/2}}\\ &= \frac{\phi_{xx}\phi_y^2 - 2\phi_x\phi_y\phi_{xy} + \phi_{yy}\phi_x^2}{(\phi_x^2+\phi_y^2)^{3/2}} \end{align}$$ |
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