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Next: 6.4.3.3 Parametric-implicit Up: 6.4.3 Third and higher Previous: 6.4.3.1 Parametric-parametric   Contents   Index

6.4.3.2 Implicit-implicit

  1. Total differentiate $ f(x,y,z)=0$ $ m$ times with respect to $ s$ , which will provide a linear equation in $ (x^{(m)}, y^{(m)}, z^{(m)})$ .
  2. Equate the projections of the $ (m+1)$ -$ th$ order derivative $ (x^{(m+1)}$ , $ y^{(m+1)}$ , $ z^{(m+1)})$ of the two implicit surfaces onto the unit normal vector (see (6.53)) to obtain a linear equation in $ (x^{(m)}, y^{(m)}, z^{(m)})$ .
  3. The third linear equation in $ (x^{(m)}, y^{(m)}, z^{(m)})$ can be obtained from $ x'x^{(m)} + y'y^{(m)}+z'z^{(m)}=c_t$ , where $ c_t$ , defined in (6.47), depends exclusively on $ \kappa$ and $ \tau$ and their derivatives.
  4. Solve the system of three linear equations for $ (x^{(m)}$ , $ y^{(m)}$ , $ z^{(m)})$ .


December 2009