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6.4.3.1 Parametric-parametric

  1. Differentiate $ {\bf c}(s) = {\bf r}^A(u_A(s),v_A(s))= {\bf r}^B(u_B(s),v_B(s))$ $ m$ times, from which we can express $ u_B^{(m)}$ and $ v_B^{(m)}$ as linear combinations of $ u_A^{(m)}$ and $ v_A^{(m)}$ (see (6.72), (6.73) for $ m=2$ ).
  2. Differentiate $ {\bf c}(s) = {\bf r}^A(u_A(s),v_A(s))= {\bf r}^B(u_B(s),v_B(s))$ $ m+1$ times and project the resulting vectors onto the normal vector $ {\bf N}$ , from which we obtain a linear equation in $ u_A^{(m)}$ , $ v_A^{(m)}$ , $ u_B^{(m)}$ , $ v_B^{(m)}$ (see (6.76) for $ m=2$ ). Substitute $ u_B^{(m)}$ and $ v_B^{(m)}$ , which are obtained from Step 1, into the resulting equation.
  3. Another additional linear equation is obtained from $ {\bf c}^{(m)}\cdot{\bf t}=c_t$ , where $ {\bf c}^{(m)}$ is the $ m$ -$ th$ order derivative of $ S^A(u_A(s), v_A(s))$ and $ c_t$ is defined in (6.47) and depends exclusively on $ \kappa$ and $ \tau$ and their derivatives (see (6.76) for $ m=2$ ).
  4. Solve the linear system for $ (u_A^{(m)}, v_A^{(m)}$ ) and substitute them into the expression of $ {\bf c}^{(m)}(s)$ in Step 1.


December 2009