A. Appendix

The equations given in this appendix are replacements for certain equations given in the body of the paper. By using these, several of Baraff & Witkin's approximations are avoided. However, my tests show that the results are less pleasing.

A..1 Per-Triangle Quantities

The first and second derivatives of the normalised vectors ${\bf {\hat w}}_u$ and ${\bf {\hat w}}_v$ are

$$\frac{\partial {\bf {\hat w}}_u}{\partial {\bf x}_{m_s}} = \frac{1}{\begin{Vmatrix}{\bf w}_u\end{Vmatrix}} \frac{\partial ... ...al {\bf x}_{m_x}} \left( {\bf I}_s- {\bf {\hat w}}_{u_s}{\bf {\hat w}}_u\right)$$ (A.1)

$$\frac{\partial {\bf {\hat w}}_v}{\partial {\bf x}_{m_s}} = \frac{1}{\begin{Vmatrix}{\bf w}_v\end{Vmatrix}} \frac{\partial ... ...al {\bf x}_{m_x}} \left( {\bf I}_s- {\bf {\hat w}}_{v_s}{\bf {\hat w}}_v\right)$$ (A.2)

$$\frac{\partial^2 {\bf {\hat w}}_u}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}} = -\frac{1}{\begin{Vmatrix}{\bf w}_u\end{Vmatrix}} \frac{\partial {... ...\partial {\bf {\hat w}}_{u_s}}{\partial {\bf x}_{n_t}} {\bf {\hat w}}_u \right)$$

$$- \frac{\partial {\bf w}_{u_x}}{\partial {\bf x}_{m_x}} \frac{\pa... ...\bf {\hat w}}_u) {\bf {\hat w}}_{u_t}}{\begin{Vmatrix}{\bf w}_u\end{Vmatrix}^2}$$ (A.3)

$$\frac{\partial^2 {\bf {\hat w}}_v}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}} = -\frac{1}{\begin{Vmatrix}{\bf w}_v\end{Vmatrix}} \frac{\partial {... ...\partial {\bf {\hat w}}_{v_s}}{\partial {\bf x}_{n_t}} {\bf {\hat w}}_v \right)$$

$$- \frac{\partial {\bf w}_{v_x}}{\partial {\bf x}_{m_x}} \frac{\pa... ...\bf {\hat w}}_v) {\bf {\hat w}}_{v_t}}{\begin{Vmatrix}{\bf w}_v\end{Vmatrix}^2}$$ (A.4)

A..2 Shear Condition

If we remove the assumption that the ${\bf {\hat w}}_u$ and ${\bf {\hat w}}_v$ vectors are unstretched, then equation (19) should be replaced with

$$C = \alpha {\bf {\hat w}}_u\cdot {\bf {\hat w}}_v$$ (A.5)

$$\frac{\partial C}{\partial {\bf x}_{m_s}} = \alpha \left( \frac{\partial {\bf {\hat w}}_u}{\partial {\bf x}_{... ...hat w}}_u\cdot \frac{\partial {\bf {\hat w}}_v}{\partial {\bf x}_{m_s}} \right)$$ (A.6)

$$\frac{\partial^2 C}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}} = \alpha \bigg( \frac{\partial^2 {\bf {\hat w}}_u}{\partial {\bf x}... ...l {\bf x}_{m_s}} \cdot \frac{\partial {\bf {\hat w}}_v}{\partial {\bf x}_{n_t}}$$

$$+ \frac{\partial {\bf {\hat w}}_u}{\partial {\bf x}_{n_t}} \cdot ... ...tial^2 {\bf {\hat w}}_u}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}} \biggr)$$ (A.7)

A..3 Bend Condition

If we remove the assumption that the normal and edge vectors have constant length, we must replace equations (40) and (43) with those shown below.

$$\frac{\partial {\bf {\hat n}}^A}{\partial {\bf x}_{m_s}} = \left({\bf I}- {\bf {\hat n}}^A({\bf {\hat n}}^A)^T\right) \frac{ S_s( {\bf q}^A_m ) }{ \begin{Vmatrix}{\bf n}^A\end{Vmatrix} }$$ (A.8)

$$\frac{\partial {\bf {\hat n}}^B}{\partial {\bf x}_{m_s}} = \left({\bf I}- {\bf {\hat n}}^B({\bf {\hat n}}^B)^T\right) \frac{ S_s( {\bf q}^B_m ) }{ \begin{Vmatrix}{\bf n}^B\end{Vmatrix} }$$ (A.9)

$$\frac{\partial {\bf {\hat e}}}{\partial {\bf x}_{m_s}} = ({\bf I}- {\bf {\hat e}}{\bf {\hat e}}^T) \frac{ q^e_m {\bf I}_s}{ \begin{Vmatrix}{\bf e}\end{Vmatrix} }$$ (A.10)

$$\frac{\partial^2 {\bf {\hat n}}^A}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}}$$

$$= - \left( \frac{\partial {\bf {\hat n}}^A}{\partial {\bf x}_{n_t}}... ...right)^T \right) \frac{S_s({\bf q}^A_m)}{\begin{Vmatrix}{\bf n}^A\end{Vmatrix}}$$

$$+ \frac{{\bf I}- {\bf {\hat n}}^A({\bf {\hat n}}^A)^T}{\begin{Vma... ...{Vmatrix} S_s\left( \frac{\partial {\bf q}^A_m}{\partial {\bf x}_{n_t}} \right)$$

$$\quad\quad\quad\quad\quad\quad\quad - \left( S_t({\bf q}^A_n) \cdot {\bf {\hat n}}^A \right) S_s({\bf q}^A_m) \bigg)$$

$$\frac{\partial^2 {\bf {\hat n}}^B}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}}$$

$$= - \left( \frac{\partial {\bf {\hat n}}^B}{\partial {\bf x}_{n_t}}... ...right)^T \right) \frac{S_s({\bf q}^B_m)}{\begin{Vmatrix}{\bf n}^B\end{Vmatrix}}$$

$$+ \frac{{\bf I}- {\bf {\hat n}}^B({\bf {\hat n}}^B)^T}{\begin{Vma... ...{Vmatrix} S_s\left( \frac{\partial {\bf q}^B_m}{\partial {\bf x}_{n_t}} \right)$$

$$\quad\quad\quad\quad\quad\quad\quad - \left( S_t({\bf q}^B_n) \cdot {\bf {\hat n}}^B \right) S_s({\bf q}^B_m) \bigg)$$

$$\frac{\partial^2 {\bf {\hat e}}}{\partial {\bf x}_{m_s} \partial {\bf x}_{n_t}}$$

$$= - \left( \frac{\partial {\bf {\hat e}}}{\partial {\bf x}_{n_t}} {... ...{n_t}} \right)^T \right) \frac{S_s(q^e_m)}{\begin{Vmatrix}{\bf e}\end{Vmatrix}}$$

$$- \frac{{\bf I}- {\bf {\hat e}}{\bf {\hat e}}^T}{\begin{Vmatrix}{\bf e}\end{Vmatrix}^2} q^e_m q^e_n {\bf {\hat e}}_t {\bf I}_s$$ (A.11)