Case of implicit surfaces
- Undeformed implicit surface \(S=\{p\in\mathbb{R}^3\;|\;\;g(p)=0\}\)
- Given a vector field \(u\)
- The implicit surface \(S'\) advected by \(u\) is defined as
-
- Note: Only velocity normal to the surface changes the shape
- \(g\) is a space-time function \(g(p,t)\)
- \(g(p+u\mathrm{d}t,t+\mathrm{d}t)=g(p,t)=0\)
-
- \(\Rightarrow\) The differential \(\mathrm{d}g = 0\)
- \(\Rightarrow \frac{\partial g}{\partial t}\,\mathrm{d}t+\nabla g \cdot \mathrm{d}p=0\)
- \(\Rightarrow \frac{\partial g}{\partial t}+\nabla g \cdot \underbrace{\frac{\mathrm{d}p}{\mathrm{d}t}}_{u}=0\)
