##### 参数化曲面 - 参数化曲面 - **参数化曲面**是用[[实函数|向量函数]]或[[参数方程]]来表示[[空间曲面|曲面]]上点的坐标, 具有可定向性的称为可定向曲面, 关注几何量有[[切向量]], [[法向量]], [[切平面]], [[曲面面积]] - $\mathbf{r}(u,v) = f(u,v)\mathbf{i}+g(u,v)\mathbf{j}+ h(u,v)\mathbf{k}$ , $\left\{\begin{matrix} x=f(u,v) \\ y=g(u,v) \\ z=h(u,v)\end{matrix}\right.$ , $u,v\in\mathbb{R}$ - ![[opentext_数学_参数化曲面.png]] >[!example]- 参数化曲面: 半球 > - $\mathbf{r}(u,v) = \sin(u)\cos(v)\mathbf{i} + \sin(u)\sin(v)\mathbf{j}+ \cos(u)\mathbf{k}$ , $\displaystyle u\in[0,\frac{\pi}{2}]$ , $v\in[0,2\pi]$ > - $\displaystyle\mathbf{r}(\frac{\pi}{4},\frac{\pi}{4})=(\frac{1}{2},\frac{1}{2},\frac{\sqrt{2}}{2})$ > - $\mathbf{r}_u=\cos(u)\cos(v)\mathbf{i} + \cos(u)\sin(v)\mathbf{j} -\sin(u)\mathbf{k}$ > - $\displaystyle\mathbf{r}_u(\frac{\pi}{4},\frac{\pi}{4})=(\frac{1}{2},\frac{1}{2},-\frac{\sqrt{2}}{2})$ > - $\mathbf{r}_v=-\sin(u)\sin(v)\mathbf{i} + \sin(u)\cos(v)\mathbf{j}$ > - $\displaystyle\mathbf{r}_v(\frac{\pi}{4},\frac{\pi}{4})=(-\frac{1}{2},\frac{1}{2},0)$ > - $\mathbf{N}=\mathbf{r}_u​×\mathbf{r}_v= \sin^2(u)\cos(v)\mathbf{i} + \sin^2(u)\sin(v)\mathbf{j}+ \sin(u)\cos(u)\mathbf{k}$ > - $\displaystyle\mathbf{N}(\frac{\pi}{4},\frac{\pi}{4})=(\frac{\sqrt{2}}{4},\frac{\sqrt{2}}{4},\frac{1}{2})$ > - $\displaystyle\frac{\sqrt{2}}{4}(x-\frac{1}{2})+\frac{\sqrt{2}}{4}(y-\frac{1}{2})+\frac{1}{2}(z-\frac{\sqrt{2}}{2})=0$ > - ![[opentext_数学_半球曲面.png|400]]