# 用Mathematica软件求解三维图形投影的边界线方程Solving the Boundary Equation of 3D Projection with Mathematica

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In the case of a given surface equation, we need to obtain the equation of the boundary line of the projection of multiple surfaces in three coordinate planes. We can use Mathematica to draw space graphics to visualize the problem. Through the Mathematica software function of visualization technology and choice of options, we can obtain the ideal space graph, easy to solve the boundary equation problem and solve other related projection problems, such as the simplification of the integral solution process.

1. 引言

2. Mathematica绘制空间图形的基本方法

Mathematica软件中绘制空间曲面的基本函数为Plot3D，命令Plot3D主要用于绘制二元函数z = f(x, y)的图形，该命令的基本格式为Plot3D [f[x, y], {x, x1, x2}, {y, y1, y2}, 选项]。其中f[x, y]是x，y的二元函数，x1，x2表示x的作图范围，y1，y2表示y的作图范围。Plot3D有许多选项，其中常用的如PlotPoints和ViewPoint。PlotPoints的默认值为15，可以增加一些点以使曲面更加精致。选项ViewPoint用于选择图形的视点，默认值为ViewPoint- > {1.3, −2.4, 2.0}。

3. 曲面方程的参数化

Figure 1. Paraboloid of revolution

Figure 2. Sphere

RegionPlot3D [xyz < 1, {x, −5, 5}, {y, −5, 5}, {z, −5, 5}, PlotStyle → Directive [Yellow, Opacity [0.5]], Mesh → None]。

4. 确定投影方程

4.1. 演示空间曲面及其投影的生成过程

4.2. 确定投影边界线方程并积分

Figure 3. Region border diagram

Figure 4. Three-dimensional projection drawing process

XOY面： $y=x,y={x}^{2};$

XOZ面： $z=\frac{1}{4}{x}^{2},x=0,z=\frac{1}{2}{x}^{2};$

YOZ面： $z=\frac{1}{4}{y}^{2},y=0,z=\frac{1}{2}{y}^{2}.$

$\begin{array}{c}\underset{D}{\iint }f\left(x,y\right)\text{d}x\text{d}y=\underset{D}{\iint }\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\text{d}x\text{d}y={\int }_{\partial D}P\text{d}x+Q\text{d}y\\ =\underset{a}{\overset{b}{\int }}\text{d}x\underset{y1\left(x\right)}{\overset{y2\left(x\right)}{\int }}f\left(x,y\right)\text{d}y=\underset{a}{\overset{b}{\int }}\text{d}y\underset{x1\left(y\right)}{\overset{x2\left(y\right)}{\int }}f\left(x,y\right)\text{d}x\end{array}$

5. 总结

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