区域矩
版本 11 对区域的多项式矩的支持为区域的比较、分类及性质计算提供了强大而灵活的工具.
符号式计算区域矩.
In[1]:=
![Click for copyable input](assets.zh/region-moments/In_54.png)
RegionMoment[Disk[], {0, 0}]
Out[1]=
![](assets.zh/region-moments/O_38.png)
In[2]:=
![Click for copyable input](assets.zh/region-moments/In_55.png)
RegionMoment[CapsuleShape[], {2, 0, 0}]
Out[2]=
![](assets.zh/region-moments/O_39.png)
In[3]:=
![Click for copyable input](assets.zh/region-moments/In_56.png)
RegionMoment[Cone[{{0, 0, 0}, {0, 0, 1}}, r], {2, 0, 0}]
Out[3]=
![](assets.zh/region-moments/O_40.png)
假设有一个参数未知,但知道所有零阶矩和一阶矩都为 1 的区域,求每个参数的数值.
定义区域以及对其参数的假设.
In[4]:=
![Click for copyable input](assets.zh/region-moments/In_57.png)
$Assumptions = r > 0 && x > 0 && y > 0 && z > 0;
In[5]:=
![Click for copyable input](assets.zh/region-moments/In_58.png)
cyl = Cylinder[{{0, 0, 0}, {x, y, z}}, r];
计算其零阶矩和一阶矩.
In[6]:=
![Click for copyable input](assets.zh/region-moments/In_59.png)
cfs = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
In[7]:=
![Click for copyable input](assets.zh/region-moments/In_60.png)
{m0, m100, m010, m001} = Table[RegionMoment[cyl, c], {c, cfs}]
Out[7]=
![](assets.zh/region-moments/O_41.png)
如果所有零阶矩和一阶矩是 1,求解参数.
In[8]:=
![Click for copyable input](assets.zh/region-moments/In_61.png)
sol = Solve[{m0 == 1, m100 == 1, m010 == 1, m001 == 1, $Assumptions}]
Out[8]=
![](assets.zh/region-moments/O_42.png)
得到区域.
In[9]:=
![Click for copyable input](assets.zh/region-moments/In_62.png)
cyl /. sol
Out[9]=
![](assets.zh/region-moments/O_43.png)
半径的近似值.
In[10]:=
![Click for copyable input](assets.zh/region-moments/In_63.png)
N[cyl /. sol]
Out[10]=
![](assets.zh/region-moments/O_44.png)