```python
import sympy as sy
sy.init_printing(use_unicode=False)
```


```python
x1,x2,x3,x4=sy.symbols('x(1:5)')
y1,y2,y3,y4=sy.symbols('y(1:5)')
```


```python
xs=[x1,x2,x3,x4]
ys=[y1,y2,y3,y4]
```


```python
b1x,b2x,b3x,b4x=sy.symbols('b_{(1:5)x}')
b1y,b2y,b3y,b4y=sy.symbols('b_{(1:5)y}')
e1x,e2x,e3x,e4x=sy.symbols('e_{(1:5)x}')
e1y,e2y,e3y,e4y=sy.symbols('e_{(1:5)y}')
```


```python
b1 = [b1x,b1y]
e1 = [e1x,e1y]

b2 = [b2x,b2y]
e2 = [e2x,e2y]

b3 = [b3x,b3y]
e3 = [e3x,e3y]

b4 = [b4x,b4y]
e4 = [e4x,e4y]
```


```python
#example
B1 = [1, -1]
E1 = [-1, 1]

B2 = [-1, 2]
E2 = [1, 4]

B3 = [2, 4]
E3 = [4, 2]

B4 = [4, 1]
E4 = [2, -1]
```


```python
def on_segment(b,e,x,y):
    return (e[0]-b[0])*(y-b[1])-(e[1]-b[1])*(x-b[0])

def orthogonal(lx, ly, cx, cy, rx, ry):
    return (lx-cx)*(rx-cx)+(ly-cy)*(ry-cy)
```


```python
f1=orthogonal(x1, y1, x2, y2, x3, y3)
f2=orthogonal(x2, y2, x3, y3, x4, y4)
f3=orthogonal(x3, y3, x4, y4, x1, y1)
f4=orthogonal(x4, y4, x1, y1, x2, y2)
fs=[f1,f2,f3,f4]
fs
```


```python
g1=on_segment(b1,e1,x1,y1)
g2=on_segment(b2,e2,x2,y2)
g3=on_segment(b3,e3,x3,y3)
g4=on_segment(b4,e4,x4,y4)
gs=[g1,g2,g3,g4]
gs
```


```python
lf1,lf2,lf3,lf4=sy.symbols('\lambda_{f_(1:5)}')
lfs=[lf1,lf2,lf3,lf4]
lfs
```


```python
lg1,lg2,lg3,lg4=sy.symbols('\lambda_{g_(1:5)}')
lgs=[lg1,lg2,lg3,lg4]
lgs
```


```python
area=(x4-x1)*(y2-y1)-(x2-x1)*(y4-y1)
```


```python
area
```


```python
Phi=area-sy.Matrix(lfs).dot(sy.Matrix(fs))-sy.Matrix(lgs).dot(sy.Matrix(gs))
Phi
```


```python
[sy.diff(Phi,x1),
sy.diff(Phi,y1),
sy.diff(Phi,x2),
sy.diff(Phi,y2),
sy.diff(Phi,x3),
sy.diff(Phi,y3),
sy.diff(Phi,x4),
sy.diff(Phi,y4)]
```


```python
[sy.diff(Phi,lf1),
sy.diff(Phi,lf2),
sy.diff(Phi,lf3),
sy.diff(Phi,lf4),
sy.diff(Phi,lg1),
sy.diff(Phi,lg2),
sy.diff(Phi,lg3),
sy.diff(Phi,lg4)]
```


```python

```

2b6f86803c28a960a7175a668fa27f601957c12f
ipynb
Jupyter Notebook
optimizer/geoetryModel/InnerRectangle.ipynb
terasakisatoshi/pythonCodes
baee095ecee96f6b5ec6431267cdc6c40512a542
MIT
optimizer/geoetryModel/InnerRectangle.ipynb
terasakisatoshi/pythonCodes
baee095ecee96f6b5ec6431267cdc6c40512a542
MIT
optimizer/geoetryModel/InnerRectangle.ipynb
terasakisatoshi/pythonCodes
baee095ecee96f6b5ec6431267cdc6c40512a542
MIT
Qwen/Qwen-72B
1. YES
2. YES