from __future__ import division
from matplotlib import  patches

fig, ax = subplots()
fig.set_figheight(5)
x = arange(6)
y= matrix([[2],
           [3]])
ax.plot(x,x)
ax.plot(*y,marker='s',color='r')
ax.add_patch(patches.Circle(y,radius=.5,alpha=0.75,color='pink'))
ax.annotate('Find point along\n line closest\n to red square',
            fontsize=12,xy=(2.5,2.5),
            xytext=(3,1.5),
            arrowprops={'facecolor':'blue'})
ax.axis(xmax=5,ymax=5)
ax.set_aspect(1)
ax.grid()

fig, ax = subplots()
fig.set_figheight(5)

x = arange(6)
y= matrix([[2],
           [3]])
ax.plot(x,x)
ax.plot(*y,marker='o',color='r')
ax.add_patch(patches.Circle(y,radius=1/sqrt(2.),alpha=0.75,color='pink'))

v = matrix([[1],[1]])  
Pv = v*v.T/ (v.T*v) # projection operator
ax.plot(*(Pv*y),marker='s',color='g')
ax.add_line( Line2D( (y[0,0], 2.5), (y[1,0],2.5) ,color='g',linestyle='--'))
ax.add_line( Line2D( (y[0,0], 0), (y[1,0],0) ,color='r',linestyle='--'))
ax.annotate( 'Closest point is\nperpedicular\nto line and tangent\nto circle',
              fontsize=12,xy=(2.6,2.5),
              xytext=(3,1.5),
              arrowprops={'facecolor':'blue'})

ax.text(.7,1.5,r'$\mathbf{y}$',fontsize=24,color='r')
ax.set_aspect(1)
ax.grid()

theta = 120/180.*pi   # rotation angle for ellipse
v = matrix([[1],[1]])

# rotation matrix
U = matrix([[ cos(theta), sin(theta)],
            [-sin(theta), cos(theta)]])

# diagonal weight matrix
S = matrix([[5,0], # change diagonals to define axes of ellipse
            [0,1]])
Q = U.T*S*U

Pv = (v)*v.T*(Q)/(v.T*(Q)*v)      # projection operator
err = sqrt((y-Pv*y).T*Q*(y-Pv*y)) # error length
xhat = Pv*y                       # closest point on line

fig, ax = subplots()
fig.set_figheight(5)

ax.plot(*y,marker='o',color='r')
ax.plot(x,x)
ax.plot(*(xhat),marker='s',color='r')
ax.add_patch( patches.Ellipse(y,err*2/sqrt(S[0,0]),err*2/sqrt(S[1,1]),
                                angle=theta/pi*180,color='pink',
                                alpha=0.5))
ax.add_line( Line2D( (y[0,0], 0), (y[1,0],0) ,color='r',linestyle='--'))
ax.add_line( Line2D( (y[0,0],xhat[0,0]), 
                     (y[1,0],xhat[1,0]) ,
                      color='g',linestyle='--'))
ax.annotate( '''Closest point is
tangent to the 
ellipse and 
"perpendicular" 
to the line 
in the sense of the 
weighted/rotated
distance
''',
              fontsize=12,xy=(xhat[0,0],xhat[1,0]),
              xytext=(3.5,1.5),
              arrowprops={'facecolor':'blue'})

ax.axis(xmax=6,ymax=6)
ax.set_aspect(1)
ax.grid()