# coding: utf-8

# # Point-normal form and signed distance

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import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt


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def draw_arrow(v, **kwargs):
    pt.arrow(0, 0, v[0], v[1], length_includes_head=True,
             head_width=0.1, **kwargs)
    
def set_up_plot():
    pt.xlim([-2,2])
    pt.ylim([-2,2])
    pt.gca().set_aspect("equal")
    pt.grid()


# Let's grab some vector $v$:

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np.random.seed(18)
v = np.array([1.6, 0.4])
v


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set_up_plot()
draw_arrow(v, color="blue")


# ------------
# Now let's find a unit normal vector $n$:

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b = np.random.randn(2)
b = b - b.dot(v)/v.dot(v)*v
n = b/la.norm(b, 2)


# Check:

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print("Norm =", la.norm(n))
print("n.v =", n.dot(v))


# Plot:

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set_up_plot()
draw_arrow(v, color="blue")
draw_arrow(n, color="red")


# Now let's grab a point $p$ and compute $r$:

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np.random.seed(355342)
p = np.random.randn(2)

r = p.dot(n)


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set_up_plot()
draw_arrow(v, color="blue")
draw_arrow(n, color="red")
draw_arrow(p)


# Next, evaluate signed distance on a grid of points in the plane:

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xs = np.linspace(-2, 2, 100)
ys = np.linspace(-2, 2, 100)


# make an grid of points in the plane
xys = np.empty((2, 100, 100))
xys[0] = xs
xys[1] = xs.reshape(-1, 1)

signed_dist = (
    xys[0]*n[0] + xys[1]*n[1] - r
    )


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set_up_plot()
draw_arrow(v, color="blue")
draw_arrow(n, color="red")
draw_arrow(p)

pt.imshow(signed_dist, extent=(-2, 2, 2, -2), cmap="RdBu", vmin=-3, vmax=3)
pt.colorbar()


# *   Where is the line parallel to $v$ through $p$? (that satisfies $x\cdot n -r = 0$)?
# 
#     (Hint: Look for plot values = 0.)
# *   Can you read off the distance of the line from the origin?
# *   Can you read off the distance of the line from $v$?
# *   Can you read off the distance between the line and the top left corner of the plot?
# *   How was each such distance computed?

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