def newtonRaphsonRootFind(bez, point, u):
"""
Newton's root finding algorithm calculates f(x)=0 by reiterating
x_n+1 = x_n - f(x_n)/f'(x_n)
We are trying to find curve parameter u for some point p that minimizes
the distance from that point to the curve. Distance point to curve is d=q(u)-p.
At minimum distance the point is perpendicular to the curve.
We are solving
f = q(u)-p * q'(u) = 0
with
f' = q'(u) * q'(u) + q(u)-p * q''(u)
gives
u_n+1 = u_n - |q(u_n)-p * q'(u_n)| / |q'(u_n)**2 + q(u_n)-p * q''(u_n)|
"""
d = bezier.q(bez, u) - point
numerator = (d * bezier.qprime(bez, u)).sum()
denominator = (bezier.qprime(bez, u)**2 +
d * bezier.qprimeprime(bez, u)).sum()
if denominator == 0.0:
return u
else:
return u - numerator / denominator
def newtonRaphsonRootFind(bez, point, u):
"""
Newton's root finding algorithm calculates f(x)=0 by reiterating
x_n+1 = x_n - f(x_n)/f'(x_n)
We are trying to find curve parameter u for some point p that minimizes
the distance from that point to the curve. Distance point to curve is d=q(u)-p.
At minimum distance the point is perpendicular to the curve.
We are solving
f = q(u)-p * q'(u) = 0
with
f' = q'(u) * q'(u) + q(u)-p * q''(u)
gives
u_n+1 = u_n - |q(u_n)-p * q'(u_n)| / |q'(u_n)**2 + q(u_n)-p * q''(u_n)|
"""
d = bezier.q(bez, u)-point
numerator = (d * bezier.qprime(bez, u)).sum()
denominator = (bezier.qprime(bez, u)**2 + d * bezier.qprimeprime(bez, u)).sum()
if denominator == 0.0:
return u
else:
return u - numerator/denominator