I have 4 points in R4 and their co-ordinates:
P1:[x1, y1, z1, w1]
P2:[x2, y2, z2, w2]
P3:[x3, y3, z3, w3]
P4:[x4, y4, z4, w4]
How do I now define a hyperplane out of these points in Python?
Also given that I have the equations of 4 hyperplanes how do I get their intersection (which should be a point)?
Thanks! O.
Equation of an hyperplane.
An hyper plane in Rn can be described, given a non zero constant K and a set of coefficients a ={a_1 ... a_n}, as the set of points x =(x_1 .. x_n) that solve the equation
Sum(a_n * x_n) = k
choosing k=1 in R4, and with
X= ( P1;P2;P3;P4)
You can solve your coefficients a by doing
X a = 1
a = X^-1 * 1
Part 2 is more of of the same.
Having 4 sets of equations
x a = k
the point that belongs to them all can be solved as
x = k ' A^-1
On numpy that's:
import numpy as np
def hyper4(p1,p2,p3,p4):
X=np.matrix([p1,p2,p3,p4])
k=np.ones((4,1))
a=numpy.matrix.dot(np.linalg.inv(X), k)
print "equation is x * %s = 1" % a
return a
Usage:
hyper4([0,0,1,1],[0,3,3,0],[0,5,2,0],[1,0,0,7])
And for the point
a1=hyper4(P1,P2,P3,P4)
a2=hyper4(P5,P6,P7,P8)
a3=hyper4(P9,P10,P11,P12)
a4=hyper4(P13,P14,P15,P16)
A=np.hstack([a1,a2,a3,a4])
k=np.ones((1,4))
x=numpy.matrix.dot(k, np.linalg.inv(A))
print "your point is %s" % x
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