How may I project vectors onto a plane defined by its orthogonal vector in Python?

3d, linear-algebra, math, python

Solution

Take `(d, e, f)` and subtract off the projection of it onto the normalized normal to the plane (in your case `(a, b, c)`). So:

v = (d, e, f)
        - sum((d, e, f) *. (a, b, c)) * (a, b, c) / sum((a, b, c) *. (a, b, c))

Here, by `*.` I mean the component-wise product. So this would mean:

sum([x * y for x, y in zip([d, e, f], [a, b, c])])

or

d * a + e * b + f * c

if you just want to be clear but pedantic

and similarly for `(a, b, c) *. (a, b, c)`. Thus, in Python:

from math import sqrt

def dot_product(x, y):
    return sum([x[i] * y[i] for i in range(len(x))])

def norm(x):
    return sqrt(dot_product(x, x))

def normalize(x):
    return [x[i] / norm(x) for i in range(len(x))]

def project_onto_plane(x, n):
    d = dot_product(x, n) / norm(n)
    p = [d * normalize(n)[i] for i in range(len(n))]
    return [x[i] - p[i] for i in range(len(x))]

Then you can say:

p = project_onto_plane([3, 4, 5], [1, 2, 3])

Problem

I have a plane, `plane A`, defined by its orthogonal vector, say `(a, b, c)`. (i.e. the vector `(a, b, c)` is orthogonal to `plane A`) I wish to project a vector `(d, e, f)` onto `plane A`. How can I do it in Python? I think there must be some easy ways.

Original source