CODEWITHSUNDEEP
algorithmmathgeometry
vicrion
vicrion

Reputation: 1733

Algorithm to find a vector parallel to a plane and perpendicular to another vector

I'm trying to derive a formula to extract vector u.

vector-plane-vector setup image

I'm given some initial data:

  1. Plane F with the method to extract its normal n = F->normal().
  2. vector c that does not lie within the plane F and passes through some point E that also does not lie within the plane F.

And some constrains to use:

  1. The desired vector u is perpendicular to the vector c.
  2. Vector u is also perpendicular to some vector r which is not given. The vector r is parallel to the plane F and also perpendicular to the vector c. Therefore, we can say the vectors c, r and u are orthogonal.

Let's denote * as dot product, and ^ operator is cross product between two 3d vectors.

The calculation of the vector u is easy by using cross product: vec3 u = c^r. So, my whole task is narrowed down to how to find the vector r which is parallel to a given plane F and at the same time perpendicular to the given vector c.

Because we know that r is parallel to F, we can use plane's normal and dot product: n*r = 0. Since r is unknown, an infinite number of lines can satisfy the aforementioned equation. So, we can also use the condition that r is perpendicular to c: r*c = 0.

To summarize, there are two dot-product equations that should help us to find the vector r: 

r*c = 0;
r*n = 0;

However, I am having hard time trying to figure out how to obtain the vector r coordinates provided the two equations, in algorithmic way. Assuming r = (x, y, z) and we want to find x, y and z; it does not seem possible from only two equations: 

x*c.x + y*c.y + z*c.z = 0;
x*n.x + y*n.y + z*n.z = 0;

I feel like I'm missing something, e.g., I need a third constrain. Is there anything else needed to extract x, y and z? Or do I have a flaw in my logic?

Upvotes: 1

Views: 2555

Answers (1)

Peter de Rivaz
Peter de Rivaz

Reputation: 33509

You can find the vector r by computing the cross product of n and c.

This will be automatically satisfy r.c=r.n=0

You are right that your two equations will have multiple solutions. The other solutions are given by any scalar multiple of r.

Upvotes: 2

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