[Math] How to find the distance of a point from a plane while moving along a line

geometrylinear algebra

Let's say I've got a plane P, a point A and a vector V, how do I find the distance from A to P moving along the direction to which V points?
Also, once I've got that, how do I get the point of intersection between the line indicated by V and the plane?

Thanks

P.S.: of course this is in an N-dimensional case

Best Answer

I don't know about the $N$-dimensional space, but in the real world of Homogeneous coordinates and and projective geometry your quantities have coordinates

  • $P = \left[\vec{n},-d\right]$
  • $A = \left(\vec{a},\delta\right)$
  • $V = \left(\vec{v}, 0\right)$

The parametrized point location is $A+t V$ and the distance to the plane is

$$ \rho = (A+t V)^T (P\,) = (\vec{a}+t\,\vec{v})\cdot\vec{n}-\delta d $$ when $\vec{n}$ and $\vec{v}$ are unit vectors, and and $\delta=1$.

Example

A plane along the YZ axes at a distance of 5 from the origin is

  • $ P =\left[\vec{n},-d\right] = [ (1,0,0), -5 ] $

A starting point $\vec{a} = (0,2,0)$ moves along the $\vec{v}=(1,-2,0)$ axis. The point is located at

  • $ A = \left(\vec{a} + t \vec{v},\delta\right) = \left((t,2-2\,t,0), 1\right)$

and the distance is

  • $ \rho = (t,2-2 t,0)\cdot(1,0,0)+(-5)*1 = t-5 $

So the intersection is when $t=5$ when $Q = A + t V = \left((5,-8,0),1\right) $ or at $(5,-8,0)$ coordinates.

I suppose for the $N$-dimensional case the homogeneous coordinates are of $N+1$ size with the appropriate dot products and such defined.

There is a way to construct a homogeneous line using pluecker coordinates from the point and direction, and then directly intersect the line with the plane to yield point Q.

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