How to generate planes equations containing a certain ray?

Question

I want to generate planes equations ($Ax + By + Cz + D = 0$) that contain a ray $R$ with origin $O$ and a direction $D$.

Answer 1

You can use the dot product to find an equation of $(x,y,z)$ perpendicular to $D = (D_x,D_y,D_z)$. This equation gives a plane at the origin. So you just need to translate the equation the point of interest:

$$ (x,y,z) \cdot (D_x,D_y,D_z) = 0 $$

$$ (x-tD_x,y-tD_y,z-tDz) \cdot (D_x,D_y,D_z) = 0$$

Answer 2

First pick a direction $E$ that is not parallel to $D$. Then you can express a plane parametrically by $P = O + tD + uE$. Write this in coordinates, expressing $P=(x,y,z)$ and choosing coordinates $D = (d_1,d_2,d_3)$ and $E = (e_1,e_2,e_3)$: $$x = t d_1 + u e_1 $$ $$y = t d_2 + u e_2 $$ $$z = t d_3 + u e_3 $$ Now just eliminate the variables $t,u$, and the one equation that you have left over can be rewritten in the form $Ax + By + Cz + D = 0$.

Answer 3

Hint:

Find two different such planes $P_0=0$ and $P_1=0$ and form the combination

$$(1-\lambda)P_0+\lambda P1=0$$ to get all planes.

To create these two planes, take the origin, a point in the given direction and two extra points.