# Real World 3D Geometry Question: Finding the Intersection Point of a Line and a Plane

My team and I are developing a product which includes LASER systems and 3D geometry. We are stuck with a problem and I have tried my best to frame it into mathematical question. Here it goes,

$$P_1=(x_1, y_1, z_1)$$ and $$P_2=(x_2, y_2, z_2)$$ are two arbitrary points in space. Two lines with angles given with respective to $$XY$$, $$YZ$$ & $$XZ$$ planes $$L_1(α_1, β_1, γ_1)$$ and $$L2(α_2, β_2, γ_2)$$ pass through $$P_1$$ & $$P_2$$ respectively.

Lines $$L_1$$ and $$L_2$$ intersect a plane $$R$$ at points $$R_1$$ and $$R_2$$ respectively. The distance between $$R_1$$ and $$R_2$$ is $$d$$ and the plane $$R$$ is parallel to plane YZ.

Reference Image

Given, a third point $$P_3=(x_3, y_3, z_3)$$ & a line $$L_3(α_3, β_3, γ_3)$$ which intersects the plane $$R$$ at point $$R_3$$, find the $$y$$ and $$z$$ coordinates of $$R_3=(x_r, y_r, z_r)$$.

We are getting the precise values of $$x_1$$, $$y_1$$, $$z_1$$, $$x_2$$, $$y_2$$, $$z_2$$, $$α_1$$, $$β_1$$, $$γ_1$$, $$α_2$$, $$β_2$$, $$γ_2$$, $$d$$, $$x_3$$, $$y_3$$, $$z_3$$, $$α_3$$, $$β_3$$, $$γ_3$$ via sensors. Find $$y_r$$ & $$z_r$$.

• What have you tried already? Do you have the parametric equation for $L_1$ in terms of $x_1,y_1,z_1,\alpha_1,\beta_1,\gamma_1$ and a parameter along the line called $t_1$? – AHusain Oct 12 '18 at 0:42
• Why not start with taking the equation of the plane as say, x=k because it is parallel to y-z plane. Then we can project the two points on the plane. Also I am assuming we know where the third point intersects the plane. So that gives us the angles in the triangle formed with three points (projected). And then we can calculate the y and z coordinates of the third point. – Rudrani Angira Oct 12 '18 at 0:42
• $P_1$ and $P_2$ and associated lines are irrelevant. Use similar triangles to find $R_3$. – amd Oct 12 '18 at 1:20