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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.

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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 $y_r$ & $z_r$.

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  • $\begingroup$ 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 $t_1$? $\endgroup$ Commented Oct 12, 2018 at 0:42
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    $\begingroup$ 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. $\endgroup$ Commented Oct 12, 2018 at 0:42
  • $\begingroup$ $P_1$ and $P_2$ and associated lines are irrelevant. Use similar triangles to find $R_3$. $\endgroup$ Commented Oct 12, 2018 at 1:20

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I have written the solution in paper. The key idea is that you first find out the equation of plane R. Then you find the point of intersection between L3 and R to obtain yr and zr.

The parametric equations for the lines and planes are included in the paper. Have a look!

First Page
Second Page

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Step 1. Find L1 equation Step 2. Find L2 equation Step 3. Find at what value of X, line L1 and line L2 are at distance d Thus now you know what is the X value of the plane Step 4. Find L3 equation [Step 5. Find y,z with X satisfying L3 That's the point

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