# How to find the intersection of a line and a plane with interpolation ( given two points in the opposite side of the plane)

I have two points in the opposite side of a plane (P1,P2) in 3D space, and i know their signed distances to the plane(D1,D2). how can i use interpolation to calculate the point that is the intersection of the line of P1P2 and the plane.

thanks

• Based on the two distances you create an equation of that point - ratios based on the relative distances. Now you have three equation and three unknowns. Solve the system to get the (x, y, z) of the intersection point. – Moti Oct 30 '15 at 16:41
• I could help you, but only without interpolation. Do you want that answer? – Ruts Oct 30 '15 at 16:48
• This may help: en.wikipedia.org/wiki/Line%E2%80%93plane_intersection – NoChance Oct 30 '15 at 16:48
• @Moti what do you mean by equation of the point ? – Nimakhin Oct 30 '15 at 16:49
• @Ruts is it replacing the line equation in the plane ? – Nimakhin Oct 30 '15 at 16:50

The values of the points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$. Let assume ratio of distance from point 1, $d_1$ and point 2, $d_2$ is r. Than for each dimension $P(: x, y, z)$ you calculate the target point value: $P_0 = (P_1-P_2) \times r$ (meaning $P$ takes the values of $x, y, z$)
• If you will do some investigation of the concept you will get the simplified form for each dimension: $r=\frac {x_1-x_0}{x_0-x_2}$ where r the ratio of distances. Now you calculate $x_0$ – Moti Oct 30 '15 at 21:03