# How to find the height of a 2D coordinate on a four-sided 3D polygon plane?

How do I find the height of a given 2D coordinate on a four-sided 3D polygon plane? The polygon has no volume. I'm trying to match 3D terrain vectors to a 3D polygon. I'll always know that the 2D version of the 3D poly contains the 2D coordinate, but I need to get the height at that 2D coordinate on the polygon surface. How can I figure out the height of point F1 in the image example?

• For clarification: are you trying to find $y$ such that $(312,y,190)$ would be a point determined by the plane determined by the points $P_1$, $P_2$, $P_3$, $P_4$? Nov 1 '13 at 22:18
• I think so... points P1,P2,P3,P4 will always represent the four points of the polygon (in the order they are presented on the image). Nov 1 '13 at 22:25

You can get a normal vector to the plane.

$n = (P_4 -P_1) \times (P_2 - P_1)$

From there you can find the point-normal form of the plane.

$(p-P_1) \cdot n = 0$

You want the intersection of this plane and the line $l =<\!312,0,190\!> + <\!0,1,0\!>\,t$ , so solve $(<\!312,0,190\!> + <\!0,1,0\!>\,t - P_1) \cdot n = 0$ for $t$:

$t = \Large{(P_1 - <312,0,190>) \cdot n \over <0,1,0> \cdot n}$

Then plug that value into your line equation to find your point.

For the 4 points of your figure to represent an actual quadrilateral in a 3-space, they would have to be coplanar. Stack the coördinates of the 4 points vertically, forming a matrix of 3 columns and 4 rows. Make a 4th column of ones. You now have a 4×4 matrix. If the determinant of the matrix evaluates to 0, then the 4 points are coplanar — i.e, all 4 points occupy the same plane. The determinant occasioned by this process does not vanish, so the four given points are not coplanar.
If they were, however, we would continue as follows: Alter the matrix by replacing one of the 4 points with the point that has the unknown coördinate. Evaluate the determinant of this matrix. This will yield a linear expression in your sought-after unknown coördinate. Equate the expression to 0 and solve the equation and — BINGO! «Senex Ægypti Parvi» dsm5442@gmail.com