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I have the $2D$ coordinate $(x_i, y_i)$ of a point $i$ on a plane $Ax + By + Cz + D = 0$. The parameters of the plane $(A, B, C, D)$ are also known. How can I find the $3D$ coordinate of that point $(x, y, z)$?

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Is that supposed to be $Cz$ and not $Cx$? And do you mean to say that the $x,y$ pair is the same as the $x,y$ in $Ax+By+Cz+D=0$ If so, then $z$ is the only unknown in the expression, and you could solve for it. If not, then there is too much ambiguity here about how the plane's coordinates relate to the plane's equation. – rschwieb Oct 14 '13 at 13:18
You wrote $Ax+By+Cx+D=0$. The equation of the plane is $Ax+By+Cz+D=0$. Are you sure your equation is correct? – Riccardo.Alestra Oct 14 '13 at 13:21
Sorry, I corrected the equation of the plane. I have the local position of a point on a plane, and I know the global equation of the plane, I want to know whether I can obtain the global position of the mentioned point. – Safir Oct 14 '13 at 13:28
Unless you know something about the local corrdinate system on the plane, you're trying to do the impossible. The equation of the plane tells you nothing about any local corrdinate system that it might (or might not) have. – bubba Oct 14 '13 at 13:39
Don't you think there are infinite many possibilities? – Shuchang Oct 14 '13 at 14:21
up vote 1 down vote accepted

Let's call the given point $\mathbf{P}$.

There are two possible interpretations of your question:

First case

You know the global coordinates $(x,y)$ of the point $\mathbf{P}$. Then the unknown $z$-coordinate of $\mathbf{P}$ can be calculated from $$ z = \frac{D - Ax - By}{C} $$ assuming that $C \ne 0$.

Second case

We have an origin point $\mathbf{Q}$ and two coordinate axis vectors $\mathbf{H}$ and $\mathbf{K}$ lying in the plane, and we know the coordinates $(u,v)$ of $\mathbf{P}$ with respect to this coordinate system. In other words, we know that $$ \mathbf{P} = \mathbf{Q} + u\mathbf{H} + v\mathbf{K} $$ Then, looking at individual coordinates, we have $$ x = P_x = Q_x + uH_x + vK_x \\ y = P_y = Q_y + uH_y + vK_y \\ z = P_z = Q_z + uH_z + vK_z $$ If you don't have any knowledge of the origin and axes of the $\mathbf{Q}$-$\mathbf{H}$-$\mathbf{K}$ coordinate system, then the known coordinates $(u,v)$ are meaningless, so you don't know the location of the point $\mathbf{P}$.

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