This Wikipedia article on distance from a point to a plane says the following:

If what is desired is the distance from a point not at the origin to the nearest point on a plane, this can be found by a change of variables that moves the origin to coincide with the given point.


Suppose we wish to find the nearest point on a plane to the point $(X_0, Y_0, Z_0)$, where the plane is given by $aX + bY + cZ = D$. We define $x = X - x_0$, $y = Y - Y_0$, $z = Z - Z_0$, and $d = D - aX_0 - bY_0 - cZ_0$, to obtain $ax + by + cz = d$ as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin.

So what's being described here is a translation, right?

  • $\begingroup$ @AnuragA But it does a change of variables to move the origin, without any rotation of the object in question; is that not a translation? $\endgroup$ – The Pointer Jul 19 at 7:05
  • $\begingroup$ Yes, it is a translation. $\endgroup$ – Aretino Jul 19 at 9:45

Short answer: yes, you can view this as a translation.

Long answer: I would say there are two equivalent ways of viewing this.

  • One is saying that yes, you move the plane. You perform a translation and end up with a new plane which is parallel to the original plane.

  • Another way would say that no, you keep the plane as it is, and instead change the coordinate system, by moving its point of origin.

The text appears to be more geared towards the latter. You might consider this a translation of the coordinate system itself, if you want. On the whole, all of these distinctions are fairly irrelevant since they boil down to the same algebraic result. So you are free to pick whatever view is most appealing to you, as the resulting computations will be the same. All that matters is the relationship between plane and coordinate system, and whether you consider it a new plane in the old coordinate system, or the old plane in a new coordinate system is irrelevant.


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