How to map 3D coordinates to 2D plane

Question

I'm writing a OpenGL application in which there is a rectangle placed on a 3D object. The 3D object can move around and rotate and the rectangle follows these movements.

What I would like to do is to point with the mouse towards the rectangle so that a dot would appear in that point on the rectangle, and I want the point to follow it as the 3D object that "holds" the rectangle moves around.

I know how to find the intersection on a plane, and I know how to find the world coordinates of the contact point. What I need is a way to convert the world coordinates to the 2D local coordinate system of the rectangle.

For example, suppose I have the plane positioned like this in the 3D world, with a given orientation (sorry, I can't really draw properly):

The black point in the center is the origin of the plane, while the blue point is the one I would like to find. The numbers near the points are their world coordinates; in this example the Z axis "comes out" of the screen.

I would like to map the coordinates of the blue point in the local coordinate system of the plane, like this:

I know that somehow this shouldn't be hard, but I can't find a way at all. Any hints would be appreciated!

Possible duplicate (language agnostic): [Graphics - equation to convert 3d point to 2d projection](http://stackoverflow.com/questions/5276315/graphics-equation-to-convert-3d-point-to-2d-projection) — Thomas Matthews, Dec 04 '16 at 21:37

score 0 · Answer 1 · answered Dec 04 '16 at 20:33

0

You probably already use a transformation matrix to render the rectangle. If you have the 3D position of the point, just transform it with the inverse transformation matrix and you get a 3D position in the rectangle's space. If the local system is aligned with the rectangle, one of the coordinates should always be zero and you can drop it.

answered Dec 04 '16 at 20:33

Nico Schertler

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The problem is that I'm using a particular framework for this application. It is based on OpenGL, but I have very little access to how things are done in the background. I don't manually manipulate tranformation matrices, the framework does it in the background and I can't interfere. For each object I can get their model matrix (computed by the framework), but I can't set my own. I can also convert coordinates from their local coordinate system to world coordinates, but not the opposite. Because of these limitations I was looking for a formula or something like that – Beriol Dec 04 '16 at 21:45
The object's model matrix is exactly what you need. – Nico Schertler Dec 05 '16 at 00:51

score 0 · Answer 2 · edited May 23 '17 at 12:08

I found what I needed in this post here.

Basically the solution for me is to project the point onto the x and y axis of the local coordinate system of the rectangle.

In pseudo code it's like this:

// I compute the direction that go from the origin of the rectangle to its x axis, in world coordinates, and take the norm; same thing for the y axis
var xDir = Norm(mRectangle.LocalToWorld([1, 0, 0]) - mRectangle.LocalToWorld([0, 0, 0]));
var yDir = Norm(mRectangle.LocalToWorld([0, 1, 0]) - mRectangle.LocalToWorld([0, 0, 0]));

// I compute the dot product
mLocalPosition.x = Dot(xDir, (contactPoint - mRectangle.LocalToWorld([0, 0, 0])));
mLocalPosition.y = Dot(yDir, (contactPoint - mRectangle.LocalToWorld([0, 0, 0])));

// I can now set the position of the point converting the local coordinates found to world coordinates
mPoint.SetPosition(mRectangle.LocalToWorld([mLocalPosition.x, mLocalPosition.y, 0]));

How to map 3D coordinates to 2D plane

2 Answers2