Question

The following is what I am trying to figure out.

Question – Explain how you can project a 2D point onto a plane to create a 3D point.

I want to know how I would go about figuring this out. I have looked through a computational geometry book and looked for anything that may relate to what I’m trying to figure out. There was no information given along with the question about the computational geometry problem. The thing is I don’t know anything about computational geometry< so figuring this out is beyond my knowledge.

Can anyone point me in the right direction?

Answer 1

If I understood this correctly you want to project points on the 2D plane onto a plane with a different orientation. I’m also going to assume that you are looking for the orthogonal projection (i.e. all points from the xy-plane are to be projected onto the closest point on the target plane).

So we have the equations of the two planes and the point we want to project:

The original 2D plane: z = 0, with the normal vector n1 = (0, 0, 1)

The target plane: ax + by + cz + d = 0 with the normal vector n2 = (a, b, c)

Point P: (e, f, 0) which obviously lies in the xy-plane

Now, we want to travel from point P in the direction of the normal of the target plane (because this will give us the closest point on the target plane). Hence we form an equation for the line which starts at point P, and which is parallel to the normal vector of the target plane.

Line L: (x,y,z) = (e,f,0) + t(a,b,c) = (e+ta, f+tb, tc) , where t is a real valued parameter

Next, we want to find a point on the line L which also lies on the target plane. Hence, we plug the line equation into the equation for the target plane and receive:

a(e+ta) + b(f+tb) + c * tc + d= 0

ae + bf + d + t(a² + b² + c ²) = 0

t = – (ae + bf + d) / (a² + b² + c ²)

hence the projected point will be:

P_projected = (e + ka, f + kb, kc), where k = – (ae + bf + d) / (a² + b² + c ²)

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With all the variables in the solution about, it might be a bit hard to grasp if you are new to the area. But really, it is rather simple. The things you have to learn are:

• The standard equation for a plane: ax + by + cz + d = 0
• How to extract the normal vector from the equation of the plane
• What the normal vector is (a vector which is perpendicular to all position vectors in the plane)
• The parametric representation of a line: (x, y, z) = u + t*v* , where u is a point which lies on the line, t is a real valued parameter and v is a vector parallel to the line.
• The understanding that the closest path between a point and a plane will be parallel to the normal of the plane

If you grasped the above concepts, computing the projection is simple:

Compute the normal vector of the target plane. Starting from the point that you want to project, travel parallel to the computed normal vector until you reach the plane.