Obtain 4x4 projection matrix that maps R3 to 3x+2y+z=1 plane

[s3a]

Homework Statement

Obtain a 4×4 projection matrix that maps $ℝ^3$ to the plane 3x + 2y = 1. Assume that the centre of projection i.e. eye is at (0,0,0).

The problem that my problem is strongly based on and its solution are #3, here. (I'm referring to the first way of solving the problem in the solution - I'll make another post for the other method.):
https://www.docdroid.net/RtcGmQj/midterm.pdf#page=4

2. Homework Equations
https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations

The Attempt at a Solution

Here is my (handwritten) work and confusions:
https://www.docdroid.net/hp65se2/myworkandconfusions.pdf

Any help in getting me to fully understand this problem's solution would be GREATLY appreciated!

Edit:
I made a mistake for what wanted to state that R is (because I assumed that theta_x = theta_y = theta_z, by accident). I attached R.png and RxRyRz.png to show my corrections. a = theta_x, b = theta_y, c = theta_z

 

[andrewkirk]

A couple of quick points:

1. In your title the equation of the plane is different from the one in the first line of your post. They are both planes, but they are different planes.

2. Why do you want a 4 x 4 matrix? Given that the input and the output of the projection are both in $\mathbb R^3$, you need a 3 x 3 projection matrix.

3. The specification about the 'centre of projection' seems superfluous and I have not encountered the term before. A projection is fully defined simply by defining the subspace that is the range of the projection, and that is given in your plane equation.

 

[Stephen Tashi]

2. Why do you want a 4 x 4 matrix? Given that the input and the output of the projection are both in $\mathbb R^3$, you need a 3 x 3 projection matrix.

Apparently the problem wants a 4x4 matrix to do perspective projection in homogeneous coordinates. https://en.wikipedia.org/wiki/3D_projection
In that case, the example from the midterm isn't completely relevant.

 

[s3a]

andrewkirk:
About the plane equation being wrong in the post (so, correct in the title), sorry; I was very tired when I made this post. (I can't edit my post anymore.)

About the superfluous part, that's helpful.

Stephen Tashi:
What do you mean when you say the example from the midterm isn't completely relevant?

Everyone:
So, what, specifically, are R (as well as R^T, which would follow easily), M and the final matrix that is being sought by the question? (Is the matrix provided in the solution midterm.pdf M or the result of the matrix multiplication I mentioned in the myworkandconfusions.pdf file, R^T M R p^, where R = RxRyRz, so that what I believe is being sought is (RxRyRz)^T M (RxRyRz) p^?)

I don't mind doing all the tedious stuff, but I'd appreciate it (a lot) if I could be directed with what I should do, in addition to having my work verified when I do what you guys tell me to do (because I am so confused with these topics, and I want to notice the patterns in the multitude of problems I plan on doing).

 

[Stephen Tashi]

Stephen Tashi:
What do you mean when you say the example from the midterm isn't completely relevant?

I was wrong. It is relevant to problem stated in the title of your post.

Everyone:
So, what, specifically, are R (as well as R^T, which would follow easily), M and the final matrix that is being sought by the question?

That's a good question! The answer sheet to the midterm assumes we can figure out R from the hint:

Let $R$ be the rotation that maps the plane normal (3,2,1) to the z-axis. Such a matrix was given lecture 2, page 4...

We need to look at lecture 2 page 4. I think an answer in the form of symbolic angles $a,b,c$ wouldn't be given full credit. The values of $a,b,c$ should be expressed as functions of the numbers given in the problem. The order of multiplying rotation matrices does matter.

The answer sheet says:

Let $M$ be a perspective projection onto the $z =\sqrt{ 3^2 + 2^2 + 1^2}$ plane (lecture 4, page 2).

I assume the context for that projection is that the eye is at (0,0,0) and looking down the positive z-axis. The plane of projection is $z = \sqrt{3^2 + 2^2 + 1^2}$. Some presentations of computer graphics use a left handed coordinate system and others use the standard right handed coordinate system, so that convention needs to be sorted out.

Are you familiar with how to calculate where the line from $(x,y,z)$ to $(0,0,0)$ intersects the plane of projection at $(x_p,y_p, f)?$

$f = \sqrt{3^2 + 2^2 + 1^2}$
$x/z = x_p/f$,
$y/z = y_p/f$.

and how the formulae result from analyzing similar triangles?

(Is the matrix provided in the solution midterm.pdf M or the result of the matrix multiplication I mentioned in the myworkandconfusions.pdf file, R^T M R p^

The numerical matrix given in the answer sheet is the final result of $R^T M R$. You don't need the "p^".

where R = RxRyRz

I'd have to see the lecture notes to know what form the rotation matrix is supposed to take. Another forum member may know.

(because I am so confused with these topics, and I want to notice the patterns in the multitude of problems I plan on doing).

Do you understand, the 3 general concepts involved in the problem: - 1) homogeneous coordinates 2) Using matrices to change coordinates from one coordinate system to another. 3) Perspective projection

I don't know whether you are taking a course in computer graphics or whether you are taking a course in pure mathematics that just happens to emphasize projective geometry. Computer graphics has specialized terminology that I, myself, have not studied.