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

In summary, the conversation discusses a problem that involves finding a 4x4 projection matrix to map ℝ^3 to a plane with the equation 3x + 2y = 1, assuming the center of projection is at (0,0,0). The provided links include equations for rotation matrices and a handwritten attempt at solving the problem. The solution is given in the form of a midterm paper, which provides a hint to find the rotation matrix R and the projection matrix M. These matrices are then combined using matrix multiplication to obtain the final 4x4 matrix sought by the problem. The conversation also touches on understanding concepts such as homogeneous coordinates, coordinate system transformation, and perspective projection.
  • #1
s3a
818
8

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
 

Attachments

  • R.png
    R.png
    1.9 KB · Views: 465
  • RxRyRz.png
    RxRyRz.png
    6.8 KB · Views: 434
Last edited:
Physics news on Phys.org
  • #2
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.
 
  • #3
andrewkirk said:
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.
 
  • #4
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).
 
  • #5
s3a said:
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.
 

1. How is a projection matrix used to map R3 to a plane?

A projection matrix is used to transform points in a three-dimensional space, known as R3, to a two-dimensional plane. This is achieved by multiplying the coordinates of the points in R3 by a specific matrix, known as the projection matrix, which projects the points onto the plane.

2. What is the purpose of obtaining a 4x4 projection matrix?

The purpose of obtaining a 4x4 projection matrix is to accurately map points in R3 to a specific plane, in this case the plane represented by the equation 3x+2y+z=1. This matrix is necessary because it allows for the translation, rotation, and scaling of the points in R3 to be taken into account when projecting them onto the plane.

3. How is the equation of the plane used in obtaining the projection matrix?

The equation of the plane, 3x+2y+z=1, is used to determine the coefficients of the projection matrix. These coefficients correspond to the scaling factors for the x, y, and z coordinates of the points in R3. By using these coefficients, the projection matrix can accurately map points in R3 to the specified plane.

4. Can a 4x4 projection matrix be used for any plane in R3?

Yes, a 4x4 projection matrix can be used for any plane in R3. This is because the projection matrix takes into account the translation, rotation, and scaling of the points in R3, allowing for accurate mapping to any plane. However, the specific coefficients of the matrix will vary depending on the equation of the plane.

5. What are the steps involved in obtaining a 4x4 projection matrix?

The steps involved in obtaining a 4x4 projection matrix include: 1) determining the coefficients of the equation of the plane, 2) creating a 4x4 matrix with these coefficients, 3) taking the inverse of this matrix, 4) multiplying it by the 3x3 identity matrix, and 5) multiplying the resulting matrix by a translation matrix if necessary. This will result in a 4x4 projection matrix that maps points in R3 to the specified plane.

Similar threads

  • Math Proof Training and Practice
6
Replies
175
Views
19K
Back
Top