Rotate Cube 80° at C(2,4,5): Point (6,7,9) Result

In summary: [1__0__0__2]__[1____0__ _____0____0]___[1__0__0__-2]__[x] [0__1__0__4]__[0___cos80__-sin80___0]___[0__1__0__-4]__[y][0__0__1__5]__[0___sin80___cos80___0]___[0__0__1__-5]_[z][0__0__0__1]__[0_____0______0_____1]___[0__0__0__1]__[1]
  • #1
lampshader
39
0

Homework Statement



Suppose that you want to rotate 80 degrees a cube in three dimensional space centered at the point C(2,4,5) about the x-axis. In other words, you want a pitch 80 degrees centered at the point C(2,4,5). Suppose that you want to rotate a vertex of the cube, the point (x,y,z) = (6,7,9) at an 80 degree pitch centered at point C. What is the resulting point (x',y',z')?



Homework Equations



Pitch rotation for x-axis:

[1____ 0______0_____0]
[0___cos80___-sin80__0]
[0___sin80____cos80__0]
[0____ 0______0_____1]


and Scaling with respect to the center point (although I do not need to scale the object):


[x'] [1__0__0__xc] [1____0_____0____0] [1__0__0__-xc] [x]
[y'] [0__1__0__yc] [0__cos80 _-sin80__0] [0__1__0__-yc] [y]
[z']= [0__0__1__zc] [0__sin80__cos80__0] [0__0__1__-zc] [z]
[1] [0__0__0__1] [0____0_____0____1] [0__0__0__1] [1]


The Attempt at a Solution



What I did was I went ahead and replaced the scaling matrix (middle) with the pitch rotation matrix..I don't think this is right, because my answers need "6" as the resulting x-component.

Could someone just show me the matrix formula to use for this problem?
 
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  • #2
Start by translating the cube so that its center is at (0, 0, 0).

That is, given any point (x, y, z) change it to (x- 2, y- 4, z- 5).

Now rotate about the x-axis 80 degrees:
[tex]\begin{bmatrix}1 & 0 & 0 \\ 0 & cos(80) & -sin(80) \\0 & sin(80) & cos(80)\end{bmatrix}[/tex]

Then translate back: change the new (x, y, z) to (x+ 2, y+ 4, z+ 5).

That appears to be exactly what you have done, using "projective" notation. That certainly will give "6" as the resulting component. Multiplying
[tex]\begin{bmatrix}1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & -5 \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix} 6 \\ 7 \\ 9\end{bmatrix}[/tex]
will give x component 6- 2= 4, multiplying by the rotation matrix will not change that, and multiplying by the third matrix gives 4+ 2= 6 again.
 
  • #3
I'm afriad I might be doing something wrong..Let me show exactly what I do step by step:

I am translating the object to the origin using the stack:

[1__0__0__2]__[1____0__ _____0____0]___[1__0__0__-2]__[x]
[0__1__0__4]__[0___cos80__-sin80___0]___[0__1__0__-4]__[y]
[0__0__1__5]__[0___sin80___cos80___0]___[0__0__1__-5]_[z]
[0__0__0__1]__[0_____0______0_____1]___[0__0__0__1]__[1]

Multiplying matrices from left to right until I get down to a single matrix:


[1___0______0________0_] [6]
[0__0.173__-0.985__-1.617] = [-9.271]
[0__0.985__0.173___0.195] [8.647]
[0____0______0_______1_] [1]

The only answer that is even close is: (6, 0.582, 8.649)

But I don't want to guess, I want to know.
 
Last edited:
  • #4
Why are you guessing? If you have done your multiplications correctly, that answer is right.

As a check, try multiplying the first matrix on the right by [x, y, z, 1], the the next, then the third (which is the way I would have multiplied to begin with).
 

1. How do you rotate a cube by 80 degrees at a specific point?

To rotate a cube by 80 degrees at a specific point, you will need to use a rotation matrix. The rotation matrix is a mathematical tool that is used to rotate objects in 3-dimensional space. In this case, you will need to determine the coordinates of the point you want to rotate the cube around and apply the rotation matrix to the cube's vertices.

2. What is the purpose of rotating a cube by 80 degrees at a specific point?

The purpose of rotating a cube by 80 degrees at a specific point is to change its orientation in 3-dimensional space. This can be useful in various applications such as computer graphics, physics simulations, and geometry problems.

3. How do you determine the coordinates of the rotated cube after a 80-degree rotation at a specific point?

To determine the coordinates of the rotated cube, you will need to apply the rotation matrix to each of its vertices. The resulting coordinates will be the new coordinates of the cube after the rotation.

4. Can you rotate a cube by 80 degrees at any point?

Yes, you can rotate a cube by 80 degrees at any point as long as you know the coordinates of the point and apply the correct rotation matrix. However, the resulting rotation may not be visible or may look distorted depending on the orientation of the cube and the point of rotation.

5. Is there a difference between rotating a cube by 80 degrees at a specific point and rotating the entire cube by 80 degrees?

Yes, there is a difference between rotating a cube by 80 degrees at a specific point and rotating the entire cube by 80 degrees. When rotating at a specific point, the cube will rotate around that point while maintaining its original orientation. On the other hand, when rotating the entire cube, it will rotate around its center point and change its orientation in 3-dimensional space.

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