How Do You Calculate the Transformation Matrix for a Quadrilateral in 3D Space?

[snarky23]

Homework Statement

Consider the quadrilateral (namely Q) in R^3 formed by the points
(1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3).

a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of 3/2, then translating it along the vector (-2, 1, -1)?

b) Find the matrix that transforms Q into R.

The Attempt at a Solution

Okay, so what I did for (a) was I used the matrix

cosθ -sinθ 0
sinθ cosθ 0
0 0 1

Then substituted 45 for θ.

After this, multiplied the identity matrix for R^3 by 3/2 and then multiplied it by the matrix with 45 substituted for θ.

Then T(x,y,z) = (-2+3sqrt(2)x/4-3sqrt(2)y/4,1+3sqrt(2)x/4+3sqrt(2)y/4,-1+3z/2).

I substituted each of the quadrilateral points in for T(x, y, z) to come up with the four points and got:


(3/(2√2) - 2, 3/(2√2) + 1, -1),
(3/√2 - 2, 3/√2 + 1, -1),
(-2, 3/√2 + 1, 3.5),
(3/(2√2) - 2, 9/(2√2) + 1, 3.5)

I was wondering if someone could show me how to find the matrix that transforms Q into R. It would be greatly appreciated!

 

[HallsofIvy]

Homework Statement

Consider the quadrilateral (namely Q) in R^3 formed by the points
(1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3).

a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of 3/2, then translating it along the vector (-2, 1, -1)?

b) Find the matrix that transforms Q into R.

The Attempt at a Solution

Okay, so what I did for (a) was I used the matrix

cosθ -sinθ 0
sinθ cosθ 0
0 0 1

Then substituted 45 for θ.

After this, multiplied the identity matrix for R^3 by 3/2 and then multiplied it by the matrix with 45 substituted for θ.

Then T(x,y,z) = (-2+3sqrt(2)x/4-3sqrt(2)y/4,1+3sqrt(2)x/4+3sqrt(2)y/4,-1+3z/2).

I substituted each of the quadrilateral points in for T(x, y, z) to come up with the four points and got:


(3/(2√2) - 2, 3/(2√2) + 1, -1),
(3/√2 - 2, 3/√2 + 1, -1),
(-2, 3/√2 + 1, 3.5),
(3/(2√2) - 2, 9/(2√2) + 1, 3.5)

I was wondering if someone could show me how to find the matrix that transforms Q into R. It would be greatly appreciated!

Translation in 3 dimensions cannot be written as a 3 by 3 matrix- it is a vector addition. However, using "projective coordinates", you can write any such transformation as a 4 by 4 matrix. Have you done anything with that?