The isometry f maps points (0,0), (1,0) and (0,1) into (3,3), (3,4) and (4,3).

  1. Determine f in the form f = Ax+b

  2. Are there any fixed points of f, and hence determine whether f is a translation, rotation, reflection or glide-reflection.

Attempt: I equated the transformed coordinates x' with transformation Ax+b to get $$ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ and $$ b = [3,3]^{T}$$ Is this correct?

Also I couldn't understand part 2, can you please explain it? Thank You.

Edit: For the second part, I deduced the transformation was glide- reflection. The transformation f first translates along line $y=x$ by 3 units and then reflects along the same line, hence it appears to be a glide - reflection. As for the fixed points of f, what does in this context? Are the common points among original and transformed coordinates called fixed points or is there any other criteria.

  • $\begingroup$ I think there is a mistake. It looks like the points are mapped in the wrong order. $\endgroup$ – Kavi Rama Murthy Jul 3 at 9:34
  • $\begingroup$ @KaviRamaMurthy you are right, thank you. I edited now. $\endgroup$ – DubsVeer23 Jul 3 at 9:36

Your matrix $A$ is not correct. Correct is

$$A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$

Hence $f(x)=x+(3,3)^T.$ This shows that $f$ has no fixed points.

Can you proceed ?

  • $\begingroup$ I edited the original question. Do you still the same A? Also I am clueless about part 2, where do I begin? Thank You. $\endgroup$ – DubsVeer23 Jul 3 at 9:40

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