Visualising transformations

  • MHB
  • Thread starter lemonthree
  • Start date
  • #1
lemonthree
51
0
Given that
\(\displaystyle
H_{1} \)= reflection about the line y = x + 1;
\(\displaystyle H_{2} \)= counterclockwise rotation of pi/2 about the point (1,0);
\(\displaystyle H_{3} \)= translation by 1 - i.



What is the image of the triangle and arrow under the map \(\displaystyle H = H_{1} *H_{2} * H_{3} \)?

I need help visualising the above transformation H. I know that \(\displaystyle H = H_{1} *H_{2} * H_{3} \), so we must perform the transformations from right to left (H3 first, then H2, and lastly H1). Now, in a question that provides the equation for \(\displaystyle H_{1},H_{2} , H_{3} \), I believe I can solve this by substituting the values into \(\displaystyle H_{1},H_{2} , H_{3} \) respectively. However, in this case, the question does not provide the equation and it is up to my visualisation.

I have attached photos showing my steps for each respectively. I have no trouble with translation (H3) as it is relatively easy to visualise.
However, I struggle a little for the rotation about (1,0). I can't quite visualise "pivoting" the arrow and triangle about (1,0)...I would like to ask for some tips to aid in visualising such rotations better.
 

Attachments

  • original.png
    original.png
    5.7 KB · Views: 11
  • H3.png
    H3.png
    5.8 KB · Views: 13
  • H2.png
    H2.png
    4.3 KB · Views: 18
  • H1.png
    H1.png
    7.6 KB · Views: 13
  • answer.JPG
    answer.JPG
    14.9 KB · Views: 10

Answers and Replies

  • #2
skeeter
1,104
1
Red triangle shows the correct rotation of $\dfrac{\pi}{2}$ CCW. Yours appears to be in the CW direction.

54E20B4D-C6B8-4531-B74B-74F50F2F0EB6.jpeg
 

Suggested for: Visualising transformations

  • Last Post
Replies
3
Views
674
Replies
25
Views
945
  • Last Post
Replies
2
Views
668
  • Last Post
Replies
0
Views
474
Replies
4
Views
655
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
3
Views
821
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
1
Views
930
  • Last Post
Replies
25
Views
911
Top