How do you rotate around a given point?

Question

In my text book I am working on Transformation Geometry, but am having trouble with Rotation.

The examples given state that when rotating 90 Degrees Anticlockwise around the point (0,0), (3,1) becomes (-1, 3). It also says that when rotating 90 Degrees Clockwise around the point (0,0), (3, 1) becomes (1, -3)

I understand this so far, but when it asks me to rotate A(-1, 2), B(-1, 5) and C(-3, 5) 90 Degrees Clockwise around the point (0,2), I can't understand how to do it.

My question is, how do I rotate around a point when that point is not (0,0)?

Answer 1

As you already know, to rotate the point (Px, Py) around thew centre, (0, 0) you have the traditional four main rotations:



(Px, Py) rotated 90 degrees counterclockwise gives (-Py, Px)

(Px, Py) rotated 180 degrees gives (-Px, -Py)

(Px, Py) rotated 90 degrees clockwise gives (Py, -Px)



To rotate around a point, you first have to "move" the centre to that point, perform the rotation and then "move" the centre back. To do this, you have to subtract the point of rotation from the point to rotate, rotate it as above, then add the point of rotation back again.



If you want to rotate the point (Px, Py) around the point (Rx, Ry) you get the following:



(Px, Py) rotated 90 degrees counterclockwise around (Rx, Ry) gives (Rx - (Py - Ry), Ry + (Px - Rx))

= (Rx + Ry - Py, -Rx + Ry + Px)

(Px, Py) rotated 180 degrees around (Rx, Ry) gives (Rx - (Px - Rx) , Ry - (Py - Ry)) = (2Rx - Px, 2Ry - Py)

(Px, Py) rotated 90 degrees clockwise around (Rx, Ry) gives (Rx + (Py - Ry), Ry - (Px - Rx)) = (Rx - Ry + Py, Rx + Ry - Px)



To show you one of cases, take (Px, Py) rotated 90 degrees counterclockwise around (Rx, Ry):



1. Subtract the point of rotation (Px, Py) - (Rx, Ry) = (Px - Rx, Py - Ry)

2. Rotate as you normally would, giving (-(Py - Ry), Px - Rx)

3. Add the point of rotation back (-(Py - Ry), Px - Rx) + (Rx, Ry) = (Rx - (Py - Ry), Ry + (Px - Rx)) = (Rx + Ry - Py, -Rx + Ry + Px)

Answer 2

say your shape is a triangle and one of the corners is point A you get some tracing paper and place it over the triangle marking a point on where the corner labelled A is. you then place a pencil or something on the point of (0,2) and spin this tracing paper around the amount of degrees and the direction it says (always keeping your pencil on the tracing paper holding it down) you then mark where the dot you orginally made for point A now (on your page not on the tracing paper, just slightly lift up the tracing paper to do this) is and do the same with points c and b and you should have a newly formed trinangle in a different position however on the same grid.



hopes this helps and is clear enough :)

Answer 3

To rotate P(x,y) about C(h,k) use the following rules:

1) 90 degrees counterclockwise, (x,y) gives (-y+h+k , x-h+k).

2) 90 degrees clockwise, (x,y) gives (y+h-k , -x+h+k).

3) 180 degrees, (x,y) gives (-x+2h , -y+2k).

Faraj Razem

Answer 4

first translate the matrix over a point say A(h.k).. then rotate it to a degree say 90 degree and at last translate it to origin(h,k).