Rotation

Worked example

1. 1

   $\text{centre }(2,1),\text{corner }(3,2)$ — Start from the centre. Every move is counted from it, never from the origin.

2. 2

   $3-2=1\text{ right},2-1=1\text{ up}$ — Take the centre's coordinates away from the corner's (or count squares) to see how far across and up the corner sits.

3. 3

   $1\text{ left},1\text{ down}$ — A half turn puts the image on the opposite side of the centre, the same distance away. Reverse both moves.

4. 4

   $(3,2)\to (2-1,1-1)=(1,0)$ — Start at the centre and make the reversed moves: 1 left changes x to 2 − 1, 1 down changes y to 1 − 1.

5. 5

   $(5,2):5-2=3\text{ right},2-1=1\text{ up}$ — Second corner: its distance from the centre, worked out the same way.

6. 6

   $3\text{ left},1\text{ down}$ — Reverse both moves for the half turn.

7. 7

   $(5,2)\to (2+(-3),1+(-1))=(-1,0)$ — Add the reversed moves to the centre. Brackets round the negatives stop the signs slipping.

8. 8

   $(3,3):3-2=1\text{ right},3-1=2\text{ up}$ — Third corner: its distance from the centre.

9. 9

   $1\text{ left},2\text{ down}$ — Reverse both moves for the half turn.

10. 10

    $(3,3)\to (2+(-1),1+(-2))=(1,-1)$ — Add the reversed moves to the centre: 2 − 1 = 1 across and 1 − 2 = −1 up.

Worked example

1. 1

   $(x,y)\to (y,-x)$ — The 90° clockwise rule about (0, 0): swap the two numbers, then change the sign of the number that is now second.

2. 2

   $(1,2)\to (2,1)$ — Corner (1, 2): swap the two numbers.

3. 3

   $(2,1)\to (2,-1)$ — Change the sign of the second number. 1 becomes −1.

4. 4

   $(3,2)\to (2,3)$ — Corner (3, 2): swap the two numbers.

5. 5

   $(2,3)\to (2,-3)$ — Change the sign of the second number. 3 becomes −3.

6. 6

   $(1,4)\to (4,1)$ — Corner (1, 4): swap the two numbers.

7. 7

   $(4,1)\to (4,-1)$ — Change the sign of the second number. 1 becomes −1.

8. 8

   $(-3,2)\to (2,-3)$ — A corner with a negative number: swap first, exactly as before.

9. 9

   $-(-3)=3,\text{so }(2,-3)\to (2,3)$ — Changing the sign of −3 makes it +3: a minus in front of a minus is a plus.

Worked example

1. 1

   $\text{centre }(2,1),\text{corner }(5,2)$ — Quarter turn anticlockwise. Start from the centre, because the origin rules do not work here.

2. 2

   $5-2=3\text{ right},2-1=1\text{ up}$ — Corner minus centre gives the moves from the centre to the corner.

3. 3

   $3\text{ right}\to 3\text{ up}$ — A quarter turn anticlockwise points what faced right up.

4. 4

   $1\text{ up}\to 1\text{ left}$ — It points what faced up to the left.

5. 5

   $(5,2)\to (2-1,1+3)=(1,4)$ — From the centre go 1 left (2 − 1) and 3 up (1 + 3).

6. 6

   $(3,2):1\text{ right},1\text{ up}$ — Second corner: its moves from the centre.

7. 7

   $1\text{ up},1\text{ left}$ — Right becomes up and up becomes left.

8. 8

   $(3,2)\to (2-1,1+1)=(1,2)$ — From the centre go 1 left (2 − 1) and 1 up (1 + 1).

9. 9

   $(3,3):1\text{ right},2\text{ up}$ — Third corner: its moves from the centre.

10. 10

    $1\text{ up},2\text{ left}$ — Right becomes up and up becomes left.

11. 11

    $(3,3)\to (2-2,1+1)=(0,2)$ — From the centre go 2 left (2 − 2) and 1 up (1 + 1).

Worked example

1. 1

   $\text{centre }(1,2),\text{corner }(4,3)$ — Now a quarter turn clockwise. Again, start from the centre.

2. 2

   $4-1=3\text{ right},3-2=1\text{ up}$ — Corner minus centre gives the moves from the centre to the corner.

3. 3

   $3\text{ right}\to 3\text{ down}$ — A quarter turn clockwise points what faced right down.

4. 4

   $1\text{ up}\to 1\text{ right}$ — It points what faced up to the right.

5. 5

   $(4,3)\to (1+1,2+(-3))=(2,-1)$ — From the centre go 1 right (1 + 1) and 3 down (2 + (−3)). Put the negative in brackets.

6. 6

   $(2,3):1\text{ right},1\text{ up}\to 1\text{ down},1\text{ right}$ — Second corner, 2 − 1 = 1 and 3 − 2 = 1 from the centre. Right becomes down and up becomes right.

7. 7

   $(2,3)\to (1+1,2+(-1))=(2,1)$ — From the centre go 1 right (1 + 1) and 1 down (2 + (−1)).