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# Delphi - rotations in 2D

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A rotation of a point around the origin is done as:
|xnew| |cos(phi) sin(phi)| |xold|
|ynew| := |-sin(phi) cos(phi)| * |yold|
or :
xnew:=xold*cos(phi)+yold*sin(phi)
ynew:=-xold*sin(phi)+yold*cos(phi)
the rotation normally is counter clock wise for increasing
phi, due to the Y axis starting at top left, the rotation
now is clockwise. The rotation angle phi is measured in radians.
For a rotation around an arbitrary point, the coordinate
system has to be shifted :
Be the rotation center (x0,y0) :
xnew:=(xold-x0)*cos(phi)+(yold-y0)*sin(phi)+ x0
ynew:=-(xold-x0)*sin(phi)+(yold-y0)*cos(phi)+ y0

### notes

Rotation of a graph is done by rotating the vectors.

An image is rotated by rotating the pixels :
Scan the new image : for x, for y.
Assign the (x,y)pixel the color found in the old image at
the rotated position. Should the rotated position be outside
the old image, fill black or white.
centerx:=image.width div 2;
centery:=image.height div 2;
for x:=0 to image.width-1 do begin
for y:=0 to image.height-1 do begin
oldx:=(x-centerx)*cos(-phi)+(y-centery)*sin(-phi)+ centerx;
oldy:=-(x-centerx)*sin(-phi)+(ycentery)*cos(-phi)+ centery;
if outsideimage(oldx,oldy) then color:=clwhite
else color:=oldimage.canvas.pixels[olx,oldy];
image.canvas.pixels[x,y]:=color;
end;
end;
Using the pixel property is the slowest possible approach.
Use the scanlines to read and write the image, and rotate
the array where the pixels are in RAM.

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last updated: 23.june.99

Copyright (99,2000) Ing.Büro R.Tschaggelar