rotation about a point

hello community, I'm slowly losing my mind trying to do something rather simple. I would like to rotate set of 2-d points about an arbibtrary point. I know that it amounts to an affine rotation and translation, but I cannot figure out the equation for the translation.

Below is my attempt on an example given by steve eddins. (http://blogs.mathworks.com/steve/2006/02/14/spatial-transformations-maketform-tformfwd-and-tforminv/)

it doesn't work. (note, the x and y vectors are too long to include here, here's the link for the data http://blogs.mathworks.com/images/steve/37/george.mat)

%point of rotation

xx = 0.2 yy = 0

%angle of rotation

t = pi/4;

%transform matrix !! this is where i'm messing up

A2 = [cos(t) -sin(t) 0; sin(t) cos(t) 0; xx-cos(t)*xx+sin(t)*yy yy-sin(t)*xx-cos(t)*yy 1];

%matlab tool for applying affine transform

tform2 = maketform('affine', A2); uv2 = tformfwd(tform2, [x y]);

%plot of results.

subplot(1,2,1) plot(x,y), axis ij, axis equal, axis([-2 2 -2 2]), grid on, title('George')

subplot(1,2,2) plot(uv2(:,1), uv2(:,2)), axis ij, axis equal, axis([-2 2 -2 2]) grid on title('Rotated by 45\circ')

thank you for your time! veritas

[Jan]

It would be *very* helpful if you explain "it doesn't work" with any details. Please try to insert a specific question.

[David]

sorry, I guess I just meant that it was failing to do the point rotation, thus something is wrong with the matrix (specifically the translation as mentioned above)...and I'm not sure what the error is.

Answer 1

[David]

i found my error and here's the solution. thanks again for all your help!

clf xx = 0 yy = -0.7

t = pi/4; A = [cos(t) -sin(t) xx-cos(t)*xx+sin(t)*yy; sin(t) cos(t) yy-sin(t)*xx-cos(t)*yy; 0 0 1]

data = []; data = [x'; y'; ones(1,length(x))]

rotdat = A*data

subplot(1,2,1) plot(x,y), axis ij, axis equal, axis([-2 2 -2 2]), grid on, title('George')

subplot(1,2,2) plot(rotdat(1,:), rotdat(2,:)), axis ij, axis equal, axis([-2 2 -2 2]) grid on title('Rotated by 45\circ')

Answer 2

[Matt Fig]

Perhaps an example would work. You can use the function ROTATE to do what you want. I have shown an example, with minimal commenting so you should follow along carefully:

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EDIT Fixed some coding errors.

S1 = subplot(1,2,1);
L = plot([1 2 3],[1 2 1],'b*');
title('Before Rotation')
xlim([-4 4])
axis equal
P = [0,0,1];  % Rotation vector
Rp = [.2 0]; % Point about which to rotate.

Xd = get(L,'xdata');
Yd = get(L,'ydata');
S2 = subplot(1,2,2);
L2 = plot(Xd,Yd,'*');
title('After Rotation')
hold on
plot(Rp(1),Rp(2),'*r')
XL = get(S1,'xlim');
YL = get(S1,'ylim');
Dx = diff(XL);
Dy = diff(YL);

for ii = 1:72
    set(S2,'xlim',[Rp(1)-Dx Rp(1)+Dx],'ylim',[Rp(2)-Dy Rp(2)+Dy])
    rotate(L2,P,5) % rotate 5 degrees at a time 
    xlim(XL)
    ylim(YL)
    pause(.05)
end

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EDIT2

For a vectorized solution, you cold do something like this (with a nod to Paulo):

D = load('george');  % Load the data.
theta = pi/4;  % Rotation angle.
plot(D.x,D.y);hold on
axis([-2 2 -2 2]);
P = [0 -.71];  % Rotation pivot point.
plot(P(1),P(2),'*k','MarkerSize',10);  % Plot Pivot point.
r = [cos(theta) -sin(theta); sin(theta) cos(theta)];
T = bsxfun(@(x,y) r*x,[D.x.'-P(1);D.y.'-P(2)],false(1,length(D.x)));
plot(T(1,:)+P(1),T(2,:)+P(2),'r');
title('Original in Blue, Rotated in Red')

[David]

thank you for the code! it seems like it will work.

The reason I'm trying to use the affine transform matrix is that I have a lot of data to run this on and thus a vectorized solution would be optimal.

I'm quite sure that I'm making an error in my calculation on the translation component of my code, so I was hoping someone could help with this part.

[Matt Fig]

Yes, you basically have to first translate the points to the origin, then calculate the rotation matrix, then translate back.

[gwoo]

Why couldn't this be:

T = (r * [D.x.' - P(1); D.y.' - P(2)])';

I'm not sure what the need for bsxfun is, unless 11 years ago one couldn't do that kind of expression without it.

Answer 3

[Paulo Silva]

just for fun

theta=0;
load george
clf;hold on
plot(x,y);
axis([-2 2 -2 2]);
title('select rotation point');
[x0,y0]=ginput(1);
plot(x0,y0,'x','MarkerSize',10);
hp=plot(nan,nan,'r');
x=x-x0;y=y-y0;
for theta=0:pi/12:2*pi
r = [cos(theta) sin(theta); -sin(theta) cos(theta)];
xyr=[];for ii=1:numel(x),xyr=[xyr r*[x(ii);y(ii)]];,end
x1 = xyr(1,:)+x0;y1 = xyr(2,:)+y0;
set(hp,'XData',x1);set(hp,'YData',y1);
title(['Original (blue) Rotated by ' num2str(180*theta/pi) '\circ (red)'])
pause(0.1)
end

[David]

thx paulo! maybe you could see my above reply?