How do I draw an ellipse with arbitrary orientation pixel by pixel?

Question

Hi,

I have to draw an ellipse of arbitrary size and orientation pixel by pixel. It seems pretty easy to draw an ellipse whose major and minor axes align with the x and y axes, but rotating the ellipse by an arbitrary angle seems trickier. Initially I though it might work to draw the unrotated ellipse and apply a rotation matrix to each point, but it seems as though that could cause errors do to rounding, and I need rather high precision.

Is my suspicion about this method correct? How could I accomplish this task more precisely?

I'm programming in C++ (although that shouldn't really matter since this is a more algorithm-oriented question).

Edit: as David pointed out, I guess I may really be wondering how to do pixel interpolation.

Answer 1

Use:

x = X cos(a) - Y sin(a)
y = Y cos(a) + X sin(a)

Where a is the angle of anticlockwise rotation, (x, y) are the new coordinates, and (X, Y) are the old.

You should use floats to preserve precision. Just go through every point, apply the transformation, and voilà.

Edit: after some searching, here's some code from Microsoft: http://research.microsoft.com/en-us/um/people/awf/graphics/bres-ellipse.html that draws rastered conic sections.

Answer 2

Bresenham (famous for his line drawing algorithm) also has an algorithm for drawing an ellipse. You can try to google bresenham ellipse.