[Ipe-discuss] Ellipse: transformation matrix must be transposed?

[Pennings, Gijs]

On Sun, Aug 27, 2023 at 21:53, Otfried Cheong via Ipe-discuss wrote:
> On Wed, Aug 23, 2023, at 07:23, Pennings, Gijs via Ipe-discuss wrote:
> > I'm using IPE to draw ellipses, but I don't fully understand the path
> > representation.
> >
> > Consider the ellipse described by x^2/4 + y^2/36 = 1, and rotate it
> > counterclockwise by pi/6. Then, we compute the transformation matrix by
> > multiplying the rotation matrix with the scale matrix (the operations
> > are applied right to left). I find A = ((1.732, -3), (1, 5.196)). This
> > seems correct; see the attached screenshot.
> >
> > However, to get the desired result in IPE, it seems that this matrix
> > must be transposed (i.e., specified as the path "1.732 1 -3 5.196 0 0
> > e"). Otherwise, the ellipse isn't rotated (since rotation would be
> > applied first, which has no effect on a unit circle?).
> > Is this observation correct and is it intended behavior?
> 
> Your observation is indeed entirely correct - but there is a very simple
> explanation.
> 
> The PDF (or Postscript) format for a matrix is a sequence of six numbers a b c d
> tx ty,
> which represent that linear transformation
> x' = a x + c y + tx
> y' = b x + d y + ty
> (See Section 4.2.3 Transformation Matrices in the PDF reference manual.)
> 
> Ipe uses the same format (so that we don't have to distinguish between
> writing to XML and writing to PDF), so the path specification "1.732 1 -3
> 5.196 0 0 e" uses exactly the matrix A you computed correctly.

Ah, I see. I'm not familiar with the PDF spec, but that makes perfect sense. Thanks!

Kind regards,
Gijs