Ellipse under an affine transformation

[Paul Giaccone]

Am I correct in thinking that an ellipse maps to an ellipse under a 2D affine
transformation? My calculations seem to suggest that this is the case but I'd
like some confirmation.

Mathematically:

(x, y) = (a cos theta, b sin theta) for 0 <= theta <= 2 * pi

describes the canonical ellipse parametrically. However, the ellipse may also
be rotated through an angle phi, giving the form

(x, y) = (A cos(theta - alpha_1), B sin(theta - alpha_2)

where

A = sqrt((a cos theta)^2 + (b sin theta)^2)
B = sqrt((a sin theta)^2 + (b cos theta)^2)
alpha_1 = arctan((b / a) tan phi)
alpha_2 = arctan((-a / b) cot phi)

Now when transpose(x, y, 1) is transformed by an affine transformation of the
form

(c d e)
(f g h)

where c, d, e, f, g and h are real constants, is the result still in the
general
form for an ellipse described above (the translation due to (e, f) may be
ignored, of course)?

--
Paul Giaccone
Vision Group
k94...@kingston.ac.uk
Kingston University

[Don Girod]

Paul Giaccone (cs_j675) wrote:
: Am I correct in thinking that an ellipse maps to an ellipse under a 2D affine

: transformation? My calculations seem to suggest that this is the case but I'd
: like some confirmation.

:
Yes. The image of an ellipse under a linear (hence also affine) trans-
formation is again an ellipse. For the case of the linear transformation
this is seen most clearly from considering the singular value factorization
of a matrix for the transformation. Every linear transformation can
be written as USV where U and V are orthogonal and S is diagonal, which
shows that circles map to ellipses.

[Sinan Karasu]

In article <4glvm9$q...@niktow.canisius.edu>,

Don Girod <gi...@niktow.canisius.edu> wrote:
>
>Yes. The image of an ellipse under a linear (hence also affine) trans-
>formation is again an ellipse. For the case of the linear transformation
>this is seen most clearly from considering the singular value factorization
>of a matrix for the transformation. Every linear transformation can
>be written as USV where U and V are orthogonal and S is diagonal, which
>shows that circles map to ellipses.

Or simpler let A be the transformation that takes it to a circle,
then T=T.A^{-1}.A
But we know that T.A^{-1} takes a circle to an ellipse.

Sinan

--
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[Don Girod]

Sinan Karasu (si...@u.washington.edu) wrote:
: In article <4glvm9$q...@niktow.canisius.edu>,

Ah, yes, but how do you KNOW that a linear transformation takes a circle
to an ellipse? I agree, everybody knows it but the proof is not all
that elementary. The singular value decomposition is one way to
prove it; there are others, but they all take some work. You
might try, for example, to find a formula in terms of a,b,c,d for
the vectors which are the semiaxes of the ellipse determined by
matrix [ a b ]
c d acting on the unit circle.