[Qt-interest] How to multiply a QTransform by float without affecting the scale?

[K. Frank]

Hi Geovani!

On Tue, Jan 11, 2011 at 10:02 PM, Luiz Geovani Vier <lgvier at gmail.com> wrote:
> Hello all,
> So I have a QTransform containing rotation and translation.
> I'd like to animate a transformation by multiplying the old matrix by 1.0 -
> [0..1] and new matrix by [0..1]. (btw, Is there a better way to do that?)
> However, when I multiply QTransform by a float, that also affects the
> scaling (e.g. * 0.1 = very tiny item :-)
> How should I be handling this so that only rotation and translation is
> applied?

The problem, as you've found out, is that if R1 and R2 are rotations
(more precisely, rotation matrices), then the interpolated matrix,
R = (1 - alpha) R1  +  alpha R2 is not a rotation.

What you need to do instead is interpolate the angles by which
you are rotating.

(However, interpolating translations, v = (1 - alpha) v1  +  alpha v2,
where v1 and v2 are translation vectors, is perfectly fine, and gives
you what you want.)

If the QTransform matrix consists of only a rotation and translation,
then it will be of a special form, and the { {m11, m12}, {m21, m22} }
piece of the matrix will be of the form

    { { cos (theta), sin (theta) }, { -sin (theta), cos (theta) } }

where theta is the angle by which you are rotating.

Given an angle theta, Qt lets you create a QTransform representing
a rotation by that angle (using the QTransform::rotate() member
function), but does not appear to let you extract the rotation angle
from a given QTransform.

You have to extract the angle by hand.  Assuming that the QTransform
is a pure rotation plus translation, then the rotation angle is

   theta  =  arctan (m12 / m11)   (for m11 > 0)
   theta  =  arctan (m12 / m11)  +  180 deg.   (for m11 < 0)

(Arctan is the inverse tangent function.)

So, if you want to interpolate between two QTransforms, T1 and T2,
that are both a rotation plus a translation, you need to extract the
two rotation angles, theta1 and theta2, as above, and the two translation
vectors, v1 = (T1.m31(), T1.m32()) and v2 = (T2.m31(), T2.m32).

Then interpolate the angles

   theta  =  (1 - alpha) theta1  +  alpha theta2

interpolate the translations

   v  =  (v_x, v_y)  = (1 - alpha) v1  +  alpha v2

and construct a QTransform from the interpolated values, e.g.,

   QTransform().rotate (theta).translate (v_x, v_y)

Given which functions are (and are not) provided by Qt, I think that
this is the most straightforward way to do the interpolations for the
animation you want.

> Thanks,
> -Geovani

Good luck.


K. Frank