Quaternion rotation demonstration program
qrot demonstrates the problem with
using linear interpolation of Euler angles when animating a rotating object
in three dimensions, and how great circle interpolation of unit quaternions
gives smooth rotation paths.
qrot uses the OpenGL graphics library, which is normally
present in Linux and Windows. It may optionally write ASCII
text files, but does not otherwise alter its environment.
When the program is started, a three dimensional object and a Cartesian grid
are displayed. Click on Move object... menu item to show the
'Object Transformations' dialogue.
The sliders allow you to rotate the object about the X, Y and Z axes, in that fixed
order. The Euler angles are shown in radians. The grid showing parts of the
XY and XZ planes may be turned on and off to help with orientation. The size
of the grid squares is 100 units.
Press 'r' or click the box on the 'Object Transformations' dialogue
to show the rotation axes, which are represented
by a marked ring perpendicular to the axis. Try rotating the X axis (red)
with the Y rotn slider until it's coplanar with the Z axis (blue)
and notice that you lose a degree of rotational freedom. This is known as 'Gimbal Lock'.
Press 'm' or click the box on the 'Object Transformations' dialogue to show a gimble mount. Note that because of
the construction of the gimbal its natural rotation order is XZY, so moving
the sliders on the 'Object Transformations' dialogue sometimes
has a surprising effect.
When either the rotation axes or the gimbal are enabled the object's translations are suppressed.
This dialogue adjusts your viewpoint. The light is positioned at (1000,
1000, -1000) and does not move. It is never visible.
You can specify two or more key positions of the object and observe the program
generate a smooth path between them.
Try these steps to reveal the use of the buttons:
Click on the Move object... and Animate object...
menu items to activate the dialogues.
- Click on Animation files and demos... and press Demo 1
to load and run an animation.
- Press the Play/Stop/Rewind button. The object will move between
the two keys.
- Adjust the duration of the animation with the length slider.
Making your own animations
Change the Alpha slider to an appropriate point in the animation.
- Rotate the object to a new orientation using the sliders.
- Press the Add/Change key button. If alpha is already at a key point,
the key will be replaced by the current object position, otherwise a new
key will be created.
- Repeat 1 - 3 until you have all the key points you want.
- Navigate between the keys with the Next key and
Previous key buttons.
- Delete key removes a key if alpha is actually at a key point,
apart from the keys at 0 and 1.
The object's translations are suppressed during animations.
This is the default method, and linearly interpolates each Euler rotation
between the key positions. The intermediate positions are usually not what
This option performs a piecewise linear great-circle quaternion interpolation
between the keys. The object rotates in the simplest possible way between
each key, but the Euler angles often bounce around all over the place.
This smooths out the kinks at the key points by splining on the unit quaternion
sphere. It is only different from 'Quaternion linear' if there
are more than two key-points in the animation.
You can save an animation to a file by pressing the Save file
button on the Animation files and demos... dialogue box, and load
it again later by pressing Load file.
These are pre-recorded animations loaded from the Animation files and
demos... dialogue box.
Demo 1 is a rotation about the Z axis only. Linear interpolation of
the Euler angles works just as well as great circle interpolation of quaternions
in this case.
Demo 2 illustrates why linear interpolation of Euler angles fails.
Linear quaternion interpolation produces a smooth simple rotation.
Demo 3 has four key points. Is quaternion splining an improvement
over piece-wise linear interpolation in this case?
Demo 4 also has four key points. This time quaternion splining does
seem to usefully smooth out the kinks.
Demo 5 demonstrates gimbal flip. It is a very simple move when the rotation
order is XYZ, but complex when it is XZY.