Jump to content
Posted 02 June 2011 - 09:58 PM
Posted 03 June 2011 - 05:37 AM
Posted 03 June 2011 - 06:03 AM
IMHO this is not really true (but I must admit to that I may misunderstand the meaning of "rate of change"): A quaternion cannot be used to express a rate of change, because this would mean a unit of "angle over time", e.g. degrees per second. Although you can implicitly attach such a unit to a quaternion, the periodicity of sine and cosine allows to store only angles from a range of 360°. As a rate of change, however, you would need to be able to store angles greater than 360° as well. Hence quaternions can be used to store orientations and "small rotations" only. Opposed to this, e.g. the axis/angle rotation representation is able to store any angle.
... That's all simple enough but one of the things that makes quaternions so great is that given a quaternion that describes an object's rate of change in orientation and the object's current orientation you can multiply the current orientation by the rate of change in orientation to calculate the new orientation. ...
Posted 03 June 2011 - 08:43 AM
Posted 03 June 2011 - 02:08 PM
The objects your are describing are not "septernions": You are talking about the orientation-preserving isometries of the three-dimensional Euclidean space, which form a 6-dimensional Lie group. An element of this group can be described by just a quaternion and a translation, and you can define the operation of composition fairly easily (multiply the quaternions, add the translations, but you have to rotate the translation using the quaternion of the other isometry; if you are careful you'll get it right).
haegarr is right to point out that "rate of change" in rotation should normally be thought of as pseudovector (angular velocity), not a quaternion. If you integrate the angular velocity for a time delta_t, you can think of the result as a quaternion, and then what you are saying is correct.
Posted 03 June 2011 - 02:39 PM
Posted 04 June 2011 - 03:34 AM