1,101 research outputs found
Visualizing Quaternion Multiplication
Quaternion rotation is a powerful tool for rotating vectors in 3-D; as a result, it has been used in various engineering fields, such as navigations, robotics, and computer graphics. However, understanding it geometrically remains challenging, because it requires visualizing 4-D spaces, which makes exploiting its physical meaning intractable. In this paper, we provide a new geometric interpretation of quaternion multiplication using a movable 3-D space model, which is useful for describing quaternion algebra in a visual way. By interpreting the axis for the scalar part of quaternion as a 1-D translation axis of 3-D vector space, we visualize quaternion multiplication and describe it as a combined effect of translation, scaling, and rotation of a 3-D vector space. We then present how quaternion rotation formulas and the derivative of quaternions can be formulated and described under the proposed approach.112sciescopu
Minkowski products of unit quaternion sets
The Minkowski product of unit quaternion sets is introduced and analyzed,
motivated by the desire to characterize the overall variation of compounded
spatial rotations that result from individual rotations subject to known
uncertainties in their rotation axes and angles. For a special type of unit
quaternion set, the spherical caps of the 3-sphere in ,
closure under the Minkowski product is achieved. Products of sets characterized
by fixing either the rotation axis or rotation angle, and allowing the other to
vary over a given domain, are also analyzed. Two methods for visualizing unit
quaternion sets and their Minkowski products in are also
discussed, based on stereographic projection and the Lie algebra formulation.
Finally, some general principles for identifying Minkowski product boundary
points are discussed in the case of full-dimension set operands.Comment: 29 pages, 1 figur
How efficiently can one untangle a double-twist? Waving is believing!
It has long been known to mathematicians and physicists that while a full
rotation in three-dimensional Euclidean space causes tangling, two rotations
can be untangled. Formally, an untangling is a based nullhomotopy of the
double-twist loop in the special orthogonal group of rotations. We study a
particularly simple, geometrically defined untangling procedure, leading to new
conclusions regarding the minimum possible complexity of untanglings. We
animate and analyze how our untangling operates on frames in 3-space, and teach
readers in a video how to wave the nullhomotopy with their hands.Comment: To appear in The Mathematical Intelligencer. For supplemental videos,
see http://www.math.iupui.edu/~dramras/double-tip.html , or
https://www.youtube.com/playlist?list=PLAfnEXvHU52ldJaOye-8kZV_C1CjxGx2C .
For a supplemental virtual reality experience, see
http://meglab.wikidot.com/visualizatio
Tracking Articulator Movements Using Orientation Measurements
This paper introduces a new method to track articulator movements, specifically jaw position and angle, using 5 degree of freedom (5 DOF) orientation data. The approach uses a quaternion rotation method to accomplish this jaw tracking during speech using a single senor on the mandibular incisor. Data were collected using the NDI Wave Speech Research System for one pilot subject with various speech tasks. The degree of jaw rotation from the proposed approach is compared with traditional geometric calculation. Results show that the quaternion based method is able to describe jaw angle trajectory and gives more accurate and smooth estimation of jaw kinematics
The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles
The gradient of potential vorticity (PV) is an important quantity because of
the way PV (denoted as ) tends to accumulate locally in the oceans and
atmospheres. Recent analysis by the authors has shown that the vector quantity
\bdB = \bnabla q\times \bnabla\theta for the three-dimensional incompressible
rotating Euler equations evolves according to the same stretching equation as
for \bom the vorticity and \bB, the magnetic field in magnetohydrodynamics
(MHD). The \bdB-vector therefore acts like the vorticity \bom in Euler's
equations and the \bB-field in MHD. For example, it allows various analogies,
such as stretching dynamics, helicity, superhelicity and cross helicity. In
addition, using quaternionic analysis, the dynamics of the \bdB-vector
naturally allow the construction of an orthonormal frame attached to fluid
particles\,; this is designated as a quaternion frame. The alignment dynamics
of this frame are particularly relevant to the three-axis rotations that
particles undergo as they traverse regions of a flow when the PV gradient
\bnabla q is large.Comment: Dedicated to Raymond Hide on the occasion of his 80th birthda
Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics
After a review of the isentropic compressible magnetohydrodynamics (ICMHD)
equations, a quaternionic framework for studying the alignment dynamics of a
general fluid flow is explained and applied to the ICMHD equations.Comment: 12 pages, 2 figures, submitted to a Focus Issue of New Journal of
Physics on "Magnetohydrodynamics and the Dynamo Problem" J-F Pinton, A
Pouquet, E Dormy and S Cowley, editor
Lagrangian particle paths and ortho-normal quaternion frames
Experimentalists now measure intense rotations of Lagrangian particles in
turbulent flows by tracking their trajectories and Lagrangian-average velocity
gradients at high Reynolds numbers. This paper formulates the dynamics of an
orthonormal frame attached to each Lagrangian fluid particle undergoing
three-axis rotations, by using quaternions in combination with Ertel's theorem
for frozen-in vorticity. The method is applicable to a wide range of Lagrangian
flows including the three-dimensional Euler equations and its variants such as
ideal MHD. The applicability of the quaterionic frame description to Lagrangian
averaged velocity gradient dynamics is also demonstrated.Comment: 9 pages, one figure, revise
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