In this paper we show that the Clifford Algebra of four dimensional Euclidean space yields a set of hypercomplex numbers called “double quaternions.” Interpolation formulas developed to generate Bezier-style quaternion curves are shown to be applicable to double quaternions by simply interpolating the components separately. The resulting double quaternion curves are independent of the coordinate frame in which the key frames are specified. Double quaternions represent rotations in E4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology that is coordinate frame invariant to a desired degree of accuracy within a bounded region of three dimensional space. Examples demonstrate the application of this theory to computing distances between spatial displacement, determining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.

1.
Bottema, O., and Roth, B., 1979 Theoretical Kinematics, Dover Publications, Inc., New York, pp. 558.
2.
Clifford, W.K., 1873, “Preliminary Sketch of Biquaternions,” In Mathematical papers, edited by R. Tucker, Macmillan, London, 1882, pp. 658.
3.
Curtis, M.L., 1984, Matrix Groups, Springer-Verlag, New York, NY.
4.
deCasteljau, F., 1963, “Outillage me´thodes calcul.” Andre´ Citroe¨n Automobiles SA, Paris.
5.
Etzel, K.R., and McCarthy, J.M., 1996, “A Metric on Spatial Displacements Using Biquaternions on SO(4),” IEEE Robotics and Automation Conference, Minneapolis, MN, April 1996.
6.
Etzel, K.R., 1996, “Biquaternion Theory and Applications to Spatial Kinematics,” M.S. Thesis, University of California at Irvine.
7.
Ge, Q. J., 1994, “On the Matrix Algebra Realization of the Theory of Biquaternions,” Proc. of the 1994 ASME Mechanisms Conference, DE-Vol. 70, pp. 425–432.
8.
Ge
Q. J.
, and
Ravani
B.
,
1994
Geometric Construction of Be´zier Motions
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
116
, pp.
749
755
.
9.
Hamilton, W. R., 1860, Elements of Quaternions, Dublin. Reprinted by Chelsea Pub., New York, 1969.
10.
Johnstone, J., and Williams, J., 1995, “Rational Control of Orientation for Animation,” Proceedings of Graphics Interface, pp. 179–186.
11.
Juttler
B.
,
1994
, “
Visualization of Moving Objects Using Dual Quaternion Curves
,”
Computers & Graphics
, Vol.
18
, pp.
315
326
.
12.
Martinez
J.
, and
Duffy
J.
,
1995
, “
On the Metrics of Rigid Body Displacements for Infinite and Finite Bodies
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
117
, No.
1
, pp.
41
47
.
13.
McCarthy
J. M.
,
1983
, “
Planar and Spatial Rigid Motion as Special Cases of Spherical and 3-Spherical Motion
,”
Journal of Mech., Trans., and Auto. in Design
, Vol.
105
, No.
3
, pp.
569
575
.
14.
McCarthy, J. M., 1990, An Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA pp. 130.
15.
Park
F. C.
,
1995
, “
Distance Metrics on the Rigid Body Motions with Applications to Mechanism Design
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
117
, No.
1
, pp.
48
54
.
16.
Park, F. C., and Ravani, B., 1995, “Be´zier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications,” Proc. of the 1994 ASME Mechanisms Conference, DE-Vol. 70, pp. 15–20.
17.
Paul, R., 1981, Robot Manipulators, MIT Press, Cambridge, MA.
18.
Pletinckx
D.
,
1989
, “
Quaternion Calculus as a Basic Tool in Computer Graphics
,”
The Visual Computer
, Vol.
5
, pp.
2
13
.
19.
Ravani
B.
, and
Roth
B.
,
1984
, “
Mappings of Spatial Kinematics
,”
ASME JOURNAL OF MECHANISMS, TRANSMISSIONS, AND AUTOMATION IN DESIGN
, Vol.
106
, pp.
341
347
, September 1984.
20.
Shoemake
K.
,
1985
, “
Animating Rotation with Quaternion Curves
,”
ACM Siggraph
, Vol.
19
, No.
3
, pp.
245
254
.
21.
Study, E., Die Geometric der Dynamen, Leipzig, 1903, pp. 437.
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