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Abstract
In this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our work is made even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our work easier. This simplicity is achieved by obtaining triplets from quaternions.
Highlights
Introduction quaternion QisA real quaternion is defined that Q = w + xi + yj + zk where w, x, y, z are real numbers and i2 = j2 = k2 = ijk = −1 ij = k, jk = i, ki = j ji = −k, kj = −i, ik = −j.The norm of a real quaternion Q is |Q|2 = QQ = w2 + x2 + y2 + z2.| Q |2 = Q Q = ( Q + ε Q∗ )( Q + ε Q∗ ) = Q Q + ε (Q Q∗ + Q∗ Q )= ŵ 2 + x2 + ŷ2 + ẑ2
Triplets are obtained from the real quaternions
Dual triplets are obtained from the dual quaternions
Summary
A real quaternion is defined that Q = w + xi + yj + zk where w, x, y, z are real numbers and i2 = j2 = k2 = ijk = −1 ij = k, jk = i, ki = j ji = −k, kj = −i, ik = −j. The norm of a real quaternion Q is |Q|2 = QQ = w2 + x2 + y2 + z2. The dual quaternion set is indicated by Ĥ [3]. The triplets is in a three-dimensional space. They can be obtained from arbitrary quaternions in fourdimensional space. The set of quaternions is indicated by H
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