Modulating signals are represented as a complex vector using a transformation equation.
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Thus, the Lorentz transformation equations take the form
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For example, the basic transformation equations become
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He also derived the transformation equations which formed the basis of the special relativity theory of Albert Einstein.
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We then simply use the Lorentz transformation equations to see when and where the observer in S sees these two events as occurring:
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The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors.
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How would the Lorentz transformation equations change if the two reference frames were moving relative to each other at an angle with respect to the set of axes.
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Note that in both these projections ( which are based on various ellipsoids ) the transformation equations for x and y and the expression for the scale factor are complicated functions of both latitude and longitude.
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When you look at the transformation equations, you will notice that this can be only be valid for two events satisfying ?t = ?t = ?x'= ?x = 0, in other words for equations of the form 0 = 0.
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In terms of dual quaternions and the homogeneous coordinates of a point \ textbf { P } : ( P _ 1, P _ 2, P _ 3, P _ 4 ) of the object, the transformation equation in terms of quaternions is given by ( see for details)
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