Dual numbers

Dual numbers

Rigid-Body Motion

Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021

Gregory S. Chirikjian, Alexander B. Kyatkin

Analogous to this, the "dual numbers" are used in kinematics so that rigid-body motions can be expressed more compactly than with 4 × 4 matrices (see, for example, [24]). Analogous to the imaginary number i2= −1, a dual number, ϵ^, is defined so that ϵ^≠0 and ϵ^2=0. Of course no real number satisfies these conditions, but taking these as defining properties of ϵ^ allows us to define a new arithmetic of dual numbers. For instance, calculating the product of two numbers with real and dual parts results in (a1+ϵ^a2)(b1+ϵ^b2)=a1b1+ϵ^(b1a2+a1b2).

Multidual Sensitivity Method in Ray-Tracing Transport Simulations

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Journal Information

Published in Nuclear Science and Engineering, 2021

Matthew R. Balcer, Harry Millwater, Jeffrey A. Favorite

Alternatively, a dual number can be used for the nonreal step. Dual numbers are a subset of generalized complex numbers.20,21 A dual number has the form , where and are real numbers and represents the nonreal component. The nonreal component of a dual number is defined under the constraint that , which implies for , and . Extraction of the nonreal part of a dual number is denoted by . A dual step, , in Eq. (1) gives

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Rigid-Body Motion

Multidual Sensitivity Method in Ray-Tracing Transport Simulations