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| Bidang (detail), Iban people. Upper Rajang river, Kapit Division of Sarawak, 20th century, 110 x 61 cm. Ikat technique. From the Teo Family collection, Kuching. Photograph by D Dunlop. |
Let particle P be described by an ordered chain of events
$\Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \; \ldots \; \sf{P}_{\it{k}} \; \ldots \; \right)$
that is repetitive so that $\Psi$ may also be written as
$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2} , \sf{\Omega}_{3} \; \ldots \; \right)$
where each orbital cycle $\sf{\Omega}$ is composed of $N$ sub-orbital events
$\sf{\Omega} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \; \ldots \; \sf{P}_{ \it{N}} \right)$
Let P be described by $\delta _{\hat{m}}$, $\delta _{\hat{e}}$ and $\delta _{z}$ which specify its spatial orientation, $R$ its orbital radius and $\lambda$ its wavelength. We use these characteristics to define the following numbers for describing sub-orbital events
$\begin{align} &dx \equiv \delta _{\hat{m}} \frac{R}{N } =\frac{ R \cos{\! 2 \theta } }{N } \\ \ \\ &dy \equiv \delta _{\hat{e}} \frac{R}{N } = - \frac{ R \sin{\! 2 \theta } }{N } \\ \ \\ &dz \equiv \delta _{z} \frac{ \lambda }{\,N \,} = \frac{\lambda }{2 \pi} d \theta \end{align}$
where $\theta$ is the phase angle. Then using the spatial axes $\hat{x}$, $\hat{y}$ and $\hat{z}$ the displacement vector is defined as
$d\bar{r} \equiv dx \, \hat{x} + dy \, \hat{y} + dz \, \hat{z}$
If we switch to implicitly using Cartesian basis vectors, we can express the displacement as an ordered set
$d\bar{r} = \left( dx, dy, dz \right)$
WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that $N$ is large enough to make an approximation to spatial continuity, then allowing the use of calculus.
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Displacement |
