= Section orientation
<Section orientation> specifies how a cut is directed relative to a specimen or reference coordinate system. A flat cut is described by a unit <normal vector> $\mathbf n$, together with an in-plane direction when image axes matter. Reversing $\mathbf n$ leaves the unoriented <plane> unchanged, although choosing a normal direction is useful for ordering an <image stack>. <Section orientation> affects which structures intersect a section and their apparent dimensions, even without <deformation (mechanics)>.
An original layered-material example complements the circular-fibre geometry in <microtomy>. Model parallel interfaces in a <polymer> laminate by
$$
\mathbf m\cdot\mathbf r=kL,\qquad k\in\mathbb Z,
$$
where $\mathbf m$ is a unit <normal vector> to the layers and $L$ is their true normal spacing. Let $\phi$ be the acute angle between the layer normal $\mathbf m$ and the section normal $\mathbf n$. For $0<\phi\leq\pi/2$, the <orthogonal projection> of $\mathbf m$ into the section has length $\sin\phi$. Its unit direction is
$$
\mathbf q=\frac{\mathbf m-(\mathbf m\cdot\mathbf n)\mathbf n}{\sin\phi}.
$$
Moving distance $d$ along $\mathbf q$ changes $\mathbf m\cdot\mathbf r$ by $d\sin\phi$. Hence adjacent interface traces in the section are separated by
$$
\boxed{d=\frac{L}{\sin\phi}}.
$$
For chosen $L=2\,\mathrm{\mu m}$ and $\phi=30^\circ$, their apparent spacing is $4\,\mathrm{\mu m}$. At $\phi=90^\circ$, the section contains the layer normal and $d=L$. At $\phi=0$, the section is parallel to the layers and does not show this family of separated interface traces; the spacing formula has no finite value there.
A useful physical check sections a reference laminate at known orientations, images it with an <optical microscope> using established length calibration, and measures trace spacing perpendicular to the traces. Recording orientation prevents an oblique cut from being mistaken for a change in layer spacing. The ideal calculation assumes flat interfaces, negligible <physical section thickness> and no cutting distortion. Manufacturer materials guidance explicitly discusses <polymers> and <composite materials>, as well as compression during sectioning.<Ultramicrotomy Techniques for Materials Sectioning by Leica Microsystems>[Opening materials discussion; How to prepare high-quality sections of polymers]{ref} Those physical effects must be assessed separately from the geometric prediction.
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