= Microtomy
\b[Microtomy] prepares thin physical sections with a <microtome>, exposing interior structure for <microscopy>. The knife separates a layer from a supported specimen while a controlled feed determines the nominal depth increment. <Physical section thickness>, <section orientation> and cutting-induced <deformation (mechanics)> all enter the interpretation of an image. <Ultramicrotomy> is the specialized thin-section technique used in applications such as <transmission electron microscopy>.
The preparation method also serves materials research. Yale's instrument description lists a Leica Ultra UC7 with a trimming mill and glass knife maker, and identifies <polymers>, <material fibres> and soft metals among suitable specimen materials.<Microtome at the Yale Institute for Nanoscience and Quantum Engineering>[About the instrument, accessories and material list]{ref} Manufacturer guidance discusses <polymer> blends and <composite materials>, along with the possibility of <section compression> and cutting artifacts.<Ultramicrotomy Techniques for Materials Sectioning by Leica Microsystems>[Opening materials discussion; How to prepare high-quality sections of polymers]{ref}
A useful teaching experiment would compare cuts through a straight, known circular <material fibre> at different <section orientations>. A <normal vector> to the cut aligned with the fibre axis gives a circular profile. An oblique cut gives an <ellipse> even when the fibre has suffered no <deformation (mechanics)>. Image the recovered sections with an <optical microscope>, measure profile dimensions with an established <image stack axis calibration>, and record the angle between the section's <normal vector> and the fibre axis. This tests the geometry of the section rather than assuming that every measured elongation is a physical distortion.
For an original ideal calculation, model the fibre boundary as an infinite <circular cylinder> of radius $R$, with axis along the unit <vector> $\mathbf u=(\sin\theta,0,\cos\theta)$, where $0\leq\theta<\pi/2$. Parallel section <planes> have equation $z=z_0$. The squared <Euclidean distance> to the axis is $(x\cos\theta-z\sin\theta)^2+y^2$, obtained by subtracting the squared axial <orthogonal projection> from $x^2+y^2+z^2$. At the section <plane>, the boundary therefore satisfies
$$
\frac{(x-z_0\tan\theta)^2}{(R/\cos\theta)^2}+\frac{y^2}{R^2}=1.
$$
This <oblique section of a circular cylinder> is an <ellipse> with <semi-major and semi-minor axes> and <area>
$$
\boxed{a=\frac{R}{\cos\theta},\qquad b=R,\qquad A=\frac{\pi R^2}{\cos\theta}}.
$$
The <area of an ellipse> gives the last expression. For chosen $R=20\,\mathrm{\mu m}$ and $\theta=45^\circ$, the <semi-major and semi-minor axes> are approximately $28.28\,\mathrm{\mu m}$ and $20.00\,\mathrm{\mu m}$. The profile <area> is $\sqrt2$ times the perpendicular-profile <area>, an increase of about 41.4 percent entirely explained by <section orientation>.
The <center of an ellipse> in this model has $x$ coordinate $x_c(z_0)=z_0\tan\theta$. In an ordered <image stack>, two sampled section <planes> separated by $h$ consequently have lateral centre displacement $\Delta x=h\tan\theta$. At the chosen $h=20\,\mathrm{\mu m}$ and $\theta=45^\circ$, this displacement is $20\,\mathrm{\mu m}$. Here $h$ is the spacing between sampled planes, which can differ from the <physical section thickness> if some sections are not imaged.
\Image[/microtomy-oblique-cylinder-section.png]
{title=Oblique <microtomy> sections of an ideal <circular cylinder>}
{description=Original chosen geometry. Side view shows the axis and two sampled section planes; the right panel compares centred profiles. The model treats sections as mathematical planes and omits finite-thickness blur and deformation.}
The calculation assumes a circular, straight fibre, known section geometry and negligible <physical section thickness> relative to resolved features. One elliptical profile alone cannot distinguish an oblique circular fibre from a differently shaped fibre or an unknown <deformation (mechanics)>. Comparing known reference features and following their positions through an <image stack> provides additional evidence. <Image registration> can align images, but an inferred shift may contain real specimen geometry as well as instrument motion; removing every shift would erase the tilt information in this example.
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