Index of sphericity

File:Index of sphericity.png

547x547px

The Index of Sphericity ($I_{s}$) is a novel mathematical method described by Paul Gagniuc in 2015, that allows the quantification of various 2D irregular shapes. Let us consider that an unknown object intersects a two-dimensional plane. By studying the shape left by the unknown object on the two-dimensional plane, we can determine how spherical the unknown object is.

The values used for finding the Index of Sphericity are the perimeter and the area of an irregular shape measured on a two-dimensional plane. The Index of Sphericity ($I_{s}$) considers two parameters: 1) the shape perimeter ($P_{m}$) measured on a 2D surface and 2) a relative parameter, namely the mean shape diameter ($d_{m}$). The mean shape diameter ($d_{m}$) is determined from the perimeter and the area of the irregular shape. Thus, the shape diameter can be calculated using the perimeter ($d_{ip}$) or the area ($d_{ia}$):

$d_{ip} = \frac{P_{m}}{\pi }$

$d_{ia} = \sqrt{\frac{4\times a_{m}}{\pi }}$

If the shape is irregular then the two formulas will provide different results. Thus, the average between the two results will be the mean diameter ($d_{m}$):

$d_{m} = \frac{d_{ip}+d_{ia}}{2}$

However, as a first reference point, the ratio between the circle perimeter ($C_{p}$) and diameter ($C_{d}$) was considered, namely $\pi$.

$\pi = \frac{Cp}{Cd}$

Next, the same steps have been applied for the ratio ($I_{\pi }$) between the perimeter and the mean diameter of the irregular shape:

$I_{\pi } = \frac{P_{m}}{d_{m}}$

The distance from the ideal proportions are quantified through the Index of Sphericity ($I_{s}$), in this case as a ratio between $\pi$ and $I_{\pi }$:

$I_{s} = \frac{I_{\pi }}{\pi }$

Thus, $I_{s}$ roughly measures how much an object shape deviates from the ideal shape of a circle (and consequently from an ideal sphere). As the value of $I_{s}$ tends more towards 1, the object is more spherical.

First use

The first application of this method was the quantification of islet sphericity throughout the human pancreas, by using morphometric measurements from histological slides.

File:Gagniuc Sphericity index.PNG

Quantification of islet sphericity throughout the human pancreas.

See also

• Sphericity
• Shape
• Circle
• Roundness