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'''Shape factors''' are [[Dimensionless quantity | dimensionless quantities]] used in [[image analysis]] and [[Microscope | microscopy]] that numerically describe the shape of a particle, independent of its size. Shape factors are calculated from measured [[dimension]]s, such as [[diameter]], [[Chord (geometry) | chord]] lengths, [[area]], [[perimeter]], [[centroid]], [[Moment (mathematics) | moments]], etc. The dimensions of the particles are usually [[Measure (mathematics) | measured]] from two-dimensional [[Cross section (geometry) | cross-sections]] or [[Orthographic projection | projections]], as in a microscope field, but shape factors also apply to three-dimensional objects. The particles could be the grains in a [[Metallography | metallurgical]] or [[Ceramography | ceramic microstructure]], or the microorganisms in a [[Microbiological culture | culture]], for example. The dimensionless quantities often represent the degree of [[Deviation (statistics) | deviation]] from an ideal shape, such as a [[Roundness | circle]], sphere or equilateral [[polyhedron]].<ref>L. Wojnar & K.J. Kurzydłowski, et al., ''Practical Guide to Image Analysis'', [[ASM International]], 2000, p 157-160, ISBN 0-87170-688-1.</ref> Shape factors are often ''normalized'', that is, the value ranges from zero to one. A shape factor equal to one usually represents an ideal case or maximum symmetry, such as a circle, sphere, square or cube.
*The most common shape factor is the [[aspect ratio]], a function of the largest diameter and the smallest diameter [[Orthogonality | orthogonal]] to it:
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The normalized aspect ratio varies from approaching zero for a very elongated particle, such as a grain in a cold-worked metal, to near unity for an equiaxed grain. The reciprocal of the right side of the above equation is also used, such that the AR varies from one to approaching infinity.
Another very common shape factor is the circularity, a function of the perimeter ''P'' and the area ''A'':
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The circularity of a circle is 1<!-- The number 1. NOT "...is one of them" -->, and much less than one for a [[starfish]] footprint. The reciprocal of the circularity equation is also used, such that ''f''<sub>circ</sub> varies from one for a circle to infinity.
The less-common elongation shape factor is defined as the square root of the ratio of the two [[Moment of inertia | second moments]] ''i<sub>n</sub>'' of the particle around its principal axes. <ref name="Exner">H.E. Exner & H.P. Hougardy, ''Quantitative Image Analysis of Microstructures'', DGM Informationsgesellschaft mbH, 1988, p 33-39, ISBN 3-88355-132-5.</ref>
:<math>f_\text{elong} = \sqrt{\frac{i_2}{i_1}}</math>
The ''[[Compactness measure of a shape | compactness]] shape factor'' is a function of the polar second moment ''i<sub>n</sub>'' of a particle and a circle of equal area ''A''. <ref name="Exner"/>
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The ''f''<sub>comp</sub> of a circle is one, and much less than one for the cross-section of an [[I-beam]].
The waviness shape factor of the perimeter is a function of the convex portion ''P''<sub>cvx</sub> of the perimeter to the total. <ref name="Exner"/>
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