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'''Shape factors''' are [[dimensionless quantity|dimensionless quantities]] used in [[image analysis]] and [[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 (object)|circle]], sphere or equilateral [[polyhedron]].<ref>L. Wojnar & K.J. Kurzydłowski, et al., ''Practical Guide to Image Analysis'', [[ASM International (society)|ASM International]], 2000, p 157-160, {{ISBN
▲'''Shape factors''' are [[dimensionless quantity|dimensionless quantities]] used in [[image analysis]] and [[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 (object)|circle]], sphere or equilateral [[polyhedron]].<ref>L. Wojnar & K.J. Kurzydłowski, et al., ''Practical Guide to Image Analysis'', [[ASM International (society)|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.
==Aspect ratio==
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==Elongation shape factor==
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
:<math>f_\text{elong} = \sqrt{\frac{i_2}{i_1}}</math>
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==An application of shape factors==
[[Greenland]], the largest island in the world, has an area of 2,166,086 km<sup>2</sup>; a coastline (perimeter) of 39,330 km; a
:<math>A_R = \frac{1290}{2670} = 0.483</math>
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