Geometrical and morphometrical tools for the inclusion analysis of metallic alloys

Inclusion cleanliness in the metallic alloys

Open Access

Table 3

Shape functionals for simply connected compact sets. A, P, r, R, ω, d, denote the area, perimeter, radii of the inscribed and circumscribed circles, minimum and maximum Feret diameters [24], respectively.

Geometrical functionals	Geometric inequalities	Morphological functionals	Extremal sets
r, R	r ≤ R	r/R	C
ω, R	ω ≤ 2R	ω/2R	C
A, R	A ≤ πR²	A/πR²	C
d, R	d ≤ 2R	d/2R	Y
r, d	2r ≤ d	2r/d	C
ω, d	ω ≤ d	ω/d	W
A, d	4A ≤ πd²	4A/πd²	C
R, d			Z
r, P	2πr ≤ P	2πr/P	C
ω, P	πω ≤ P		W
A, P	4πA ≤ P²	4πA/P²	C
d, P	2d ≤ P	2d/P	L
R, P	4R ≤ P	4R/P	L
r, A	πr² ≤ A	πr²/A	C
r, ω	2r ≤ ω	2r/ω	X

Extremal sets are the sets for which an inequality becomes an equality.

C: the disks; W: the constant width compact convex sets; L: the line segments; X: many compact convex sets; Y: many simply connected compact sets; Z every compact convex set of diameter d containing an equilateral triangle of side-length d.

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