Fitting

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Definition FITS.D1

Point-set PS1 fits in point-set PS2 if there is a rigid mapping of PS1 to some subset of PS2.

Declaration: fits(PS1, PS2 : point-set)
Formalism: fits(PS1,PS2) < = > exists(M) P(image(PS1,M),PS2)

Fact FITS.F1

(Monotonicity of fits_in) If PS1 fits in PS2, PS1A is a subset of PS1 and PS2 is a subset of PS2A, then PS1A fits in PS2A.

Characteristics: Geometry.

Formalism:
fits1(PS1,PS2) ^ P(PS1A,PS1) ^ P(PS2,PS2A) => fits1(PS1A,PS2A).

Fact FITS.F2

"fits1" is a partial ordering up to congruence.

Characteristics: Geometry.

Formalism:
congruent(P1,P2) => fits1(P1,P2).
fits1(P1,P2) ^ fits1(P2,P1) => congruent(P1,P2).
fits1(P1,P2) ^ fits1(P2,P3) => fits1(P1,P3).

Fact FITS.F3

Necessary condition for "fits1"

Formalism:
fits1(P1,P2) => diameter(P1) < = diameter(P2).

Characteristics: Geometry.

Fact FITS.F4

Sufficient condition for a point-set fitting inside a sphere.

Formalism:
fits1(P,sphere(diameter(P))).

Fact FITS.F5

Sufficient condition for one brick fitting inside another.

Formalism:
XA1 < = XB1 ^ XA2 < = XB2 ^ XA3 < = XB3 => fits1(brick(XA1,XA2,XA3),brick(XB1,XB2,XB3))