5.261. orth_on_top_of_orth

DESCRIPTIONLINKSGRAPH
Origin

Used for defining πš™πš•πšŠπšŒπšŽ_πš’πš—_πš™πš’πš›πšŠπš–πš’πš.

Constraint

πš˜πš›πšπš‘_πš˜πš—_πšπš˜πš™_𝚘𝚏_πš˜πš›πšπš‘(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1,π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2,πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼)

Type
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš˜πš›πš’-πšπšŸπšŠπš›,πšœπš’πš£-πšπšŸπšŠπš›,πšŽπš—πš-πšπšŸπšŠπš›)
Arguments
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄
πš…π™΄πšπšƒπ™Έπ™²π™°π™»_π™³π™Έπ™Όπš’πš—πš
Restrictions
|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|>0
πš›πšŽπššπšžπš’πš›πšŽ_𝚊𝚝_πš•πšŽπšŠπšœπš(2,π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄,[πš˜πš›πš’,πšœπš’πš£,πšŽπš—πš])
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πšœπš’πš£β‰₯0
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πš˜πš›πš’β‰€π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πšŽπš—πš
|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1|=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2|
πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼β‰₯1
πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼≀|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1|
πš˜πš›πšπš‘_πš•πš’πš—πš”_πš˜πš›πš’_πšœπš’πš£_πšŽπš—πš(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1)
πš˜πš›πšπš‘_πš•πš’πš—πš”_πš˜πš›πš’_πšœπš’πš£_πšŽπš—πš(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2)
Purpose

π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 is located on top of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2 which concretely means:

  • In each dimension different from πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼 the projection of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 is included in the projection of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2.

  • In the dimension πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼 the origin of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 coincide with the end of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2.

Example
πš˜πš›πš’-5 πšœπš’πš£-2 πšŽπš—πš-7,πš˜πš›πš’-3 πšœπš’πš£-3 πšŽπš—πš-6,πš˜πš›πš’-3 πšœπš’πš£-5 πšŽπš—πš-8,πš˜πš›πš’-1 πšœπš’πš£-2 πšŽπš—πš-3,2

As illustrated by FigureΒ 5.261.1 the orthotope π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 (rectangle R1 coloured in pink) is on top of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2 (rectangle R2 coloured in blue) according to the hypothesis that the vertical dimension corresponds to dimension 2 (i.e.,Β πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼=2). This stands from the fact that the following conditions hold:

  • π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2[2].πš˜πš›πš’+π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2[2].πšœπš’πš£=1+2=π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1[2].πš˜πš›πš’,

  • π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2[1].πš˜πš›πš’=3β‰€π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1[1].πš˜πš›πš’=5,

  • π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1[1].πšŽπš—πš=7β‰€π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2[1].πšŽπš—πš=8.

Consequently, the πš˜πš›πšπš‘_πš˜πš—_πšπš˜πš™_𝚘𝚏_πš˜πš›πšπš‘ constraint holds.

Figure 5.261.1. Illustration of the relation on top of
ctrs/orth_on_top_of_orth1
Used in

πš™πš•πšŠπšŒπšŽ_πš’πš—_πš™πš’πš›πšŠπš–πš’πš.

Keywords

geometry: geometrical constraint, non-overlapping, orthotope.

Arc input(s)

π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2

Arc generator
π‘ƒπ‘…π‘‚π·π‘ˆπΆπ‘‡(=)β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1,πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ2)

Arc arity
Arc constraint(s)
β€’ πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1.πš”πšŽπš’β‰ πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼
β€’ πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ2.πš˜πš›πš’β‰€πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1.πš˜πš›πš’
β€’ πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1.πšŽπš—πšβ‰€πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ2.πšŽπš—πš
Graph property(ies)
𝐍𝐀𝐑𝐂=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1|-1

Arc input(s)

π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄2

Arc generator
π‘ƒπ‘…π‘‚π·π‘ˆπΆπ‘‡(=)β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1,πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ2)

Arc arity
Arc constraint(s)
β€’ πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1.πš”πšŽπš’=πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼
β€’ πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1.πš˜πš›πš’=πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ2.πšŽπš—πš
Graph property(ies)
𝐍𝐀𝐑𝐂=1

Graph model

The first and second graph constraints respectively express the first and second conditions stated in the Purpose slot defining the πš˜πš›πšπš‘_πš˜πš—_πšπš˜πš™_𝚘𝚏_πš˜πš›πšπš‘ constraint.

PartsΒ (A) andΒ (B) of FigureΒ 5.261.2 respectively show the initial and final graph associated with the second graph constraint of the Example slot. Since we use the 𝐍𝐀𝐑𝐂 graph property, the unique arc of the final graph is stressed in bold.

Figure 5.261.2. Initial and final graph of the πš˜πš›πšπš‘_πš˜πš—_πšπš˜πš™_𝚘𝚏_πš˜πš›πšπš‘ constraint
ctrs/orth_on_top_of_orthActrs/orth_on_top_of_orthB
(a) (b)
Signature

Consider the second graph constraint. Since all the πš”πšŽπš’ attributes of the π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄1 collection are distinct, because of the arc constraint πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ1.πš”πšŽπš’=πš…π™΄πšπšƒπ™Έπ™²π™°π™»_𝙳𝙸𝙼, and since we use the π‘ƒπ‘…π‘‚π·π‘ˆπΆπ‘‡(=) arc generator the final graph contains at most one arc. Therefore we can rewrite the graph property 𝐍𝐀𝐑𝐂=1 to 𝐍𝐀𝐑𝐂β‰₯1 and simplify 𝐍𝐀𝐑𝐂 Β― Μ² to 𝐍𝐀𝐑𝐂 Β―.