I have been working towards proposing definition for my proposed spatial inclusion relation *enveloped_by*. After quite a lot of readings and discussions, the following two possible definitions are proposed. One is by using the tangential proper part relation and the other is by using covered by relation.

**Spatial Inclusion Relation — Enveloped_by / envelopes :**

**Proposal definition 1:**

enveloped_by(b,a) / envelopes(a,b) = TPP(a,b) and SCOINC(a,b)

enveloped_by(b,a)= a is tangentialproperpartof b and two boundaries of

a and b coincide at all points such that:

b is a closed material object (whether sphere, or cylindrical),

a is a material object,

the outer boundary of b and the inner boundary of a coincide with each

other at all points (i.e. the boundaries of b and a co-exist with each

other)

SCOINC(a,b) — if two (spatial) boundaries a and b coincide, a and b

cannot spatially exist independently or cannot be located farther from

each other

**Proposal definition 2:**

enveloped_by(b,a) / envelopes(a,b) = (bCOVERED_BYa) and (aSCOINCb)

enveloped_by(b,a)= b is coveredby a and two boundaries of a and b

coincide at all points such that:

b is a closed material object (whether sphere, or cylindrical),

a is a material object, and

the outer boundary of b and the inner boundary of a coincide with each

other (i.e. the boundaries of b and a co-exist with each other)

SCOINC(a,b) — if two (spatial) boundaries a and b coincide, a and b

cannot spatially exist independently or cannot be located farther from

each other

Properties: Intransitive and assymetric

Cardinality: one to one

Relations:

nucleus is enveloped_by nuclear membrane

mitochondria is enveloped_by double layered membrane

animal cell is enveloped_by cell membrane

Enveloped_by / envelopes = enclosed_by / encloses

Enveloped_by notequalto surrounded_by because in surrounded_by the boundaries of a and b do not coincide and a and b can spatially exist independently or can be located farther from each other.

References: RO, GFO, FMA, Point-Set Topological Relations, Region Connection Calculus

I have proposed these two definitions in the OBO-discuss and OBO-Relations mailing lists for further discussions and comment from the community. Let us see.