QUALITATIVE SPATIAL MODELLING BASED ON

QUALITATIVE SPATIAL MODELLING BASED ON PARTHOOD, STRONG CONNECTION AND CONGRUENCE Stefano Borgo + , Nicola Guarino+ , and Claudio Masolo* (+) National Research Council, LADSEB-CNR, Padova, Italy (*) University of Padova, Dep. of Electronics and Computer Science, Padova, Italy {borgo, guarino, masolo}@ladseb.pd.cnr.it LADSEB-CNR Int. Rep. 03/96 - draft

1. Introduction Various logical theories aimed at the representation of commonsense spatial knowledge have been proposed in the AI community in recent years. In the spirit of [Hayes 1985], the goal has been that of establishing the logical basis of a geometry of commonsense, intended to be used for tasks as disparate as robot navigation or natural language understanding. Besides specific proposals focused on particular aspects, the most general frameworks have been perhaps those based on a combination of mereological and topological notions, which appeared in AI after the publication of Clarke’s mereotopological theory [Clarke 1981]. Among these, particular relevance for AI purposes has the so-called RCC theory [Randell and Cohn 1992, Gotts et al. 1996], which has been recently joined by other proposals originating in the area of philosophy and linguistics [Asher and Vieu 1995, Eschenbach and Heydrich 1995, Casati and Varzi 1996, Smith 1996]. These approaches differ in the primitives adopted and in the ontological assumptions about the domain. They all have in common however the use of the tools of so-called “formal ontology” [Guarino 1995] for the representation of commonsense reality: specifically, mereology and topology. Readers can refer to [Simons 1987] for a general overview of mereology, and to [Varzi 1994, Varzi 1996] for a systematic account of the subtle relations between mereology and topology. We present here some applications of the logical theory of commonsense geometry presented in [Borgo et al. 1996]. The theory is based on three distinct levels of description, each one corresponding to a basic conceptual notion: the mereological level, corresponding to the parthood

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relation; the topological level, based on surface connection (also called strong-connection or sconnection), as distinct from line- and point-connection (l- and p-connection); and the morphological level, where we build up on the basic notion of congruence. Adopting a naive conception of space where points and other lower-dimension entities don’t have a specific ontological status, we show how to deal with various kinds of connections, surfaces, boundaries, holes and polyedra within a logical theory of pointless geometry. Focusing on the possible applications of such a theory in the area of qualitative geometrical reasoning, we show in particular: •

how basic notions of elementary geometry like reflection, equivalence, perpendicularity and parallelism can be reconstructed in our framework;

•

how basic solid shapes like sphere, cube, and cylinder can be defined in an exact way;

•

how a notion of granularity can be introduced in order to characterise boundaries and approximate solid shapes;

These aspects are especially important for the long-term application perspective we have in mind, which aims to establishing a logical interface for geometry-based product description standards like ISO-10303 (STEP), with the twofold objective of making clear its semantic commitment, and at the same time facilitating the integration of logic-based intelligent programs [Guarino 1997].

2. Basic conceptual notions Let us now recall part of our definitions for each of the three levels introduced above. We skip here the details of the axiomatization, assuming the set of axioms we gave in [Borgo et al. 1996]. In any case, we could assume any other set of axioms provided that it restricts the models of our theory in order to characterise the intended meaning of the basic conceptual notions. Notice that we use here the expression “basic conceptual notion” to refer to logical predicates used as cognitive building blocks, which are not necessarily primitive: this means that two axiomatizations may differ in the primitive predicates adopted while characterising the same conceptual notions.

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2.1 Mereology At the mereological level, the basic conceptual notion is the parthood relation (denoted with ‘P’). Its axiomatization is pretty standard, and corresponds to Closed Extensional Mereology [Simons 1987, Varzi 1996]. This differs from General Extensional Mereology since the so-called “fusion axiom” (allowing for infinite sums) is not assumed. We report here some definitions used in mereology: D1. PPxy =df Pxy ∧ ¬x=y

(x is a proper part of y)

D2. Oxy =df ∃z(Pzx ∧ Pzy)

(x and y overlap)

D3. POxy =df Oxy ∧ ¬Pxy ∧ ¬Pyx

(x and y properly overlap)

D4. PNPxy =df ¬Pxy ∧ ¬Pyx

(x and y are not part each other)

D5. x+y =df ιz ∀w(Owz ↔ (Owx ∨ Owy))

(Sum)1

D6. x-y = df ιz ∀w(Pwz ↔ (Pwx ∧ ¬Owy))

(Difference)

D7. x×y =df ιz ∀w(Pwz ↔ (Pwx ∧ Pwy))

(Product)

2.2 Topology At the topological level, the basic conceptual notion adopted is that of strong-connection (sconnection) rather than the usual topological connection. Our approach differs therefore from those inspired to [Clarke 1981], most notably [Aurnague and Vieu 1993; Asher and Vieu 1995] and the so-called RCC theory [Randell and Cohn 1992; Gotts et al. 1996]. The intuition behind this choice is bound to the notion of physical continuity, which seems to us to be the most basic form of connection. In [Borgo et al. 1996], strong connection is defined in terms of a more basic primitive predicate, simple region (‘SR’), holding for a region “all in a piece” which can be occupied by a physical body. The two notions can however be defined one in terms of the other. We report here the more relevant definitions:

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The ι operator appearing above is defined contextually à la Russel: ψ[ιxφ] =df ∃y(∀x(φ ↔ x=y) ∧ ψ[y])

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D8. IPxy =df PPxy ∧ ∀z((SRz ∧ POzx) → Oz(y-x)) D9. MCPxy =df Pxy ∧ SRx ∧ ¬∃z(Pzy ∧ SRz ∧ PPxz) D10. SCxy =df ∃uv(Pux ∧ Pvy ∧ SR(u+v))

(x is an internal partof y)2 (x is a maximally connected part of y) (x and y are strong connected)

2.3 Morphology The basic notion adopted at the morphological level is the relation of congruence between regions, denoted with ‘CG’: the intended meaning is the classical one, i.e. two regions are congruent if they have the same shape and size. Using‘'CG’, we have proposed the following definition for spheres, which links together mereology, topology and morphology (Fig.1): D11. SPHx =df SRx ∧ ∀y(CGxy ∧ POxy → SR(x-y))

(x is a sphere)

The above definition has been used in [Borgo et al. 1996] to propose a first order axiomatization of congruence for our pointless geometry, which exploits the correspondence between points and classes of concentrical spheres established in [Tarski 1929].

Fig.1 According to D11, these regions are not spheres.

The following relations among spheres have been defined on the basis of Tarski’s work, and are here reported together with their informal description: ETxy

x and y are externally tangent

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The above definition of internal part is weaker than that of tangential part used for instance in the RCC theory and here defined in D15: tangential parts do not touch the boundary, while our internal parts can be either p-connected or l-connected to the boundary.

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ITxy

x is proper part of y and internally tangent to y

CNCxy

x and y are concentric

LINxyz

x, y, and z are aligned

BTWxyz

x, y, and z are aligned, and x is between y and z

IDxyz

x and y are internally diametrical relative to z (x and y are internally tangent to z, and z is between x and y )

SEGxy

x and y are not concentric (they form an s-segment)

TRIxyz

x, y, and z are not concentric and not aligned (they form an s-triangle)

PTRIxyz

x, y, and z form an s-triangle, and none of them is a part of the sum of the others (they form a proper s-triangle)

3. Reconstructing classical topological connection With the help of spheres, we are now in the position to define l- and p-connection (Fig.2), and then the usual notion of topological connection. Two regions are l-connected if they are not sconnected, and there exists a sphere that (a) overlaps both regions and (b) results in two non-sconnected pieces when both the overlapping parts are removed. Two regions are p-connected if they are not s- or l-connected, and there exists a sphere such that every sphere concentric to it overlaps both regions. Two regions are connected if they are s-, l-, or p-connected.

u x

z

LCxy

y

x

w

v

u

w

y

v

w

z

x ¬CONVx

PCxy

Fig.2 Line- and point-connection

v

x

CONVx

Fig.3 Defining convexity

D12. LCxy =df ¬SCxy ∧ ∃z(SPHz ∧ Ozx ∧ Ozy ∧ SR(z-x) ∧ SR(z-y) ∧ ¬SR(z-(x+y))) (x and y are l-connected)

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D13. PCxy =df ¬SCxy ∧ ¬LCxy ∧ ∃z(SPHz ∧ ∀u(CNCuz →(Oux ∧ Ouy))) (x and y are p-connected) D14. Cxy =df SCxy ∨ LCxy ∨ PCxy

(x and y are connected)

Topological connection can be used to introduce the classical mereo-topological definitions of tangential part, non tangential proper part and external connection D15. TPxy =df Pxy ∧ ∃z(¬Ozy ∧ Czx)

(x is a tangential part of y)

D16. NTPPxy =df PPxy ∧ ¬TPxy

(x is a non tangential proper part of y)

D17. ECxy =df ¬Oxy ∧ Cxy

(x and y are externally connected)

4. Reconstructing some geometrical notions 4.1 Convexity and convex hull Convexity is an essential property in qualitative spatial reasoning (see for instance [Faltings 1995]). It can be assumed as a primitive, like in the RCC theory where the notion of “convex hull” has an importat role [Cohn 1995]. In our framework, we can however define convexity in terms of congruence (Fig.3). We say that a region x is convex when, for each pair u, v of congruent spheres which are parts of it, any sphere w between u and v and congruent to them is also part of x: D18. CONVx =df ∀uvw(P(u+v)x ∧ CGuv ∧ CGwu ∧ BTWwuv) → Pwx

(x is convex)

Then we can say that a region x is the convex hull of y if it is the smallest convex region containing y: D19. CVHxy =df CONVx ∧ Pyx ∧ ¬∃z(CONVz ∧ Pyz ∧ PPzx)

(x is the convex hull of y)

4.2 Coplanarity, perpendicularity, and parallelism If we want to talk about right polyhedra, we need a characterisation of a “right solid angle”. In our theory, it is not hard to define when two s-segments are perpendicular or parallel, using

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congruence between s-segments to express equidistance. Obviously, we have to concentrate to coplanar s-segments. The definition of coplanarity between spheres lets us state when two ssegments are coplanar, by looking at the spheres which individuate them. We say that four spheres are coplanar when there exist four congruent spheres concentric to them such that the convex hull of their sum is a region non containing any sphere bigger than them: D20. COPLANxyzv=df ∃x’y’z’v’u(CNCxx’ ∧ CNCyy’ ∧ CNCzz’ ∧ CNCvv’ ∧ CGx’y’ ∧ CGx’z’ ∧ CGx’v’ ∧ CHVu(x’+y’+z’+v’) ∧ ¬∃ww’(SPHw ∧ SPHw’ ∧ Pwu ∧ PPw’w ∧ CGw’x’))

(x,y,z,v are coplanar spheres)

Two s-segments are perpendicular when they are coplanar and there exists a sphere z aligned with both of them, and two spheres w and w’ both equidistant from z and aligned with the first ssegment, which are also equidistant from a fourth sphere z’ aligned with the second s-segment. D21. PERPx1x2y1y2 =df COPLANx1x2y1y2 ∧ SEGx1x2 ∧ SEGy1y2 ∧ ∃zz’ww'(PNPz’z ∧ PNPw'w ∧ PNPwz ∧ PNPwz' ∧ LINzx1x2 ∧ LINzy1y2 ∧ LINwx 1x2 ∧ LINw'x1x2 ∧ LINz’y1y2 ∧ CG(w+z)(w'+z) ∧ CG(w+z’)(w'+z’)) (x1x2 and y1y2 are perpendicular s-segments) Two s-segments are parallel when they are coplanar and there is no sphere aligned with both of them: D22. PARLx1x2y1y2 = df COPLANx1x2y1y2 ∧ SEGx1x2 ∧ SEGy1y2 ∧ ¬∃z(SPHz ∧ LINzx1x2 ∧ LINzy1y2)

(x1x2 and y1y2 are parallel s-segments)

4.3 Congruence by reflection vs. congruence by roto-translation The ‘CG’ relation which we have used so far corresponds to the classical geometrical notion of isometry, which does not make any difference between an object and its specular counterpart

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(say, a right-handed vs. a left-handed screw). This is insufficient for reasoning about physical objects, where we need to be clear about the possibility for two objects to occupy the same region. To this purpose, we define here a special kind of congruence, namely congruence by reflection, as opposite to congruence by roto-translation. The main idea is to consider a particular region that fixes a sort of “oriented” plane of reflection, to act as a reference system in our space. Any sum of three pairwise non-overlapping and non-congruent spheres can be used for this purpose. We call such a region Janus3 : D23. JANUSx =df ∃yzw(PTRIyzw ∧ ¬CGzy ∧ ¬CGzw ∧ ¬CGyw ∧ x=y+z+w) (x is a Janus) Notice that the conditions of non-congruence impose an orientation on the plane individuated by the three spheres, and this fact is crucial in order to guarantee the unicity of the reflection. In fact, given a sphere y, if it is coplanar to the “Janus” x, then there exists no different sphere y ' such that CG(x+y)(x+y'), whereas, if y is not coplanar to x, then there exists exactly one different sphere y' such that CG(x+y)(x+y'). After this observation, we can easily define the congruence by reflection. A special case, which we call simple reflection, is captured by the following definition (Fig.4): D24. JCGxy =df ∃z(JANUSz ∧ ¬Cz(x+y) ∧ ∀vw((SPHv ∧ SPHw ∧ CG(z+v)(z+w) ∧ ¬v=w) → (Pvx ↔ Pwy)))

(x and y are congruent by simple reflection)

Two regions are congruent by simple reflection when there exists a Janus-region z such that, for any sphere v in one region, there exists a corresponding sphere w in the other region satisfying CG(z+v)(z+w) (and viceversa).

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The name is taken from Greek mythology: two objects reflect into each other if two-faced Janus can look at them as if they were the same.

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x

y

Fig.4 Simple reflection using Janus.

The following definition, supported by some classical results in three-dimensional space [Yale 1968], accounts for the general case. A region x is a reflection of y when there exist a sequence of three simple reflections yielding y from x. D25. RFCGxy =df ∃u1u2(JCGxu1 ∧ JCGu1u2 ∧ JCGu2y)

(x and y are congruent by reflection)

Now we can define the congruence relationship holding between two regions that can be occupied by the same physical object. Two regions are congruent by roto-translation when there exists a sequence of four simple reflections yielding y from x: D26. RTCGxy =df ∃u1u2u3(JCGxu1 ∧ JCGu1u2 ∧ JCGu2u3 ∧ JCGu3y)) (x and y are congruent by roto-translation) Notice that some regions are congruent both by reflection and by roto-traslation (spheres, cubes, and so on). 4.4 Basic solid shapes With the help of the formal machinery introduced above, we are able to define the basic solid shapes of geometry without referring to points or lines. The definition of cylinder is very easy, and it shows clearly the main idea underlying the other solid shape descriptions. A cylinder is the convex hull of a region which is the difference between the convex hull w of two congruent spheres and the spheres themselves (Fig.5):

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D27. DCYLxyz =df SEGyz ∧ CGyz ∧ ∃w(CVHw(y+z) ∧ CVHx(w-(y+z))) (x is the cylinder defined by y and z) Therefore, the definition of a cylinder is: D28. CYLx =df ∃yz(DCYLxyz)

(x is a cylinder)

Clearly, if the spheres are not congruent, we obtain a truncated cone.

y

s1

s4

s1

c1

c3

c4

s4

c2 s2

s3

h1

s2

s3

h2

z

w

w-(y+z)

x

x=h1*h2

Fig.5. From spheres to cylinders

Fig.6 Using spheres to define cube.

A possible definition of cube is obtained through the definition of cylinder. Consider four congruent, pairwise externally connected and coplanar spheres, and then the two pairs of cylinders formed by them as in Fig.6. Our cube will be the intersection of the convex hulls of these two pairs (h1 and h2). D29. DCUBExs1s2s3s4=df COPLANs1s2s3s4 ∧ PERPs1s2s2s3 ∧ ETs1s2 ∧ ¬s2=s4 ∧ DCYLc1s1s2 ∧ DCYLc2s2s3 ∧ DCYLc3s3s4 ∧ DCYLc4s4s1 ∧ CGc1c2 ∧ CGc2c3 ∧ CGc3c4 ∧CVHh1c1c3 ∧ CVHh2c2c4 ∧ x=h1×h2 (x is the cube defined by s1s2s3s4) Therefore the definition of a cube is: D30. CUBEx =df ∃s1s2s3s4(DCUBExs1s2s3s4)

(x is a cube)

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We obtain the definition of a parallelepiped with square basis by erasing the condition ‘ETs1s2’ in the definition of DCUBE. Let us now define another solid shape, which makes use of the idea of ‘thick boundary’ developed in further detail in Section 5. We introduce first the notion of half-sized sphere (Fig.7): D31. HALFxy =df SPHy ∧ ∃zw(CGzw ∧ CGxz ∧ ETzw ∧ IDzwy) (sphere x is half-sized wrt y) D32. DCONExyz =df SPHy ∧ SPHz ∧ ¬Oyz ∧ ∃p(CGpy ∧ PPpz) ∧ ∃srv(CVHs(y+z) ∧ CVHr(s-z) ∧ ∀a(Pav ↔ ∀w(Pwa→ ∃u(HALFuy ∧ Ouw ∧ TPus)))) → (¬Prv ∧ x=r-v) (x is the cone defined by y and z) and then: D33. CONEx =df ∃yz(DCONExyz)

y

(x is a cone)

y w

z

ETzw IDzwy

s

r

x

z v

Fig.7 w and z are half-sized sphere wrt y

Fig.8 Construction of a cone

We can explain this definition as follows (see Fig.8): given two spheres y, z with z bigger than y, consider the convex hull s of y+z and then the convex hull r of (s-z). The region r is a cone with a rounded top. Now we subtract from r all the spheres such that their size if a half of that of y and they are tangential parts of s. The remainder is a geometrical cone. Note that the vertex of the cone x is exactly the centre of the sphere y.

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5. Dealing with granularities and boundaries 5.1 Granularity Being able to reason about geometrical features like faces, edges, slots and holes is of fundamental importance. All these issues involve the notion of boundary of a region. We introduce therefore boundaries in our framework, avoiding however to rely on their classical mathematical definition. Rather, we adopt a definition more akin to the commonsense intuition, where surfaces and edges are thought of as concrete entities, and granularity considerations are invoked.4 We can easily introduce a notion of granularity within our system by fixing a particular sphere g (a new constant in our language), and defining a grain as follows: D34. Gx =df CGxg

assuming SPHg

In the following definitions, granularity will be assumed as fixed, for the sake of simplicity. It would be easy however to parametrize them with respect to granularity, making it possible to reason with different granularities within the same theory. 5.2 Boundaries With the help of grains, we "approximate" now the mathematical notion of the boundary of a region by means of a suitably thin region overlapping the "real" boundary. We say that the boundary of y is the region corresponding to the mereological sum of all regions z such that each part w of them is overlapped by a “half grain” u straddling the mathematical boundary of y. Notice that such boundary is itself a region, and it is only defined for regions bigger than a grain. Notice also that the boundary is always as thick as a grain, except in correspondence of edges and corners whereit is "fatter"(see Fig.9).

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This does not mean that we cannot define characteristics of “perfect” mathematical regions (as the example of cylinders and cubes shows): for instance, we shall see how to define a region with a perfectly flat surface, whithout being able of speaking of the (mathematical) surface itself.

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D35. Bxy =df ∃u(Gu ∧ IPuy) ∧ ∀z(Pzx ↔ ∀w(Pwz → ∃u(HALFug ∧ Ouw ∧ POuy))) (x is the boundary of y)

y

ILINxy

Bxy

SURFxy

ENVExy

ELINxy

IDCxy

EDCxy

Fig.9 The region y occupied by a nail; its boundary; its surface; its envelope; its internal lining; its internal discontinuities; its external lining; its external discontinuities.

The definition below is crucial in what follows. D36. BRxy =df ∃zwvwbvb(Bzy ∧ Pxz ∧ Cx(y-z) ∧ TPwy ∧ ECvy ∧ Bw bw ∧ Bvbv ∧ x=w b×vb) ∧ ∀u∃v(SPHu ∧ Pux ∧ Puv ∧ Gv) → ∃w(Gw ∧ Puw ∧ Pwx) (x is a boundary region of y)

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The main intuition here is to define a piece of a boundary which preserves some of its fundamental characteristics. For this reason, we cannot consider whatever part of the boundary. From the above formulation, we can roughly say that a boundary region x of y is a piece of a boundary which is connected both with the "outside" and the “inside” (i.e. y without its boundary) of y, and is everywhere at least as thick as a grain. 5.3 Surfaces and envelopes Another intuitive notion is that of skin or thick surface, i.e. the external part of a region which is as thick as a grain (with the exception of the edges and corners). We say that a region x is the surface of y when y is bigger than a grain, x is part of y and, for any part of x , there exists a grain overlapping that part and properly overlapping y itself. D37. SURFxy =df ∃u(Gu ∧ IPuy) ∧ Pxy ∧ ∀z(Pzx ↔ ∀w(Pwz → ∃u(Gu ∧ Ouw ∧ POuy)))) (x is the surface of y) Analogous to the notion of surface is that of envelope of a region, denoting a “covering” of the region which is as thick as a grain. A region x is called the envelope of y when every part of it is overlapped by a grain that partially overlaps y: D38. ENVExy =df ∃u(Gu ∧ IPuy) ∧ ¬Oxy ∧ ∀z(Pzx ↔ ∀w(Pwz → ∃u(Gu ∧ Ouw ∧ POuy)))) (x is the envelope of y) Note that we constrain these definitions to regions bigger than a grain in agreement with the definition of boundary. 5.4 Approximation of polyhedra Once we have fixed a particular grain, it is possible to decide whether a given region is a approximation of a regular shape. Suppose for instance that we are reasoning about a particular region x. The actual shape of x can be very complicated (for instance, x may be the region occupied by a block of concrete). If our task is not specifically related to the precise shape of this block, we need a method to decide whether it can be roughly considered as a cube or not.

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Taking our grain g as a parameter, we can give the definition of 'ACUBE', approximate cube. D39. ACUBEx =df ∃yz(CUBEy ∧ SURFzy ∧ Pxy ∧ P(y-z)x)

(x is an approximate cube)

The region x is an approximate cube when its mathematical boundary is completely contained in the skin of a perfect cube (Fig.10). A similar definition can be suitably extended to the other shapes.

y

x g

z

Fig.10 x is an approximate cube

5.5 Discontinuities The use of the notions of boundaries and granularity to characterize relevant aspects of shape like discontinuity points (corners or edges) is shown in Fig.9. To capture such aspects, we introduce the notion of internal lining of a region, which is the sum of all its external grains, i.e. the grains which are tangential parts of that region. This turns out to be distinct from the above definition of surface only in the case where the region in question has edges or corners, since, in correspondence of these, there exists a region which is part of the surface but is not overlapped by any external grain. Now imagine to line internally the mathematical surface of a region with a lining whose thickness is exactly a grain: the “empty” regions remaining where the lining does not touch the surface mark discontinuity regions (depending of course on granularity). We report here the definitions of internal and external lining of a region, which we shall use to characterize its internal and external discontinuities:

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D40. ILINxy =df ∀z(Pzx ↔ ∀w(Pwz → ∃u(Gu ∧ Owu ∧ TPuy)))

(x is the internal lining of y)

D41. ELINxy =df ∀z(Pzx ↔ ∀w(Pwz → ∃u(Gu ∧ Owu ∧ ECuy))) (x is the external lining of y) Finally, we can state that the sum of all the internal (external) discontinuities is the difference between the surface (envelope) and the internal (external) lining: D42. IDCxy =df ∃zw(SURFzy ∧ ILINwy ∧ ¬z=w ∧ x=z-w) (x is the sum of the internal discontinuities of y) D43. EDCxy =df ∃zw(ENVEzy ∧ ELINwy ∧ ¬z=w ∧ x=z-w) (x is the sum of the external discontinuities of y) Notice that these sums are not boundary regions. However, we would like to refer to “solid” edges and corners as boundary regions. To this purpose, we introduce the following definitions: D44. IDBxy =def ∃wv(BRxy ∧ IDCwy ∧ Pwx) ∧ ¬∃u(BRuy ∧ Pwu ∧ PPux) (x is the set of the internal discontinuity boundary regions of y) D45. EDBxy =def ∃wv(BRxy ∧ EDCwy ∧ Pwx) ∧ ¬∃u(BRuy ∧ Pwu ∧ PPux) (x is the set of the external discontinuity boundary regions of y) The set of the internal (external) discontinuity boundary regions is the smallest boundary region containing the internal (external) discontinuities. 5.6 Properties of boundary regions We shall now define some commonsense properties of boundary regions such as planarity, flatness, and smoothness (Fig.11). We say that a boundary region is plane if, given any pairs of grains in it, every grain between them is part of the region. In this way we do impose that the region has to be completely straight. D46. PLANEx = df ∃y(BRxy ∧ ∀wvz(Gw ∧ Gv ∧ Gz ∧ PP(w+v)x ∧ BTWzwv) → Pzx) (x is a plane boundary region)

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Note that the following holds: PLANEx → CONVx Flat regions differ from plane regions for irregularities thinner than a grain. It suffices to substitute in the definition above the condition ‘Pzx’ with the less restricting ‘Ozx’, to obtain the desired definition: D47. FLATx = df ∃y(BRxy ∧ ∀wvz(Gw ∧ Gv ∧ Gz ∧ PP(w+v)x ∧ BTWzwv) → Ozx) (x is a flat boundary region) Finally, we can say when a region is smooth: D48. SMTHx = df ∃z(BRxz) ∧ ¬∃y(IDCyz ∨ EDCyz)

(x is a smooth boundary region)

PLANEx

FLATx ∧ SMTHx

FLATx ∧ ¬SMTHx ¬FLATx ∧ SMTHx ¬FLATx ∧ ¬SMTHx

Fig.11 Properties of boundary region

5.7 Concavities As a final example of application of our theory to the characterization of shape, let us show how we can classify different kinds of concavities, distinguishing among tunnels, containers, holes, internal cavities and so on. A full discussion of these concepts can be found in [Casati and Varzi 1994] and some related problems in qualitative physics are addressed in [Decuyper et al.]. We shall focus here on some shape aspects which have particular relevance in the domain of mechanical engineering, like slots, blind slots, steps, blind steps and so on (Fig.12).

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Let us make clear first some basic assumptions: a) We say that a region is a concavity with respect to another region, called “host”. We assume that the host is a simple region. This is an unworrying hypothesis in our theory because it is always possible to isolate a simple region from a more complex region; b) For most application purposes, the relevant concavities have plane surfaces. Therefore, we restrict ourselves to polyedrical hosts, excluding regions with curved surfaces. In fact, it is not clear what a “step” would mean when considering a host like a sphere or an irregular curved region. In any case, it would be possible to characterise concavities for some non-regular hosts by means of granularity considerations; c) For sake of simplicity, we limit ourself to convex concavities. However, it is possible to overcome this condition looking at solid angles.

Slot Step

Pocket Blind-hole

Blind-slot Inverted Dover-tail-slot Blind-step

Fig.12 Interesting shape aspects in mechanical engineering

We shall distinguish among different kinds of concavities by looking at the plane faces they share with their hosts. To this purpose, we need first the following definitions: D49. CBRxyz =df BRxy ∧ BRxz

(x is the common boundary of y and z)

D50. MCBRxyz =df CBRxyz ∧ ¬∃u(CBRuyz ∧ PPxu) (x is the maximal common boundary of y and z) D51. ΣCNVxy =df ¬CONVy ∧ ∃z(CVHzy ∧ x=z-y)

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(x is the sum of the concavities of y)

D52. MCCxy =df CONVx ∧ ∃z(ΣCNVzy ∧ MCPxz) (x is a maximally self-connected and convex concavity of y) We introduce the notion of a face, i.e., a maximally self-connected plane boundary region: D53. MPLANxy =def Pxy ∧ PLANEx ∧ ¬∃z(PLANEz ∧ Pzy ∧ PPxz) (x is a maximally self-connected plane of y) We now count the faces that concavities share with their hosts. For the sake of simplicity, we limit ourselves to the case of 2 or 3 faces. D54. 2FCxys1s2 =df ¬s1=s2 ∧ MCCxy ∧ MCBR(s1+s2)xy ∧ MPLANs1(s1+s2) ∧ MPLANs2(s1+s2)

(s1,s2 are the common faces of the concavity x and the host y)

D55. 3FCxys1s2s3 =df ¬s1=s2 ∧ ¬s1=s3 ∧ ¬s2=s3 ∧ MCCxy ∧ MCBR(s1+s2+s3)xy ∧ MPLANs1(s1+s2+s3) ∧ MPLANs2(s1+s2+s3) ∧ MPLANs3(s1+s2+s3) (s1,s2,s3 are the common faces of the concavity x and the host y) Distinguishing concavities by the number of their faces is not sufficient, however. We look therefore at the relation existing between these faces. To this purpose, we need some further definitions: A grain-segment (g-segment) is the convex hull of the sum of two different grains: D56. GSEGx =df ∃yz(Gy ∧ Gz ∧ SEGyz ∧ CHVx(y+z))

(x is a g-segment)

A maximal g-segment for a region is a segment that cannot be “lengthened” while remaining inside that region. D57. MSEGxy =df GSEGx ∧ Pxy ∧ ¬∃z(GSEGz ∧ Pzy ∧ PPxz)

(x is a maximal g-segment)

Two plane boundary regions and a g-segment form a solid angle when the segment is the only common segment for these regions, it does not disconnect any of them and it is maximal for both of them:

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D58. GSAxyz =df PLANEx ∧ PLANEy ∧ MSEGzx ∧ MSEGzy ∧ SR(x-z) ∧ SR(y-z) (x and y form a solid angle along z) Now we can define a step as a concavity having two plane pieces of boundary in common with its host: D59. STEPxy =df ∃s1s2l(2FCxys1s2 ∧ GSAs1s2l)

(x is a step in y)

It is easy to see that no other kind of concavity can arise from the combination of two plane boundary regions under our general assumptions. A slot is defined in an analogous way, as a convex concavity which has three faces in common with its host, and two of these do not overlap: D60. SLOTxy =df ∃s1s2s3l1l2(3FCxys1s2s3 ∧ GSAs1s2l1 ∧ GSAs2s3l2 ∧ ¬Os1s3)

(x is a slot in y)

A blind-step is a concavity which has three faces in common with its host; they meet on a grain, and overlap each other on a solid line: D61. BLIND-STEPxy =df ∃s1s2s3l1l2l3(3FCxys1s2s3 ∧ GSAs1s2l1 ∧ GSAs2s3l2 ∧ GSAs1s3l3 ∧ G(l1×l2×l3))

(x is a blind-step in y)

A straight tunnel with triangular section is a concavity with three faces which overlap each other on a solid segment, but they do not meet all together. D62. 3TUNNELxy =df ∃s1s2s3l1l2l3(3FCxys1s2s3 ∧ GSAs1s2l1 ∧ GSAs2s3l2 ∧ GSAs1s3l3 ∧ ¬Ol1l2 ∧ ¬Ol1l3 ∧ ¬Ol2l3 ∧ ¬∃z(Pzs1 ∧ Pzs2 ∧ Pzs3))

(x is a three-faced tunnel in y)

In the same way, as it should be clear now, we can go on making a distinction between blindslots, pockets, and the like.

Conclusions We have shown how to reconstruct common geometrical properties and relations through the notions of parthood, surface connection, and congruence, related to distinct levels of description (mereological, topological and morphological).

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We have pointed out how properties such as being coplanar, being perpendicular and being plane, which are related to points, lines, and surfaces in Euclidean geometry, can be exactly defined in the domain of three-dimensional regions. We have shown how to represent geometrical properties at different levels of granularity, making it possible define approximate shapes as in the case of the block of concrete. This approach can be easily generalised in order to deal with different granularities. Finally, we have applied our tools to describe some kinds of concavities which have a high relevance in the domain of mechanical engineering.

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