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Worboys and Duckham (2004)GIS: A Computing Perspective, Second Edition, CRC Press

Chapter 3

Fundamental spatialconcepts

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Worboys and Duckham (2004) GIS: A Computing Perspective, Second Edition, CRC Press

! Geometry: provides a formalrepresentation of the abstract propertiesand structures within a space

! Invariance: a group of transformations ofspace under which propositions remaintrue

! Distance- translations and rotations! Angle and parallelism- translations rotations,

and scalings

Geometry and invariance

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3.1

Euclidean space

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Euclidean Space

! Euclidean Space: coordinatized model of space! Transforms spatial properties into properties of tuples

of real numbers

! Coordinate frame consists of a fixed, distinguishedpoint (origin) and a pair of orthogonal lines (axes),intersecting in the origin

! Point objects! Line objects! Polygonal objects

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! Scalar: Addition, subtraction,and multiplication, e.g.,(x1, y1) (x2, y2) =(x1 x2, y1 y2)! Norm:! Distance: ja bj = jja-bjj! Angle between vectors:

! A point in the Cartesian plane R2 is associated with aunique pair of real numbera = (x,y) measuring distancefrom the origin in thexand ydirections. It is sometimesconvenient to think of the point a as a vector.

Points

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Lines

! The line incident with a and b is defined as the point set{a + (1 )b | 2 R}

! The line segment between a and b is defined as the pointset {a + (1 )b | 2 [0, 1]}

! The half line radiating from b and passing through a isdefined as the point set {a + (1 )b | 0}

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Polygonal objects

! Apolyline in R2 is a finite set of line segments (called edges)such that each edge end-point is shared by exactly two edges,except possibly for two points, called the extremes of thepolyline.

! If no two edges intersect at any place other than possibly at theirend-points, the polyline is simple.

! A polyline is closedif it has no extreme points.! A (simple) polygon in R2 is the area enclosed by a simple closed

polyline. This polyline forms the boundaryof the polygon. Eachend-point of an edge of the polyline is called a vertexof the

polygon.! A convexpolygon has every point intervisible! A star-shapedorsemi-convexpolygon has at least one point

that is intervisible

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Polygon objects

monotone polyline

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Transformations

! Transformations preserve particular properties ofembedded objects

! Euclidean Transformation! Similarity transformations! Affine transformations! Projective transformations! Topological transformation

! Some formulas can be provided! Translation: through real constants a and b

(x,y) ! (x+a,y+b)! Rotation: through angle about origin (x,y) ! (xcos - ysin,xsin + ycos)! Reflection: in line through origin at angle tox-axis (x,y)! (xcos2 + ysin2,xsin2- ycos2)

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3.2

Set-based geometry ofspace

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Sets

! The set based model involves:! The constituent objects to be modeled, called

elements or members

! Collection of elements, called sets! The relationship between the elements and the sets to

which they belong, termed membership

We write s 2 Sto indicate that an element s is a memberof the set S

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Sets

! A large number of modeling tools areconstructed:

! Equality! Subset: S2 T! Power set: the set of all subsets of a set, P(S)! Empty set; ;! Cardinality: the number of members in a set #S! Intersection: S T! Union: S[ T! Difference: S\T! Complement: elements that are not in the set, S

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Distinguished sets

Name Symbol Description

Booleans B Two-valued set of true/false, 1/0, oron/off

Integers Z Positive and negative numbers,including zero

Reals R Measurements on the number line

Real Plane R2 Ordered pairs of reals

Closed

interval

[a,b] All reals between a and b 9 including

a and b)Openinterval

]a,b[ All reals between a and b (excludinga and b)

Semi-openinterval

[a,b[ All reals between a and b (including aand excluding b)

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Relations

! Product: returns the set of ordered pairs, whose firstelement is a member of the first set and second elementis a member of the second set

! Binary relation: a subset of the product of two sets,whose ordered pairs show the relationships between

members of the first set and members of the second set! Reflexive relations: where every element of the set isrelated to itself

! Symmetric relations: where ifxis related to ythen yisrelated tox

! Transitive relations: where ifxis related to yand yisrelated to zthenxis related to z

! Equivalence relation: a binary relation that is reflexive,symmetric and transitive

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Functions

! Function: a type of relation which has theproperty that each member of the first setrelates to exactly one member of the second set

! f: S! T

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Functions

! Injection: any two different points in the domainare transformed to two distinct points in thecodomain

! Image: the set of all possible outputs! Surjection: when the image equals the

codomain

! Bijection: a function that is both a surjectionand an injection

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Inverse functions

! Injective function have inverse functions! Projection

! Given a point in the plane that is part of the image ofthe transformation, it is possible to reconstruct thepoint on the spheroid from which it came

! Example: A new function whose domain is the image of the UTM

maps the image back to the spheroid

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Convexity

! A set is convexif every point is visible fromevery other point within the set

! Let Sbe a set of points in the Euclidean plane! Visible:! Pointxin Sis visible from point yin Sif either

x=yor; it is possible to draw a straight-line segment betweenx

and ythat consists entirely of points ofS

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Convexity

! Observation point:! The pointxin Sis an observation point forSif every

point ofSis visible fromx

! Semi-convex:! The set Sis semi-convex (star-shaped ifSis a

polygonal region) if there is some observation point forS

! Convex:! The set Sis convex if every point ofSis an

observation point forS

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Convexity

Visibility between points x, y,and z

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3.3

Topology of Space

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Topology

! Topology: study of form; concerns properties that areinvariant under topological transformations

! Intuitively, topological transformations are rubber sheettransformations

Topological A point is at an end-point of an arcA point is on the boundary of an area

A point is in the interior/exterior of an area

An arc is simple

An area is open/closed/simple

An area is connected

Non-topological Distance between two points

Bearing of one point from another point

Length of an arc

Perimeter of an area

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Point set topology

! One way of defining a topological space is withthe idea of a neighborhood

! Let Sbe a given set of points. A topologicalspace is a collection of subsets ofS, calledneighborhoods, that satisfy the following twoconditions:

! T1 Every point in Sis in some neighborhood.! T2The intersection of any two neighborhoods of any

pointxin Scontains a neighborhood ofx! Points in the Cartesian plane and open disks(circles surrounding the points) form a topology

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Point set topology

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Usual topology

! Usual topology: naturally comes to mind withEuclidean plane and corresponds to the rubber-sheet topology

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Travel time topology

! Let Sbe the set of points in a region of the plane! Suppose:

! the region contains a transportation network and! we know the average travel time between any twopoints in the region using the network, following the

optimal route

! Assume travel time relation is symmetric! For each time tgreater than zero, define a t-zone around pointxto be the set of all points

reachable fromxin less than time t

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Travel time topology

! Let the neighborhoodsbe all t-zones around a

point

! T1 and T2 are satisfied

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Nearness

! Let Sbe a topological space! Then Shas a set of neighborhoods associated with it. Let

Cbe a subset of points in Sand can individual point in S

! Define cto be nearCif every neighborhood ofccontainssome point ofC

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Open and closed sets

! Let Sbe a topological space andXbe a subset of pointsofS.

! ThenXis open if every point ofXcan be surrounded by aneighborhood that is entirely withinX

A set that does not contain its boundary!ThenXis closed if it contains all its near points

A set that does contain its boundary

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Closure, boundary, interior

! Let Sbe a topological space andXbe a subset of points ofS

! The boundary ofXconsists of all points which are near tobothXandX0. The boundary of setXis denoted X

! The closure ofXis theunion ofXwith the set of allits near points

! denotedX! The interiorof X consists

of all points which belongtoXand are not near

points ofX0

! denotedX

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Topology and embedding space

2-space 1-space

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Topological invariants

! Properties that arepreserved by

topologicaltransformations arecalled topologicalinvariants

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Connectedness

! Let Sbe a topological space andXbe asubset of points ofS

! ThenXis connectedif whenever it ispartitioned into two non-empty disjointsubsets,A and B,

! eitherA contains a point nearB, orB contains apoint nearA, or both

! A set in a topological space ispath-connectedif any two points in the set can bejoined by a path that lies wholly in the set

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Connectedness

disconnected

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Combinatorial topology

! Remove a single face from a polyhedron and apply a 3-space topological transformation to flatten the shape ontothe plane

! Modify Eulers formula for the sphere to derive Eulersformula for the plane

!Given a cellular arrangement in the plane, with fcells, eedges, and vvertices, f e + v= 1

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Simplexes and complexes

! 0-simplex: a set consisting of a single point inthe Euclidean plane

! 1-simplex: a closed finite straight-line segment! 2-simplex: a set consisting of all the points onthe boundary and in the interior of a triangle

whose vertices are not collinear

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! Simplicial complex: simple triangular networkstructures in the Euclidean plane (two-dimensional case)

! A face of a simplex Sis a simplex whosevertices form a proper subset of the vertices ofS

! A simplicial complex Cis a finite set ofsimplexes satisfying the properties:

! 11 A face of a simplex in Cis also in C! 22 The intersection of two simplexes in Cis either

empty or is also in C

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Problem with combinatorial topology

! The more detailed connectivity of the object is notexplicitly given. Thus there is no explicit representation ofweak, strong, or simple connectedness

! The representation is not faithful, in the sense that twodifferent topological configurations may have the same

representation

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Combinatorial map

! Assume that the boundary of a cellulararrangement is decomposed into simple arcsand nodes that form a network

! Give a direction to each arc so that travelingalong the arc the object bounded by the arc is tothe right of the directed arc

! Provide a rule for the order of following the arcs:! After following an arc into a node, move

counterclockwise around the node and leave by thefirst unvisited outward arc encountered

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Combinatorial map

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Abstract graphs

! A graphG is defined as a finite non-empty setofnodes together with a set of unordered pairsof distinct nodes (called edges)

! Highly abstract! Represents connectedness between elements of the

space

! Directed graph! Labeled graph

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Abstract graphs

Connected graphEdgesPathCycle

NodesDegreeIsomorphicDirected/ non-directed

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Tree

Connected graphAcyclicNon-isomorphic

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Rooted tree

RootImmediate descendantsLeaf

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Planar graphs

! Planar graph: a graph that can be embedded inthe plane in a way that preserves its structure

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Planar graphs

! There are many topologically inequivalent planarembeddings of a planar graph in the plane

! Eulers formula: fe + v=1

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Dual G*

! Obtained by associating a node in G* with eachface in G

! Two nodes in G* are connected by an edge ifand only if their corresponding faces in G are

adjacent

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3.5

Metric spaces

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Definition

! A point-set Sis a metric space if there is adistance function d, which takes ordered pairs(s,t) of elements ofSand returns a distance thatsatisfies the following conditions

! 11 For each pairs, tin S, d(s,t) >0 ifs and tare distinctpoints and d(s,t) =0 ifs and tare identical! 22 For each pairs,tin S, the distance from s to tis equal

to the distance from tto s, d(s,t) = d(t,s)

! 3 For each tripe s,t,uin S, the sum of the distances froms to tand from tto uis always at least as large as thedistance from s to u

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Distances defined on the globe

Metric space Metric space

Metric space

Quasimetric

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3.6

Fractal geometry

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Fractal geometry

! Scale dependence: appearance andcharacteristics of many geographic and naturalphenomena depend on the scale at which theyare observed

! Straight lines and smooth curves of Euclideangeometry are not well suited to modeling self-similarityand scale dependence

! Fractals: self-similar across all scales! Defined recursively, rather than by describing their

shape directly

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Koch snowflake

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Fractal dimensions

! Self-affine fractals: constructed using affinetransformations within the generator, sorotations, reflections, and shears can be used inaddition to scaling

! Fractal dimension: an indicator of shapecomplexity;

! Lies somewhere between the Euclidean dimensions ofthe shape and its embedding space

!A shape with a high fractal dimension is complexenough to nearly fill its embedding space (spacefilling)

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