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The Mathematics of GIS

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The Mathematics of GIS The Mathematics of GIS Wolfgang Kainz Wolfgang Kainz Department of Geography and Regional Research University of Vienna Universitätsstraße 7, A-1010 Vienna, Austria E-Mail: [email protected] Version 2.1 (August 2010) Contents 0. Preface ............................................................................ XI 1. The Structure of Mathematics ........................................ 1 1.1 Brief History of Mathematics ...................................................................... 2 1.2 Sub-disciplines of Mathematics................................................................... 2 2. Propositional Logic .......................................................... 5 2.1 Assertion and Proposition ............................................................................ 6 2.2 Logical Operators ......................................................................................... 7 2.3 Types of Propositional Forms ...................................................................... 9 2.4 Applications in GIS .................................................................................... 11 2.5 Exercises ..................................................................................................... 11 3. Predicate Logic .............................................................. 13 3.1 Predicates .................................................................................................... 14 3.2 Quantifiers .................................................................................................. 15 3.3 Quantifiers and Logical Operators ............................................................ 16 3.4 Compact Notation ...................................................................................... 17 3.5 Applications in GIS .................................................................................... 18 3.6 Exercises ..................................................................................................... 18 4. Logical Inference ........................................................... 19 4.1 Logical Arguments ..................................................................................... 20 4.2 Proving Arguments Valid in Propositional Logic .................................... 20 4.2.1 Proving Arguments Valid with Truth Tables ................................................................ 21 4.2.2 Proving Arguments Valid with Rules of Inference ....................................................... 21 4.3 Proving Arguments Valid in Predicate Logic ........................................... 22 4.4 Applications in GIS .................................................................................... 23 4.5 Exercises ..................................................................................................... 23 5. Set Theory ...................................................................... 25 5.1 Sets and Elements ...................................................................................... 26 5.2 Relations between Sets .............................................................................. 27 5.3 Operations on Sets ...................................................................................... 27 5.4 Applications in GIS .................................................................................... 29 5.5 Exercises ..................................................................................................... 30 6. Relations and Functions ............................................... 31 6.1 Cartesian Product ....................................................................................... 32 6.2 Binary Relations ......................................................................................... 32 6.2.1 Relations and Predicates ................................................................................................ 33 6.2.2 Graphic Representation of Binary Relations ................................................................ 33 I II 6.2.3 Special Properties of Relations ...................................................................................... 33 6.2.3.1 Equivalence Relation .............................................................................................................. 34 6.2.3.2 Order Relation ......................................................................................................................... 35 6.2.4 Composition of Relations ............................................................................................... 35 6.3 Functions .................................................................................................... 36 6.3.1 Composition of Functions .............................................................................................. 37 6.3.2 Classes of Functions ....................................................................................................... 37 6.4 Applications in GIS .................................................................................... 40 6.5 Exercises ..................................................................................................... 42 7. Coordinate Systems and Transformations .................. 43 7.1 Coordinate Systems .................................................................................... 44 7.1.1 Cartesian Coordinate Systems ....................................................................................... 44 7.1.2 Polar Coordinate Systems .............................................................................................. 45 7.1.3 Transformations between Cartesian and Polar Coordinate Systems ........................ 45 7.1.4 Geographic Coordinate System ..................................................................................... 47 7.2 Vectors and Matrices ................................................................................. 47 7.2.1 Vectors ............................................................................................................................. 47 7.2.2 Matrices ........................................................................................................................... 51 7.3 Transformations ......................................................................................... 52 7.3.1 Geometric Transformations ........................................................................................... 52 7.3.1.1 Translation ............................................................................................................................... 52 7.3.1.2 Rotation .................................................................................................................................... 53 7.3.1.3 Scaling ...................................................................................................................................... 53 7.3.2 Combination of Transformations .................................................................................. 54 7.3.3 Homogeneous Coordinates ............................................................................................ 55 7.3.4 Transformation between Coordinate Systems .............................................................. 56 7.4 Applications in GIS .................................................................................... 57 7.5 Exercises ..................................................................................................... 58 8. Algebraic Structures ..................................................... 59 8.1 Components of an Algebra ........................................................................ 60 8.1.1 Signature and Variety ..................................................................................................... 60 8.1.2 Identity and Zero Elements ............................................................................................ 61 8.2 Varieties of Algebras .................................................................................. 61 8.2.1 Group ............................................................................................................................... 62 8.2.2 Field ................................................................................................................................. 62 8.2.3 Boolean Algebra ............................................................................................................. 63 8.2.4 Vector Space .................................................................................................................... 63 8.3 Homomorphism ......................................................................................... 64 8.4 Applications in GIS .................................................................................... 65 8.5 Exercises ..................................................................................................... 66 9. Topology ......................................................................... 67 9.1 Topological Spaces .................................................................................... 68 9.1.1 Metric Spaces and Neighborhoods ............................................................................... 68 9.1.2 Topology and Open Sets ................................................................................................. 69 9.1.3 Continuous Functions and Homeomorphisms ............................................................. 71 9.1.4
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