Elementary Surveying: An Introduction to Geomatics

1 Introduction¿ 1.1 Definition of Surveying¿ 1.2 Geomatics¿ 1.3 History of Surveying¿ 1.4 Geodetic and Plane Surveys¿ 1.5 Importance of Surveying¿ 1.6 Specialized Types of Surveys¿ 1.7 Surveying Safety¿ 1.8 Land and Geographic Information Systems¿ 1.9 Federal Surveying and Mapping Agencies¿ 1.10 The Surveying Profession¿ 1.11 Professional Surveying Organizations¿ 1.12 Surveying on the Internet¿ 1.13 Future Challenges in SurveyingProblemsBibliography2 Units, Significant Figures, and Field NotesPart I ¿Units and Significant Figures¿ 2.1 Introduction¿ 2.2 Units of Measurement 25t¿ 2.3 International System of Units (SI)¿ 2.4 Significant Figures¿ 2.5 Rounding Off NumbersPart II ¿Field Notes¿ 2.6 Field Notes¿ 2.7 General Requirements of Handwritten Field Notes¿ 2.8 Types of Field Books¿ 2.9 Kinds of Notes¿ 2.10 Arrangements of Notes¿ 2.11 Suggestions for Recording Notes¿ 2.12 Introduction to Data Collectors¿ 2.13 Transfer of Files from Data Collectors¿ 2.14 Digital Data File Management¿ 2.15 Advantages and Disadvantages of Data CollectorsProblemsBibliography3 Theory of Errors in Observations¿ 3.1 Introduction¿ 3.2 Direct and Indirect Observations¿ 3.3 Errors in Measurements¿ 3.4 Mistakes¿ 3.5 Sources of Errors in Making Observations¿ 3.6 Types of Errors¿ 3.7 Precision and Accuracy¿ 3.8 Eliminating Mistakes and Systematic Errors¿ 3.9 Probability¿ 3.10 Most Probable Value¿ 3.11 Residuals¿ 3.12 Occurrence of Random Errors¿ 3.13 General Laws of Probability¿ 3.14 Measures of Precision¿ 3.15 Interpretation of Standard Deviation¿ 3.16 The 50, 90, and 95 Percent Errors¿ 3.17 Error Propagation3.17.1 Error of a Sum3.17.2 Error of a Series3.17.3 Error in a Product3.17.4 Error in the Mean¿ 3.18 Applications¿ 3.19 Conditional Adjustment of Observations¿ 3.20 Weights of Observations¿ 3.21 Least-Squares AdjustmentProblemsBibliography4 Leveling–Theory, Methods, and EquipmentPart I ¿Leveling–Theory and Methods¿ 4.1 Introduction¿ 4.2 Definitions¿ 4.3 North American Vertical Datum¿ 4.4 Curvature and Refraction¿ 4.5 Methods for Determining Differences in Elevation4.5.1 Measuring Vertical Distances by Taping or Electronic Methods4.5.2 Differential Leveling4.5.3 Barometric Leveling4.5.4 Trigonometric LevelingPart II ¿Equipment for Differential Leveling¿ 4.6 Categories of Levels¿ 4.7 Telescopes¿ 4.8 Level Vials¿ 4.9 Tilting Levels¿ 4.10 Automatic Levels¿ 4.11 Digital Levels¿ 4.12 Tripods¿ 4.13 Hand Level¿ 4.14 Level Rods¿ 4.15 Testing and Adjusting Levels4.15.1 Requirements for Testing and Adjusting Instruments4.15.2 Adjusting for Parallax4.15.3 Testing and Adjusting Level Vials4.15.4 Preliminary Adjustment of the Horizontal Cross Hair4.15.5 Testing and Adjusting the Line of SightProblemsBibliography5 Leveling—Field Procedures and Computations¿ 5.1 Introduction¿ 5.2 Carrying and Setting Up a Level¿ 5.3 Duties of a Rodperson¿ 5.4 Differential Leveling¿ 5.5 Precision¿ 5.6 Adjustments of Simple Level Circuits¿ 5.7 Reciprocal Leveling¿ 5.8 Three-Wire Leveling¿ 5.9 Profile Leveling5.9.1 Staking and Stationing the Reference Line5.9.2 Field Procedures for Profile Leveling5.9.3 Drawing and Using the Profile¿ 5.10 Grid, Cross-Section, or Borrow-Pit Leveling¿ 5.11 Use of the Hand Level¿ 5.12 Sources of Error in Leveling5.12.1 Instrumental Errors5.12.2 Natural Errors5.12.3 Personal Errors¿ 5.13 Mistakes¿ 5.14 Reducing Errors and Eliminating MistakesProblemsBibliography6 Distance MeasurementPart I ¿Methods for Measuring Distances¿ 6.1 Introduction¿ 6.2 Summary of Methods for Making Linear Measurements¿ 6.3 Pacing¿ 6.4 Odometer Readings¿ 6.5 Optical Rangefinders¿ 6.6 Tacheometry¿ 6.7 Subtense BarPart II ¿Distance Measurements by Taping¿ 6.8 Introduction to Taping¿ 6.9 Taping Equipment and Accessories¿ 6.10 Care of Taping Equipment¿ 6.11 Taping on Level Ground6.11.1 Lining In6.11.2 Applying Tension6.11.3 Plumbing6.11.4 Marking Tape Lengths6.11.5 Reading the Tape6.11.6 Recording the Distance¿ 6.12 Horizontal Measurements on Sloping Ground¿ 6.13 Slope Measurements¿ 6.14 Sources of Error in Taping6.14.1 Incorrect Length of Tape6.14.2 Temperature Other Than Standard6.14.3 Inconsistent Pull6.14.4 Sag6.14.5 Tape Not Horizontal and Tape Off-Line6.14.6 Improper Plumbing6.14.7 Faulty Marking6.14.8 Incorrect Reading or Interpolation6.14.9 Summary of Effects of Taping Errors¿ 6.15 Tape Problems¿ 6.16 Combined Corrections in a Taping ProblemPart II ¿Electronic Distance Measurement¿ 6.17 Introduction¿ 6.18 Propagation of Electromagnetic Energy¿ 6.19 Principles of Electronic Distance Measurement¿ 6.20 Electro-Optical Instruments¿ 6.21 Total Station Instruments¿ 6.22 EDM Instruments Without Reflectors¿ 6.23 Computing Horizontal Lengths From Slope Distances6.23.1 Reduction of Short Lines by Elevation Differences6.23.2 Reduction of Short Lines by Zenith or Altitude Angle¿ 6.24 Errors in Electronic Distance Measurement6.24.1 Personal Errors6.24.2 Instrumental Errors6.24.3 Natural ErrorsProblemsBibliography7 Angles, Azimuths, and Bearings¿ 7.1 Introduction¿ 7.2 Units of Angle Measurement¿ 7.3 Kinds of Horizontal Angles¿ 7.4 Direction of a Line¿ 7.5 Azimuths¿ 7.6 Bearings¿ 7.7 Comparison of Azimuths and Bearings¿ 7.8 Computing Azimuths¿ 7.9 Computing Bearings¿ 7.10 The Compass and The Earth’s Magnetic Field¿ 7.11 Magnetic Declination¿ 7.12 Variations in Magnetic Declination¿ 7.13 Software for Determining Magnetic Declination¿ 7.14 Local Attraction¿ 7.15 Typical Magnetic Declination Problems¿ 7.16 MistakesProblemsBibliography8 Total Station Instruments: Angle MeasurementsPart I ¿Total Station Instruments¿ 8.1 Introduction¿ 8.2 Characteristics of Total Station Instruments¿ 8.3 Functions Performed by Total Station Instruments¿ 8.4 Parts of a Total Station Instrument¿ 8.5 Handling and Setting Up a Total Station Instrument¿ 8.6 Servo-Driven and Remotely Operated Total Station InstrumentsPart II ¿Angle Measurements¿ 8.7 Relationship of Angles and Distances¿ 8.8 Observing Horizontal Angles with Total Station Instruments¿ 8.9 Observing Horizontal Angles by the Direction Method¿ 8.10 Closing the Horizon¿ 8.11 Observing Deflection Angles¿ 8.12 Observing Azimuths¿ 8.13 Observing Vertical Angles¿ 8.14 Sights and Marks¿ 8.15 Prolonging a Straight Line¿ 8.16 Balancing-In¿ 8.17 Random Traverse¿ 8.18 Total Stations for Determining Elevation Differences¿ 8.19 Adjustment of Total Station Instruments and Their Accessories8.19.1 Adjustment of Plate-Level Vials8.19.2 Adjustment of Tripods8.19.3 Adjustment of Tribrachs8.19.4 Adjustment of Optical Plummets8.19.5 Adjustment of Circular Level Bubbles¿ 8.20 Sources of Error in Total Station Work8.20.1 Instrumental Errors8.20.2 Natural Errors8.20.3 Personal Errors¿ 8.21 Propagation of Random Errors in Angle Observations¿ 8.22 MistakesProblemsBibliography9 Traversing¿ 9.1 Introduction¿ 9.2 Observation of Traverse Angles or Directions9.2.1 Traversing by Interior Angles9.2.2 Traversing by Angles to the Right9.2.3 Traversing by Deflection Angles9.2.4 Traversing by Azimuths¿ 9.3 Observation of Traverse Lengths¿ 9.4 Selection of Traverse Stations¿ 9.5 Referencing Traverse Stations¿ 9.6 Traverse Field Notes¿ 9.7 Angle Misclosure¿ 9.8 Traversing with Total Station Instruments¿ 9.9 Radial Traversing¿ 9.10 Sources of Error in Traversing¿ 9.11 Mistakes in TraversingProblems10 Traverse Computations¿ 10.1 Introduction¿ 10.2 Balancing Angles¿ 10.3 Computation of Preliminary Azimuths or Bearings¿ 10.4 Departures and Latitudes¿ 10.5 Departure and Latitude Closure Conditions¿ 10.6 Traverse Linear Misclosure and Relative Precision¿ 10.7 Traverse Adjustment10.7.1 Compass (Bowditch) Rule10.7.2 Least-Squares Method¿ 10.8 Rectangular Coordinates¿ 10.9 Alternative Methods for Making Traverse Computations10.9.1 Balancing Angles by Adjusting Azimuths or Bearings10.9.2 Balancing Departures and Latitudes by Adjusting Coordinates¿ 10.10 Lengths and Directions of Lines from Departures and Latitudes or Coordinates¿ 10.11 Computing Final Adjusted Traverse Lengths and Directions¿ 10.12 Coordinate Computations in Boundary Surveys¿ 10.13 Use of Open Traverses¿ 10.14 State Plane Coordinate Systems¿ 10.15 Traverse Computations Using Computers¿ 10.16 Locating Blunders in Traverse Measurements¿ 10.17 Mistakes in Traverse ComputationsProblemsBibliography11 Coordinate Geometry in Surveying Calculations¿ 11.1 Introduction¿ 11.2 Coordinate Forms of Equations for Lines and Circles¿ 11.3 Perpendicular Distance from a Point to a Line¿ 11.4 Intersection of Two Lines, Both Having Known Directions¿ 11.5 Intersection of a Line with a Circle¿ 11.6 Intersection of Two Circles¿ 11.7 Three-Point Resection¿ 11.8 Two-Dimensional Conformal Coordinate Transformation¿ 11.9 Inaccessible Point Problem¿ 11.10 Three-Dimensional Two-Point Resection¿ 11.11 ConclusionsProblemsBibliography12 Area¿ 12.1 Introduction¿ 12.2 Methods of Measuring Area¿ 12.3 Area by Division into Simple Figures¿ 12.4 Area by Offsets from Straight Lines12.4.1 Regularly Spaced Offsets12.4.2 Irregularly Spaced Offsets¿ 12.5 Area by Coordinates¿ 12.6 Area by Double Meridian Distance Method¿ 12.7 Area of Parcels with Circular Boundaries¿ 12.8 Partitioning of Lands12.8.1 Trial and Error Method12.8.2 Use of Simple Geometric Figures12.8.3 Coordinate Method¿ 12.9 Area by Measurements from Maps12.9.1 Area by Counting Coordinate Squares12.9.2 Area by Scaled Lengths12.9.3 Area by Digitizing Coordinates12.9.4 Area by Planimeter¿ 12.10 Sources of Error in Determining Areas¿ 12.11 Mistakes in Determining AreasProblemsBibliography13 The Global Positioning System–Introduction and Principles of Operation¿ 13.1 Introduction¿ 13.2 Overview of GPS¿ 13.3 The GPS Signal¿ 13.4 Reference Coordinate Systems for GPS13.4.1 The Satellite Reference Coordinate System13.4.2 The Geocentric Coordinate System13.4.3 The Geodetic Coordinate System¿ 13.5 Fundamentals of GPS Positioning13.5.1 Code Ranging13.5.2 Carrier Phase-Shift Measurements¿ 13.6 Errors in GPS Observations13.6.1 Clock Bias13.6.2 Refraction13.6.3 Other Error Sources13.6.4 Geometry of Observed Satellites13.6.5 Selective Availability¿ 13.7 Differential GPS¿ 13.8 Kinematic GPS Methods¿ 13.9 Relative Positioning13.9.1 Single Differencing13.9.2 Double Differencing13.9.3 Triple Differencing¿ 13.10 Other Satellite Navigation Systems13.10.1 The GLONASS Constellation13.10.2 Galileo System13.10.3 Compass¿ 13.11 The FutureProblemsBibliography14 The Global Positioning System–Static surveys¿ 14.1 Introduction¿ 14.2 Field Procedures in GPS Surveys14.2.1 Static Relative Positioning14.2.2 Rapid Static Relative Positioning14.2.3 Pseudokinematic Surveys14.2.4 Kinematic Surveys¿ 14.3 Planning GPS Surveys14.3.1 Preliminary Considerations14.3.2 Selecting the Appropriate Survey Method14.3.3 Field Reconnaissance14.3.4 Developing an Observation Scheme14.3.5 Availability of Reference Stations¿ 14.4 Performing Static GPS Surveys¿ 14.5 Data Processing and Analysis14.5.1 Specifications for GPS Surveys14.5.2 Analysis of Fixed Baseline Measurements14.5.3 Analysis of Repeat Baseline Measurements14.5.4 Analysis of Loop Closures14.5.5 Baseline Network Adjustment14.5.6 The Survey Report¿ 14.6 Sources of Errors in GPS Work14.6.1 Instrumental Errors14.6.2 Natural Errors14.6.3 Personal Errors¿ 14.7 Mistakes in GPS Work¿ 14.8 Future Outlook for GPSProblemsBibliography15 The Global Positioning System–Kinematic GPS¿ 15.1 Introduction¿ 15.2 Initialization¿ 15.3 Equipment Used in Kinematic Surveys¿ 15.4 Methods Used in Kinematic Surveys¿ 15.5 Performing Post-Processed Kinematic Surveys¿ 15.6 Communication in Real-Time Kinematic Surveys¿ 15.7 Real-Time Networks¿ 15.8 Performing Real-Time Kinematic Surveys¿ 15.9 Machine Control¿ 15.10 Errors in Kinematic Surveys¿ 15.11 Mistakes in Kinematic SurveysProblemsBibliography16 Adjustments by Least Squares¿ 16.1 Introduction¿ 16.2 Fundamental Condition of Least Squares¿ 16.3 Least-Squares Adjustment by the Observation Equation Method¿ 16.4 Matrix Methods in Least-Squares Adjustment¿ 16.5 Matrix Equations for Precisions of Adjusted Quantities¿ 16.6 Least-Squares Adjustment of Leveling Circuits¿ 16.7 Propagation of Errors¿ 16.8 Least-Squares Adjustment of GPS Baseline Vectors¿ 16.9 Least-Squares Adjustment of Traditional Horizontal Plane Surveys16.9.1 Linearizing Nonlinear Equations16.9.2 The Distance Observation Equation16.9.3 The Azimuth Observation Equation16.9.4 The Angle Observation Equation16.9.5 A Traverse Example Using WOLFPACK¿ 16.10 Error Ellipses¿ 16.11 Adjustment Procedures¿ 16.12 Other Measures of Precision for Horizontal Stations¿ 16.13 ConclusionsProblemsBibliography17 Mapping Surveys¿ 17.1 Introduction¿ 17.2 Basic Methods for Performing Mapping Surveys¿ 17.3 Map Scale¿ 17.4 Control for Mapping Surveys¿ 17.5 Contours¿ 17.6 Characteristics of Contours¿ 17.7 Direct and Indirect Methods of Locating Contours17.7.1 Direct Method17.7.2 Indirect Method¿ 17.8 Digital Elevation Models and Automated Contouring Systems¿ 17.9 Basic Field Methods for Locating Topographic Details17.9.1 Radiation by Total Station17.9.2 Radiation by Stadia17.9.3 Coordinate Squares or “Grid” Method17.9.4 Offsets from a Reference Line17.9.5 Topographic Detailing with GPS17.9.6 Laser-Scanning¿ 17.10 Three-Dimensional Conformal Coordinate Transformation¿ 17.11 Selection of Field Method¿ 17.12 Working with Data Collectors and Field-to-Finish Software¿ 17.13 Hydrographic Surveys17.13.1 Equipment for Making Soundings17.13.2 Locating Soundings17.13.3 Hydrographic Mapping¿ 17.14 Sources of Error in Mapping Surveys¿ 17.15 Mistakes in Mapping SurveysProblemsBibliography18 Mapping¿ 18.1 Introduction¿ 18.2 Availability of Maps and Related Information¿ 18.3 National Mapping Program¿ 18.4 Accuracy Standards for Mapping¿ 18.5 Manual and Computer-Aided Drafting Procedures¿ 18.6 Map Design¿ 18.7 Map Layout¿ 18.8 Basic Map Plotting Procedures18.8.1 Manually Plotting by Coordinates18.8.2 Plotting Using CADD¿ 18.9 Contour Interval¿ 18.10 Plotting Contours¿ 18.11 Lettering¿ 18.12 Cartographic Map Elements¿ 18.13 Drafting Materials¿ 18.14 Automated Mapping and Computer-Aided Drafting Systems¿ 18.15 Impacts of Modern Land and Geographic Information Systems on Mapping¿ 18.16 Sources of Error in Mapping¿ 18.17 Mistakes in MappingProblemsBibliography19 Control Surveys and Geodetic Reductions¿ 19.1 Introduction¿ 19.2 The Ellipsoid and Geoid¿ 19.3 The Conventional Terrestrial Pole¿ 19.4 Geodetic Position and Ellipsoidal Radii of Curvature¿ 19.5 Geoid Undulation and Deflection of The Vertical¿ 19.6 U.S. Reference Frames19.6.1 North American Horizontal Datum of 1927 (NAD27)19.6.2 North American Horizontal Datum of 1983 (NAD83)19.6.3 Later Versions of NAD8319.6.4 National Geodetic Vertical Datum of 1929 (NGVD29)19.6.5 North American Vertical Datum of 1988 (NAVD88)19.6.6 Transforming Coordinates between Reference Frames¿ 19.7 Accuracy Standards and Specifications For Control Surveys¿ 19.8 The National Spatial Reference System¿ 19.9 Hierarchy of the National Horizontal Control Network¿ 19.10 Hierarchy of the National Vertical Control Network¿ 19.11 Control Point Descriptions¿ 19.12 Field Procedures For Traditional Horizontal Control Surveys19.12.1 Triangulation19.12.2 Precise Traverse19.12.3 Trilateration19.12.4 Combined Networks¿ 19.13 Field Procedures for Vertical Control Surveys¿ 19.14 Reduction of Field Observations to Their Geodetic Values19.14.1 Reduction of Distance Observations Using Elevations19.14.2 Reduction of Distance Observations Using Vertical Angles19.14.3 Reduction of Directions and Angles19.14.4 Leveling and Orthometric Heights¿ 19.15 Geodetic Position Computations19.15.1 Direct Geodetic Problem19.15.2 Inverse Geodetic Problem¿ 19.16 The Local Geodetic Coordinate System¿ 19.17 Three-Dimensional Coordinate Computations¿ 19.18 ConclusionsProblemsBibliography20 State Plane Coordinates¿ 20.1 Introduction¿ 20.2 Projections Used in State Plane Coordinate Systems¿ 20.3 Lambert Conformal Conic Projection¿ 20.4 Transverse Mercator Projection¿ 20.5 State Plane Coordinates in NAD27 and NAD83¿ 20.6 Computing SPCS83 Coordinates in the Lambert Conformal Conic System20.6.1 Zone Constants20.6.2 The Direct Problem20.6.3 The Inverse Problem¿ 20.7 Computing SPCS83 Coordinates in the Transverse Mercator System20.7.1 Zone Constants20.7.2 The Direct Problem20.7.2 The Inverse Problem¿ 20.8 Reduction of Distances and Angles to State Plane Coordinate Grids20.8.1 Grid Reduction of Distances20.8.2 Grid Reduction of Azimuths and Angles¿ 20.9 Computing State Plane Coordinates of Traverse Stations¿ 20.10 Surveys Extending from One Zone to Another¿ 20.11 Conversions between SPCS27 and SPCS83¿ 20.12 The Universal Transverse Mercator Projection¿ 20.13 Other Map Projections20.13.1 Oblique Stereographic Map Projection20.13.2 Oblique Mercator Map ProjectionProblemsBibliography21 Boundary Surveys¿ 21.1 Introduction¿ 21.2 Categories of Land Surveys¿ 21.3 Historical Perspectives¿ 21.4 Property Description by Metes and Bounds¿ 21.5 Property Description by Block and Lot System¿ 21.6 Property Description by Coordinates¿ 21.7 Retracement Surveys¿ 21.8 Subdivision Surveys¿ 21.9 Partitioning Land¿ 21.10 Registration of Title¿ 21.11 Adverse Possession and Easements¿ 21.12 Condominium Surveys¿ 21.13 Geographic and Land Information Systems¿ 21.14 Sources of Error in Boundary Surveys¿ 21.15 MistakesProblemsBibliography22 Surveys of the Public Lands¿ 22.1 Introduction¿ 22.2 Instructions for Surveys of the Public Lands¿ 22.3 Initial Point¿ 22.4 Principal Meridian¿ 22.5 Baseline¿ 22.6 Standard Parallels (Correction Lines)¿ 22.7 Guide Meridians¿ 22.8 Township Exteriors, Meridional (Range) Lines, and Latitudinal (Township) Lines¿ 22.9 Designation of Townships¿ 22.10 Subdivision of a Quadrangle Into Townships¿ 22.11 Subdivision of a Township Into Sections¿ 22.12 Subdivision of Sections¿ 22.13 Fractional Sections¿ 22.14 Notes¿ 22.15 Outline of Subdivision Steps¿ 22.16 Marking Corners¿ 22.17 Witness Corners¿ 22.18 Meander Corners¿ 22.19 Lost and Obliterated Corners¿ 22.20 Accuracy of Public Lands Surveys¿ 22.21 Descriptions by Township Section and Smaller Subdivision¿ 22.22 BLM Land Information System¿ 22.23 Sources of Error¿ 22.24 MistakesProblemsBibliography23 Construction Surveys¿ 23.1 Introduction¿ 23.2 Specialized Equipment for Construction Surveys23.2.1 Visible Laser-Beam Instruments23.2.2 Reflectorless Total Stations¿ 23.3 Horizontal and Vertical Control¿ 23.4 Staking Out A Pipeline¿ 23.5 Staking Pipeline Grades¿ 23.6 Staking Out a Building¿ 23.7 Staking Out Highways¿ 23.8 Other Construction Surveys¿ 23.9 Construction Surveys Using Total Station Instruments¿ 23.10 Construction Surveys Using GPS Equipment¿ 23.11 Machine Control¿ 23.12 As-Built Surveys with Laser Scanning¿ 23.13 Sources of Error in Construction Surveys¿ 23.14 MistakesProblemsBibliography24 Horizontal Curves¿ 24.1 Introduction¿ 24.2 Degree of Circular Curve¿ 24.3 Definitions and Derivation of Circular Curve Formulas¿ 24.4 Circular Curve Stationing¿ 24.5 General Procedure of Circular Curve Layout by Deflection Angles¿ 24.6 Computing Deflection Angles and Chords¿ 24.7 Notes for Circular Curve Layout by Deflection Angles and Incremental Chords¿ 24.8 Detailed Procedures for Circular Curve Layout by Deflection Angles and Incremental Chords¿ 24.9 Setups on Curve¿ 24.10 Metric Circular Curves by Deflection Angles and Incremental Chords¿ 24.11 Circular Curve Layout by Deflection Angles and Total Chords¿ 24.12 Computation of Coordinates on a Circular Curve¿ 24.13 Circular Curve Layout by Coordinates¿ 24.14 Curve Stakeout Using GPS and Robotic Total Stations¿ 24.15 Circular Curve Layout by Offsets¿ 24.16 Special Circular Curve Problems24.16.1 Passing a Circular Curve through a Fixed Point24.16.2 Intersection of a Circular Curve and a Straight Line24.16.3 Intersection of Two Circular Curves¿ 24.17 Compound and Reverse Curves¿ 24.18 Sight Distance on Horizontal Curves¿ 24.19 Spirals24.19.1 Spiral Geometry24.19.2 Spiral Calculation and Layout¿ 24.20 Computation of “As-Built” Circular Alignments¿ 24.21 Sources of Error in Laying Out Circular Curves¿ 24.22 MistakesProblemsBibliography25 Vertical Curves¿ 25.1 Introduction¿ 25.2 General Equation of a Vertical Parabolic Curve¿ 25.3 Equation of an Equal Tangent Vertical Parabolic Curve¿ 25.4 High or Low Point on a Vertical Curve¿ 25.5 Vertical Curve Computations Using the Tangent Offset Equation25.5.1 Example Computations Using the English System of Units25.5.2 Example Computations Using the Metric System¿ 25.6 Equal Tangent Property of a Parabola¿ 25.7 Curve Computations by Proportion¿ 25.8 Staking a Vertical Parabolic Curve¿ 25.9 Machine Control in Grading Operations¿ 25.10 Computations for an Unequal Tangent Vertical Curve¿ 25.11 Designing a Curve to Pass through a Fixed Point¿ 25.12 Sight Distance¿ 25.13 Sources of Error in Laying Out Vertical Curves¿ 25.14 MistakesProblemsBibliography26 Volumes¿ 26.1 Introduction¿ 26.2 Methods of Volume Measurement¿ 26.3 The Cross-Section Method¿ 26.4 Types of Cross Sections¿ 26.5 Average End Area Formula¿ 26.6 Determining End Areas26.6.1 End Areas by Simple Figures26.6.2 End Areas by Coordinates¿ 26.7 Computing Slope Intercepts¿ 26.8 Prismoidal Formula¿ 26.9 Volume Computations¿ 26.10 Unit-Area, or Borrow-Pit, Method¿ 26.11 Contour-Area Method¿ 26.12 Measuring Volumes of Water Discharge¿ 26.13 Sources of Error in Determining Volumes¿ 26.14 MistakesProblemsBibliography27 Photogrammetry¿ 27.1 Introduction¿ 27.2 Uses of Photogrammetry¿ 27.3 Aerial Cameras¿ 27.4 Types of Aerial Photographs¿ 27.5 Vertical Aerial Photographs¿ 27.6 Scale of a Vertical Photograph¿ 27.7 Ground Coordinates from a Single Vertical Photograph¿ 27.8 Relief Displacement on a Vertical Photograph¿ 27.9 Flying Height of a Vertical Photograph¿ 27.10 Stereoscopic Parallax¿ 27.11 Stereoscopic Viewing¿ 27.12 Stereoscopic Measurement of Parallax¿ 27.13 Analytical Photogrammetry¿ 27.14 Stereoscopic Plotting Instruments27.14.1 Direct Optical Projection Stereoplotters27.14.2 Mechanical Projection Stereoplotters27.14.3 Analytical Stereoplotters27.14.4 Softcopy Stereoplotters¿ 27.15 Orthophotos¿ 27.16 Ground Control for Photogrammetry¿ 27.17 Flight Planning¿ 27.19 Airborne Laser-Mapping Systems¿ 27.20 Remote Sensing¿ 27.21 Sources of Error in Photogrammetry¿ 27.22 MistakesProblemsBibliography28 Introduction to Geographic Information Systems¿ 28.1 Introduction¿ 28.2 Land Information Systems¿ 28.3 GIS Data Sources and Classifications¿ 28.4 Spatial Data28.4.1 Simple Spatial Objects28.4.2 Vector and Raster Formats28.4.3 Topology¿ 28.5 Nonspatial Data¿ 28.6 Data Format Conversions28.6.1 Vector-to-Raster Conversion28.6.2 Raster-to-Vector Conversion¿ 28.7 Creating GIS Databases28.7.1 Generating Digital Data from Field Surveys28.7.2 Digitizing from Aerial Photos with Stereoplotters28.7.3 Digitizing Existing Graphic Materials29.7.4 Keyboard Entry29.7.5 Existing Digital Data Sets29.7.6 Scanning¿ 28.8 Metadata¿ 28.9 GIS Analytical Functions28.9.1 Proximity Analysis28.9.2 Boundary Operations28.9.3 Spatial Joins28.9.4 Logical Operations28.9.5 Other GIS Functions¿ 28.10 GIS ApplicationsProblemsBibliographyA Dumpy Levels, Transits, and Theodolites¿ A.1 Introduction¿ A.2 The Dumpy Level¿ A.3 Introduction to the Transit and Theodolite¿ A.4 The TransitA.4.1 Parts of a TransitA.4.2 Circle Scales and VerniersA.4.3 Properties of the TransitA.4.4 Handling, Setting up, and using a Transit¿ A.5 The TheodoliteA.5.1 Characteristics of TheodolitesA.5.2 Repeating TheodolitesA.5.3 Directional TheodolitesA.5.4 Handling, Setting Up, and Using a TheodoliteB Example NoteformsC Astronomical Observations¿ C.1 Introduction¿ C.2 Overview of Usual Procedures for Astronomical Azimuth Determination¿ C.3 Ephemerides¿ C.4 Definitions¿ C.5 Time¿ C.6 Timing Observations¿ C.7 Computations for Azimuth from Polaris Observations by the Hour Angle Method¿ C.8 Azimuth from Solar ObservationsD Using the Worksheets on the Companion Disk¿ D.1 Introduction¿ D.2 Using the Files¿ D.3 Using the Worksheets as an Aid in LearningE Introduction to Matrices¿ E.1 Introduction¿ E.2 Definition of a Matrix¿ E.3 The Dimensions of a Marix¿ E.4 The Transpose of a Matrix¿ E.5 Matrix Addition¿ E.6 Matrix Multiplication¿ E.7 Matrix InverseF U.S. State Plane Coordinate System Defining Parameters¿ F.1 Introduction¿ F.2 Defining Parameters for States Using the Lambert Conformal Conic Map Projection¿ F.3 Defining Parameters for States Using the Transverse Mercator Map ProjectionG Answers to Selected ProblemsIndex