Feature Geometry In this lesson you will learn

Feature Geometry In this lesson you will learn: • what is feature geometry • the geometry of points, lines, & polygons • reconstructing features from feature geometry • topologically-encoded geographic data • TIGER • networks

What is feature geometry? B θ A C

Geometry of a point Point data and nothing more. What could this possibly be?

Geometry of lines terminus origin terminus

Describing lines – spline segments The Platte River, from central Wyoming to its junction with the Missouri River.

Describing lines – arc segments source: Advanced Geospatial Lab, Department of Geography, Northern Illinois University.

radius Describing lines – arc segments origin terminus rad arc center ius Geometry of an arc segment Example of a stream channel described by arc and spline segments

Geometry of polygons vertices > > > > terminus origin centroid

Describing polygons > > > centroid > > shared node > > origin > terminus

Reconstructing features from feature geometry The Geo. Database model of point, line and polygon spatial objects Polygon Line From_Pt to_Pt arc_Pt Point “x” “y” PG 01 L 001 P 0002 . P 0001 -88. 9910 43. 2140 PG 01 L 002 P 0002 P 0003 . P 0002 -88. 9907 43. 2136 PG 01 L 003 P 0003 P 0004 . P 0003 -88. 9905 43. 2144 PG 01 L 004 … … … P 0289 -88. 7644 41. 1711 PG 02 L 019 P 0290 -88. 8222 42. 4579 … Data Edge_ID … …. … L 019 P 0204 P 0205 P 0288 L 019 P 0205 P 0206 P 0289 … … … Polygon Area Line Length Orient Radius PG 01 173. 22 214. 14 L 001 13. 64 114. 22 . PG 02 Geometry Perimeter 89. 14 114. 89 L 002 10. 11 41. 41 . PG 03 144. 64 266. 37 L 003 14. 45 114. 23 . … … … L 019 76. 89 247. 23 5. 11 L 020 51. 74 189. 87 18. 73 … … …

Reconstructing features from feature geometry

Introducing topology 3642 Washburn? 3717 Washburn? 2934 Washburn? ? Image courtesy of Musée Rodin, http: //www. musee-rodin. fr

Topology: “The geometric relationship between objects located in space. ” Ian Heywood, Sarah Cornelius, and Steve Carver, An Introduction to Geographical Information Systems, 2 nd edn. , Upper Saddle River, NJ: Prentice Hall, 2002, pg. 291. “(1) The property that describes adjacency and connectivity of features. … (2) The numerical description of the relationships among geographic features, encoded by adjacency, linkage, inclusion, or proximity. …” Keith Clarke, Getting Started with Geographic Information Systems, 4 th edn. , Upper Saddle River, NJ: Prentice Hall, 2003, pg. 326. “… a branch of mathematics that is concerned with spatial properties of discrete objects that remain invariant when distorted. ” Jeff Thurston, Thomas Poiker, J. Patrick Moore, Integrated Geospatial Technologies, Hoboken, NJ: John Wiley & Sons, 2003, pg. 93. A graphic rendering of the famous, Bridges of Königsberg problem. Image courtesy of the School of Mathematics and Statistics, University of St. Andrews, Scotland. http: //www-groups. dcs. st-and. ac. uk Can a walking route be found through all parts of the city, such that each bridge is crossed only once?

Topology of lines terminus origin left face right face origin left face terminus left face

Topology of polygons > outside neighbor > > > inside > neighbor > > > neighbor

Topology and spatial data unsnapped node overshoot slivers undershoot

TIGER 2003 TIGER/Line® Files Entity Point “A point used for identifying the location of point features (or areal features collapsed to a point), such as towers, buoys, buildings, places, etc. ” Node “A zero-dimensional object that is a topological junction of two or more links or chains, or an end point of a link or chain, ” is a node. Complete Chain “A chain [a sequence of non-intersecting line segments] that explicitly references left and right polygons and start and end nodes. ” The shape points combine with the nodes to form the segments that make a complete chain. Network Chain “A chain that explicitly references start and end nodes and not left and right polygons. ” GT-Polygon “An area that is an atomic two-dimensional component of a twodimensional manifold, [which is defined as] one and only one planar graph and its twodimensional objects. ” GT-polygons are elementary polygons that are mutually exclusive and completely exhaust the surface. Classes of spatial objects in TIGER 2003. From TIGER/Line Files 2003, Technical Documentation, U. S. Bureau of the Census, Washington, D. C. : U. S. GPO. March 2004, pg 1 -6.

Topology of TIGER - node School Ave. - shape point - complete chain Court Ave. Park Pl. Aldridge H. S. GT-polygon #2 Aldridge Creek Madison St. GT-polygon #3 GT-polygon #1 Adapted from TIGER/Line Files 2003, Technical Documentation, U. S. Bureau of the Census, Washington, D. C. : U. S. GPO. March 2004, pg 1 -8. Parkside Blvd.

Topology of TIGER Record Type 1 – Complete Chain Basic Data Record from TIGER/Line Files 2003, Technical Documentation, U. S. Bureau of the Census, Washington, D. C. : U. S. GPO.

What TIGER is not, and why Kane County, Illinois TIGER/Line data overlaid on an orthophoto image; compiled by the author from public data files.

TIGER in the future TIGER 2010 and beyond Goals: 1. correct the locations of all existing street centerlines and all other map features used to orient field staff in a one-time effort to make TIGER street centerlines (and boundaries) accurate enough that • GPS will put every house address in the correct census block 100% of the time • they will be used in The National Map 2. identify and add new features and structures to the database, and remove nonexistent features and structures in an on-going maintenance process. 3. acquire and use digital files prepared and provided by state, local and tribal governments as a first priority source: U. S. Bureau of the Census, http: //www. census. gov

Networks

on m Gi lla om Th 0 1 1 The Pas 1 0 0 Flin Flon 1 0 0 Thompson 1 0 0 0 1 Gillam 1 0 0 1 0 in Fl F l ps on s Pa e Th Winnipeg W in ni pe g Network topology Gi lla m on Th om ps n Fl in F lo as e P Winnipeg 0 1: 15 1: 25 1: 50 2: 50 The Pas 1: 15 ∞ ∞ Flin Flon 1: 25 ∞ ∞ Thompson 1: 50 ∞ ∞ ∞ 0: 30 Gillam 2: 50 ∞ ∞ 0: 30 ∞ W Th in ni pe g Connectivity matrix for the AIR CANADA regional service network AIR CANADA partner service from Winnipeg. Source: Air Canada; http: //www. aircanada. ca/ Travel Cost (in hr : min) for the AIR CANADA regional service network

What you have learned In this lesson you learned: • Geometry and Geography share the same root word, geo, from Greek, meaning earth. • Points have no inherent geometric properties; lines have two essential geometric properties: length and orientation; polygons have the geometric properties of perimeter length and area within the perimeter. • The geography of complex linear features can be described by splines (short straight lines), arcs (smooth portions of circles or ellipses), or combinations of splines and arcs connected sequentially so as to maintain the continuity of feature length and overall feature orientation. • The geography of polygons can be described as an ordered series of lines comprising the perimeter of the polygon. The perimeter begins and ends at the same vertex – a property known as closure. • Topology describes the relational geography of spatial objects through: connectivity, adjacency, containment, and inclusion. The topology of line features includes connectivity, intersection, and left and right geography; the topology of polygons includes containment (inside vs. outside) and adjacency. • Topological operations are often used to verify the quality of spatial data, particularly with respect to slivers, undershoot, overshoot, and unsnapped node errors. Topology is also particularly useful when combining spatial data of vastly different scales and for certain advanced spatial data models. • The Census Bureau’s TIGER/Line files originated as topologically-encoded spatial data and are now structured on the Geometry-Topology model of U. S. Spatial Data Transfer Standards. • Network data models are commonly used as the basis for analysis of geographic movement – from transportation and communication to diffusion and migration.