Field Based Data Model

Field based model: describes the world as a collection of spatial distributions of phenomena Field based data: - Raster graphics: Theme based graphics with a matrix of cells representing the variables in all the location - Digital Elevation Model: a grid of squares with elevation values (height-map) or a as a vector-based triangulated irregular network - Satellite Image: remote sensing data

Field-based data model: every point in space is represented, so we must store infinite data Solution: sample the data and make some assumption about the behaviour of the variable between sample points

Selecting discrete objects from a continuous surface

A method of constructing new data points within the range of a discrete set of known data points: i.e. make some assumptions about the behaviour of the variable between sample points

• Linear
• Polynomial
• Spline

Finding an interpolated value at a given point based on distance squared weighting of the values of nearest data points

• Tessellation: cover a surface with a pattern of flat shapes so that there are no overlaps or gaps. AKA Tiling

• Regular Tessellation: a pattern made by repeating a regular polygon

• Irregular Tessellation: a pattern made by some irregular shapes.

GIS almost exclusively use square-based tessellations - regular gird model Features of regular grid model:

• Coordinate system: origin, region
• Rows and columns with cells
• Grid is both spatial and attribute data

Example of this is the lab excercise: Landcov

• Irregular tessellation: a tessellation for which the participating polygons are not all regular and equal

• TIN (Triangulated Irregular Networks) is the most commonly used irregular tessellations in GIS's elevation model.

• The irregularity of a TIN Allows the resolution to vary over the surface, capturing finer details where required

How to determine the triangulation?

How to estimation the z value at every location?-- an interpolation problem Three points to define a plane:αx+ βy+ γz+ δ= 0

Given a set of points of P (e.g. elevation survey points), a triangulation of P is a planar subdivision of the convex hull of P into triangles with vertices from P. Which triangulation should we use?

Delaunay triangulation: constituent triangles in a Delaunay triangulations are "as near equilateral as possible" Proximal polygon: A region R around a point p with the property that every location in R is nearer to p than to any other point Voronoi diagram: the dual of a Delaunay triangulations

Given an initial point set P (Suppose no sets of three points are collinear to avoid degenerate cases):

• The Delaunay triangulations is unique
• The external edges of the triangulation from the convex hull of P (i.e. the smallest convex set containing P)

A Simple DT construction method:

• First, constructing the Voronoi polygons
• Second, connecting all adjacent polygon centres
• Third, connect the convex hull of points

Diagram

Diagram