Fundamental spatial concepts

• What is space?

In the context of GIS, we normally use the term space to refer to geographic space: the structure and properties of the relationship between locations at the Earth's surface.

• geometry: measurement of the Earth, provides a formal representation of the abstract properties and structures within a space

• invariance: a group of transformations of space under which propositions remain true

• primitive space: collections of objects with no other structure

Euclidean space: coordinatized model of space

• enables measurements of distances and bearings between points
• transforms spatial properties into properties of tuples of real numbers.
• coordinate frame consists of a fixed origin and a pair of orthogonal lines (axes) intersecting in the origin
• point
• line
• polygon
  • monotone chain
  • monotone polygon
  • triangulation
• transformations
  • preserve particular properties of embedded objects
  • translations
  • rotation
  • reflection
• topological invariants: properties that are preserved by topological transformations

Set-based geometry of space

• elements: the constituent objects to be modeled
• sets: collection of elements
• membership: the relationship between the elements and the sets to which they belong

Topology of space:study of form

• branch of geometry
• concerns properties that are invariant under topological transformations
  • topological
    • an arc is simple
    • an area is connected
  • Non-topological
    • distance between two points
    • length of an arc
• neighborhood: one way of defining a topological space
• topology: the study of topological transformation and properties that are left invariant by them

Network spaces

• to represent a system of roads or rail
• graph: a set of unordered paris of distinct nodes
  • highly abstract
• directed graph
• labeled graph

Metric spaces

• metric: a generalization of the Euclidean metric
• distance function
  • Geodestic distance: e.g., the distance along the great circle of the Earth passing through the two city centers
  • Manhattan distance: the sum of the difference latitudes and longitudes.
  • Travel time distance: the minimum time required to travel from one city to the other
• properties
  • sensible distances cannot be allowed to be negative
  • distance is symmetric, a->b = b->a
  • triangle inequality

Fractal geometry

• scale dependence: appearance and characteristics of many geographic and natural phenomena depend on the scale at which they are observed
• straight line adn smooth curves are not well suited to modeling self-similarity and scale dependence (why)
• fractals: self-similar across all scales, fundamentally different types of geometry
  • defined recursively, rather than by describing their shape directly
  • dimensions: an indicator of shape complexity,