Vector Data Models

Specifically, this section will cover vector models and their relation to topology.

Recall a vector model is a model made with points, lines, and polygons, representing entities and features. There can also be associated attributes, which are the non-spatial components of a model.

Vector models can commonly be represented as:

Non-Topological Vector Data Model

  • Advantages
    • simple (often open file standards)
    • efficient for display and plotting
  • Disadvantages
    • can contain duplicate common boundaries (increases costs due storage and analysis)
    • can be inefficient for spatial analysis
    • editing and querying can be difficult

Can be referred to as spaghetti models.

Topological Vector Data Model

Topology: geometric properties that do not chance with changes shapes, specifically the following properties:

  • adjacency
  • connectivity
  • containment

A topologically defined model can have its shape changed (or warped) and still retain its properties.

  • Advantages
    • data storage reduced (boundary arcs stored once)
    • explicit neighbor relations maintained
      • useful for cleaning, digitizing, analysis, and queries
  • Disadvantages
    • computational overhead
    • draws slower

Topological versus Non-Topological Editing

Snapping

Given a snap tolerance, points and lines can be combined into shared boundaries. Essentially, points are automatically set to have the same coordinates. They become magnetic. Position errors can stem during build processes, especially when digitizing.

Snap tolerance must be chosen carefully:

  • undershooting: node does not quite reach other node or line
  • overshooting: lines cross over existing nodes or lines
  • rule of thumb: should be smaller than the desired positional accuracy

Advanced Data Model with Applied Topology

  • Network Models
    • routes
    • hydrology (i.e. river networks with help in flooding scenarios)
    • other models with attributes that can affect a cost
    • can get very complex
    • given a topologically regulated network, can change shape without changing properties, must follow:
      • adjacency
      • connectivity
      • containment
  • Triangular Irregular Networks (TINs)
    • create a network over a landscape with triangles to represent terrain elevation via a topological network
    • points (nodes) can be actual measured references
    • edges are formed between the nodes
    • the triangular faces are known as facets
    • adhere to the topological properties:
      • adjacency
      • connectivity
      • containment

Topology Rules (ArcGIS Specific)

Vector Operations

Vector operations are used in spatial data analysis on vector data models. Vector operations can be applied with functions and can have one-to-one or many-to-one type inputs and outputs.

Simpson’s Rule

Calculate the geometry or area from coordinates.

\[A=\frac{1}{2}\sum\limits_{i=1}^{n-1}(x_{i+1} - x_i)(y_i + y_{i+1})\]

To perform the operation, we want to “close the loop”.

Code 
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

x = [2, 3, 5, 6, 2]
y = [4, 6, 6, 4, 4]

# Create the plot
plt.figure()
plt.plot(x, y, marker='o')  # Plot the trapezoid with markers at vertices
plt.fill(x, y, 'skyblue', alpha=0.5)  # Fill the trapezoid with color

# Set plot attributes
plt.title('Trapezoid Plot')
plt.xlabel('X-axis')
plt.ylabel('Y-axis')
plt.grid(True)

# Show the plot
plt.show()

  • \((x_0, y_0): (2, 4)\)
  • \((x_1, y_1): (3, 6)\)
  • \((x_2, y_2): (5, 6)\)
  • \((x_3, y_3): (6, 4)\)
  • \((x_4, y_4): (2, 4)\)

\[A = \frac{1}{2}[(3-2)(4+6) + (5-3)(6+6) + (6-5)(6+4) + (2-6)(4+4)] = 6\]

Common Vector Operations

  • Measurements
    • length (lines): commonly measured via Euclidean distance formula
    • area (polygons): commonly measured by dividing polygons into triangles, which have easily calculable area, and summing over positive and negative results
  • Selection
    • Query features (by layer) based on attributes
    • Query features (by layer) based on spatial criteria such as topology rules (adjacency, connectivity, containment, etc.)
    • Querying features by multiple layers:
      • Set algebra: \(<, >, =, <>\)
      • Boolean algebra:
        • disjunction (OR)
        • conjunction (AND)
        • complement/negation (NOT)
  • Classification and Re-classification
    • categorize geographic objects based on a set of conditions
    • i.e. re-classification could be forest into
      • deciduous forest
      • evergreen forest
      • mixed forest
    • re-classification based on binary logical results (i.e. “West of the Mississippi”)
    • Classification Schemes
      • equal-interval
      • natural breaks
      • equal-area
    • Classification Schemes Further Explained
      • Population is assumed to normally follow a uniform distribution, which can skew how a visulation looks
      • Classification schemes are important in terms of misuse

  • Dissolve
    • combine like features based on attributes
    • i.e. combine smaller polygons into larger polygons based on criteria
    • can be used after reclassification (i.e. states west and east of the Mississippi dissolves into two total polygons)
    • don’t forget about MAUP when performing dissolves
      • note on maup: when disaggregating, this can be done with higher accuracy when using ancillary data of the smaller scale.
    • neighboring boundary shares attributes, this is the dissolve boundary
    • able to bring over summary statistics (i.e. sum of population OR different statistical characteristics)
    • population density can be tricky to dissolve.
  • Buffer
    • simple buffer: point layer \(\rightarrow\) overlap dissolved
    • compound buffer: point layer \(\rightarrow\) overlap identified
    • nested buffer (doughnut buffer): nested rings within a simple buffer
    • variable buffer: buffer changes with relation to attribute (i.e. distance from main branch of river)
    • other examples where buffer can be important:
      • combine different layers

  • Overlay
    • combine different layers
    • requires common coordinate system
    • creates a new layer
    • point in polygon
      • creates a new layer with an additional attribute of the polygon which contains the point
    • line in polygon
      • additional attribute of split between polygons
      • note that the new layer has an additional attribute and an additional row in the attribute table
    • polygon
      • intersection
        • new nodes are created in this process
        • example: residential area which intersects protected land
      • union
        • at least one of the combined conditions must be true
        • think of this as creating a Ven Diagram (i.e. the union of 2 shapes creates 3 shapes - left, intersection, right)
      • clip
        • direction matters: input attributes retained; clip feature attributes ignored
        • example: residential area which intersects protected land:
          • clip: protected
          • input: residential
      • erase
        • direction matters: input attributes retained; erase feature attributes ignored
        • example: residential area which intersects protected land:
          • erase: protected
          • input: residential
      • identity
        • one of the more “abstract” polygon operations
        • extends the identity attribute
        • note that the resulting layer (final attribute table) has null values
        • be careful in analyzing the example, as A & B are used three times in slightly different contexts
          • ultimately creates a subset
    • spatial analysis within overlay
      • using a common coordinate system, different layers with spatial and attribute data can be combined into a new layer
    • vector overlay (point, line, polygon):
      • intersecting liens are split and a node placed at the intersection point
      • topology is likely to be different
      • must be recreated for later processing
      • output typically takes the lowest dimension of the inputs
    • can use overlays to combine attributes of layers and create new tables

Overlay: Point in Polygon

Line in Polygon

Polygon on Polygon (Boolean Algebra / Clip / Erase)

Polygon Overlay: Intersection

Polygon Overlay: Union

Can be shown similarly to the above with attributes, intermediate, and final output.

Polygon Overlay: Clip

  • Clip feature must be polygons, input can be points, lines or polygons
  • Input attributes retained; clip feature attributes ignored
  • Essentially acts as a crop function

Polygon Overlay: Erase

  • Erase and input features of same type
  • Input attributes retained; erase feature attributes ignored
  • Essentially acts as an inverse crop function, removing a portion of the map

Polygon Overlay: Identity

  • computes a geometric intersection of the input features and identity features
  • essentially extends attribute features
  • the input features or portions thereof that overlap identity features will get the attributes of those identity features

Importance of Topology in Overlay

Slivers or Sliver Polygons can occur when polygons with a shared/common boundary overlap or don’t match entirely. Topological Data Sets can help fix or prevent this by preventing duplicate shared boundaries or nodes. This can also be fixed via snapping and other ArcGIS tools. Important to Note that data is changing with each operation and new layer.

Selection by Adjacency

When selecting by adjacency, there are different criteria:

  • shared line requirement
  • share node OR line requirement
  • among other spatial criteria that could define adjacency