Scientific Visualization and Animation: Examples from Geography

Computer Science Department

bernstdh@jmu.edu

From Longitude/Latitude to Cartesian Coordinates

Notation

From Longitude/Latitude to Cartesian Coordinates (cont.)

Derivation of the Conversion:
- \(\cos \phi = d / R \Rightarrow d = R \cos \phi\)
- \(\cos \lambda = p_{z}/d \Rightarrow p_{z} = d \cos \lambda = R\cos \phi \cos \lambda\)
- \(\sin \lambda = p_{x}/d \Rightarrow p_{x} = d \sin \lambda = R \cos \phi \sin \lambda\)
- \(p_{y} = R \sin \phi\)
Note:
- This is very similar to the conversion of spherical coordinates to Cartesian coordinates (but it uses the latitude not the colatitude)

From Longitude/Latitude to Cartesian Coordinates (cont.)

svaexamples/cartography/projections.c (Fragment: lonlat2xyz)

Visualizing a Globe

What We Need To Do:
- Represent geographic features
- Read country boundaries from a file
- Create longitude/latitude lines
- Draw geographic features
3D Visualization:
- We will use an orthographic visualization (i.e., we will essentially ignore \(z\) coordinates)

Visualizing a Globe (cont.)

Representing Geographic Features svaexamples/cartography/feature.c

Visualizing a Globe (cont.)

File Format:
- One record description of each file (number of polygons, number of countries)
- One record description of each polygon (polygon number, country number,number of vertices,country name)
- Each polygon has one record for each vertex (longitude, latitude)
An Important Observation:
- Each country might consist of several polygons

Visualizing a Globe (cont.)

Reading Country Borders svaexamples/cartography/initialization.c (Fragment: readPolygons)

Visualizing a Globe (cont.)

Longitude/Latitude Lines svaexamples/cartography/initialization.c (Fragment: createGrid)

Visualizing a Globe (cont.)

Drawing the Features svaexamples/cartography/globe.c (Fragment: display)

Visualizing a Globe (cont.)

Putting It All Together svaexamples/cartography/globe.c (Fragment: main)

Classical Map Projections

Objective:
- Project points on the surface of the Earth onto a map
Some Intuition:
- Shine a light onto or through a transparent Earth ` and capture the shadows cast by the opaque features
- The parameters are: the shape of the screen (called the projection surface), the position of the projection surface, and the location of the light source

Classical Map Projectins (cont.)

Projection Surfaces

Classical Map Projectins (cont.)

Light Sources

Desirable Properties of Map Projections

The Most Common:
- Conformal (i.e., angles are preserved)
- Equal Area (i.e., areas are in constant proportion)
- Equidistant (i.e., distances are in constant proportion)
An Important Mathematical Result:
- A single projection can not be both conformal and equal area

Equatorial Cylindrical Equal Area Projection

Parameters:
- \(\lambda_{0}\) is the standard longitude (i.e., the horizontal center of the projection)
Projection:
- \(x = R (\lambda - \lambda_{0}) \)
- \(y = R \sin(\phi)\)
Inverse:
- \(\lambda = \lambda_{0} + \frac{x}{R}\)
- \(\phi = \sin^{-1}(y / R)\)

Equatorial Cylindrical Equal Area Projection (cont.)

The World

Equatorial Cylindrical Equal Area Projection (cont.)

svaexamples/cartography/projections.c (Fragment: cylindricalEqualArea)

Visualizing a Map

What We Need To Do:
- Everything for visualizing a globe
- Perform the projection
An Important Observation:
- Some projections will "tear" polygons

Visualizing a Map (cont.)

Drawing the Features svaexamples/cartography/map.c (Fragment: display)

There's Always More to Learn