Representing terrain as a function of two variables — from continuous surfaces to discrete grids
2026-02-26
You should know
How coordinates locate a point on a map and how a graph can assign one value to each position.
You will learn
How a digital elevation model works, how landscapes can be treated as mathematical surfaces, and why grids are so useful in computational geography.
Why this matters
Once you can think of a landscape as a function, you can calculate slope, flow, shading, visibility, and many other terrain properties.
If this gets hard, focus on…
The simple idea that every grid cell stores one height value. The rest of the chapter is about what that idea lets us do.
The SRTM mission, flown aboard Space Shuttle Endeavour in February 2000, used a radar interferometer to measure the elevation of virtually the entire land surface of Earth in eleven days. The result — a 30-metre resolution grid of elevations covering 80% of the globe — transformed terrain analysis from a laborious manual discipline into something any laptop could compute in seconds. But behind the data product is a conceptual leap worth making explicit: a landscape is a function.
Elevation at any location is determined entirely by its coordinates. Given a position (x, y), there is exactly one elevation z. That is what a mathematical function does — it assigns a unique output to every input. For terrain, z = f(x, y) is the one-line summary on which slope calculation, hydrological flow routing, viewshed analysis, and landslide prediction are all built. Understanding what it means to represent a continuous surface as a discrete grid — and what information is preserved or lost in that digitisation — is the first step toward doing terrain analysis whose results you can actually interpret and trust.
How do we represent the shape of the land mathematically?
A landscape is not at all just flat. Mountains rise, valleys sink, hillslopes tilt. Every point on the surface has an elevation above some reference (usually sea level).
If we know the coordinates (x, y) of a point on the ground, can we write down a function that gives us the elevation z?
Yes — and this is the foundation of terrain analysis, hydrology, viewshed modelling, and landslide prediction.
The mathematical question: How do we go from a continuous surface to a computable representation?
Imagine standing at a point with coordinates (x, y) measured in meters east and north from some origin. The elevation at that point is z.
Mathematically:
z = f(x, y)
This is a function of two variables: - Input: A position (x, y) in the horizontal plane - Output: Elevation z at that position
Examples of terrain functions:
Real terrain is more complex, but the concept is the same: elevation is determined by position.
In reality, we can’t write down a simple algebraic formula for real terrain. Instead, we measure elevation at many points and store the values in a grid.
Digital Elevation Model (DEM): A raster grid where each cell contains the elevation of the terrain at that location.
Grid structure: - Rows and columns define positions - Cell size (resolution) determines how finely we sample the terrain (e.g., 10 m × 10 m cells) - Elevation values are stored as numbers (integers or floating point)
Example: A 5×5 DEM (rows from north to south, columns from west to east):
110 115 120 125 130
108 112 118 124 129
105 110 115 121 127
103 108 113 118 123
100 105 110 115 120Each number is the elevation (in meters) at the center of that cell.
To turn a grid into a function, we need to define: 1. Origin: Where is (x, y) = (0, 0)? 2. Cell size: \Delta x (east-west), \Delta y (north-south) 3. Grid extent: How many rows and columns?
Standard convention: - x increases eastward - y increases northward - Origin is typically the southwest corner of the grid
If the grid has m rows and n columns, with cell size \Delta x = \Delta y = d:
x_j = x_0 + j \cdot d, \quad j = 0, 1, \ldots, n-1 y_i = y_0 + i \cdot d, \quad i = 0, 1, \ldots, m-1
The elevation at grid cell (i, j) is z_{i,j}.
Given a grid cell index (i, j), the corresponding real-world coordinates are:
(x, y) = (x_0 + j \cdot d, \; y_0 + i \cdot d)
Example: Grid with origin (x_0, y_0) = (500000, 4500000) (UTM coordinates), cell size d = 30 m, cell (i=2, j=3):
x = 500000 + 3 \times 30 = 500090 \text{ m} y = 4500000 + 2 \times 30 = 4500060 \text{ m}
A DEM gives elevation only at grid cell centers. What about points between cells?
Bilinear interpolation: Given a point (x, y) inside a grid cell, interpolate elevation using the four surrounding cell values.
Let the four corners have elevations z_{00}, z_{01}, z_{10}, z_{11} at positions:
\begin{align} (x_0, y_0) &\rightarrow z_{00} \\ (x_0, y_1) &\rightarrow z_{01} \\ (x_1, y_0) &\rightarrow z_{10} \\ (x_1, y_1) &\rightarrow z_{11} \end{align}
For a point (x, y) with x_0 < x < x_1 and y_0 < y < y_1:
z(x, y) = (1 - s)(1 - t) z_{00} + s(1 - t) z_{10} + (1 - s)t z_{01} + st z_{11}
Where:
s = \frac{x - x_0}{x_1 - x_0}, \quad t = \frac{y - y_0}{y_1 - y_0}
This creates a piecewise continuous surface — smooth within each cell, potentially discontinuous at cell boundaries.
Problem: A 3×3 DEM has cell size 10 m and origin at (x_0, y_0) = (0, 0). Elevations (in meters):
Row 0: 100 105 110
Row 1: 102 107 112
Row 2: 105 110 115(a) Elevation at cell (1, 2)
Row 1, column 2: z_{1,2} = 112 m
(b) Coordinates of cell center
x = x_0 + j \cdot d = 0 + 2 \times 10 = 20 \text{ m} y = y_0 + i \cdot d = 0 + 1 \times 10 = 10 \text{ m}
Cell center: (20, 10) m
(c) Interpolation at (15, 5)
Point (15, 5) lies between cells: - x: between column 1 (x = 10) and column 2 (x = 20) - y: between row 0 (y = 0) and row 1 (y = 10)
Four corner elevations: - (10, 0): z = 105 (row 0, col 1) - (20, 0): z = 110 (row 0, col 2) - (10, 10): z = 107 (row 1, col 1) - (20, 10): z = 112 (row 1, col 2)
Interpolation parameters:
s = \frac{15 - 10}{20 - 10} = \frac{5}{10} = 0.5
t = \frac{5 - 0}{10 - 0} = \frac{5}{10} = 0.5
Interpolated elevation:
\begin{align} z(15, 5) &= (1 - 0.5)(1 - 0.5)(105) + (0.5)(1 - 0.5)(110) \\ &\quad + (1 - 0.5)(0.5)(107) + (0.5)(0.5)(112) \\ &= 0.25 \times 105 + 0.25 \times 110 + 0.25 \times 107 + 0.25 \times 112 \\ &= 26.25 + 27.5 + 26.75 + 28 \\ &= 108.5 \text{ m} \end{align}
Elevation at (15, 5): 108.5 m
Below is an interactive 3D visualization of a synthetic DEM.
<label>
Terrain type:
<select id="terrain-select">
<option value="cone">Cone (radial peak)</option>
<option value="saddle">Saddle point</option>
<option value="plane">Tilted plane</option>
<option value="ridge">Ridge</option>
<option value="valley">Valley</option>
</select>
</label>
<label>
Grid resolution:
<input type="range" id="resolution-slider" min="10" max="50" step="5" value="30">
<span id="resolution-value">30</span> cells
</label><p>Each cell represents one grid point in the raster. Colour encodes elevation (blue = low, red = high). This top-down view is exactly how a DEM is stored: a 2D array of elevation values.</p>Try this: - Cone: Radial symmetry, single peak - Saddle: Up in one direction, down in perpendicular direction (mountain pass) - Plane: Uniform slope (no curvature) - Ridge: Linear high feature - Valley: Linear low feature - Increase resolution: Smoother surface, more cells, same terrain shape - Decrease resolution: Coarser surface, blockier appearance
Key insight: The mathematical function is continuous, but the DEM is discrete. Higher resolution approximates the continuous surface better.
Coarse DEM (90 m cells): - Good for regional analysis - Misses small features (gullies, terraces) - Fast to process
Fine DEM (1 m cells): - Captures detailed topography - Required for local hydrology, erosion modelling - Large file sizes, slower computation
Rule of thumb: Cell size should be smaller than the features you care about.
A DEM with 10 m horizontal resolution might have: - ±1 m vertical accuracy (high-quality lidar) - ±10 m vertical accuracy (older photogrammetry)
Don’t confuse the two. Resolution is spacing; accuracy is error.
At the edge of a DEM, there are no neighboring cells beyond the boundary. Calculations requiring neighbors (slope, aspect) are undefined or require special handling (mirror, clamp, or mark as no-data).
Common error: Treating row index as x-coordinate.
Correct: - Row index i corresponds to y-coordinate (north-south) - Column index j corresponds to x-coordinate (east-west)
Always check your coordinate system convention.
Not all DEMs are regular grids: - Triangulated Irregular Networks (TINs): Points connected by triangles (better for sparse data) - Irregular point clouds: Lidar before gridding
This model focuses on regular grids (rasters) because they’re simpler and more common.
Bilinear interpolation creates a piecewise planar surface — each cell becomes a flat facet. This can introduce: - Artificial terracing - Discontinuous slopes at cell boundaries
More sophisticated interpolation (bicubic, splines) can smooth these, but at computational cost.
Real-world DEMs use projected coordinate systems (e.g., UTM) or geographic coordinates (latitude/longitude). Geographic coordinates are not uniform in distance — 1° longitude spans different distances at different latitudes.
Always check the CRS before doing distance-based calculations.
A function z = f(x, y) defines a surface in 3D space.
Partial derivatives measure how the surface changes when you move in one direction while holding the other constant:
\frac{\partial z}{\partial x} = \text{rate of change in the x-direction (eastward)}
\frac{\partial z}{\partial y} = \text{rate of change in the y-direction (northward)}
In terrain analysis: - \frac{\partial z}{\partial x} is the east-west slope component - \frac{\partial z}{\partial y} is the north-south slope component
Next model (Model 8) will show how to compute these from a DEM using finite differences and combine them into gradient and aspect.
z = f(x, y)
Read as: “z is a function of x and y.”
f(x, y) = x^2 + y^2
Evaluation: - f(3, 4) = 3^2 + 4^2 = 9 + 16 = 25 - f(0, 0) = 0 - f(-1, 2) = 1 + 4 = 5
A level curve is the set of all (x, y) points where f(x, y) equals a constant.
Example: For z = x^2 + y^2, the level curve z = 25 is the circle x^2 + y^2 = 25 (radius 5).
In terrain, level curves are contour lines — lines of constant elevation.