3/31/2015
PA199 Advanced Game Design
Lecture 7 Terrain Generation
Dr. Fotis Liarokapis 31st March 2015
Terrain Data
• Terrain data relates to the 3D configuration of the surface of the Earth
• Map data refers to data located on the surface of the Earth (2D)
• The geometry of a terrain is modeled as a 2 ^-dimensional surface
Terrain Models
• Global terrain model
- Defined by a single function interpolating all data
• Local terrain models
- Defined on a partition of the domain into patches
• They represent the terrain by means of a different function on each of the regions in which the domain is subdivided
• In general it is very difficult to find a single function that interpolates all available data
- Usually local models are used
Mathematical Terrain Models
V
A topographic surface or terrain can be mathematically modeled by the image of a real bivariate function: z = rj> (x, y) Defined over a domain D such that D c5R2 The pair T=(D, rj>) is called a mathematical terrain model
Unidimensional profile of a mathematical terrain model
Digital Terrain Models Video
l
3/31/2015
Digital Elevation Models (DEMs)
DEM Video
DEM is set of regularly or irregularly spaced height values -Terrain elevation data No other information
Elevation Data Acquisition
• Elevation data can be acquired through:
- Sampling technologies
• i.e. on-site measurements or remote sensing techniques
- Digitisation of existing contour maps
• Elevation data can be scattered (irregularly distributed) or form a regular grid
• The set of non-crossing lines can form a collection of polygonal chains
UK DEM Data Sources
•   OrdnanCe SUrVey: http://www.ordnancesurvev.co.uk/
-Landform Panorama
• Source scale: 1:50,000
• Resolution: 50m
• Vertical accuracy: ±3m -Landform Profile
• Source scale: 1:10,000
• Resolution: 10m
• Vertical accuracy: ±0.3m
Landform Panorama Landform Profile
Light Detection And Ranging (LIDAR)
• LIDAR is a remote sensing technology that measures distance by illuminating a target with a laser and analyzing the reflected light
-Uses ultraviolet, visible or near infrared light
to image objects -Can target a wide range of materials
• i.e. non-metallic objects, rocks, rain, chemical compounds, aerosols, clouds, single molecules
2
3/31/2015
Digital Terrain Models (DTMs)
DTM Examples
Digital terrain models represent an approximation of mathematical terrain models
In general, a larger number of sampled points allows for a better representation: multiresolution terrain models
Sampled model
Digital terrain model
3
3/31/2015
DTM Types
Polyhedral Terrain Models (PTM)
Polyhedral terrain models Gridded elevation models Contour maps
A PTM for a set of sampled points V can be defined on the basis of:
- A partition of the domain D into polygonal regions having their vertices at points in V
- A function f that is linear over each region of the partition
• The image of f over each polygonal region is a planar patch to guarantee continuity of the surface along the common edges
PTM Properties
• They can be used for any type of sampled pointset
- Regularly and irregularly distributed
• They can adapt to the irregularity of terrains
• They represent continuous surfaces
Triangulated Irregular Networks (TINs)
• TINs are the most
commonly used PTMs -Each polygon of the
domain partition is a
triangle
TINs Example
V
TINs for Regular Data
Regular sampling is enough in areas where the terrain elevation is more or less constant
Example of a TIN based on irregularly distributed data
4
3/31/2015
TINs Properties
TIN Video
They guarantee the existence of a planar patch for each region (triangle) of the domain subdivision (three points define a plane)
-The resulting surface interpolates all elevation data
The most commonly used triangulations are Delaunay triangulations
Why Delaunay Triangulations
They generate the most equiangular triangles in the domain subdivision -Thus minimising numerical problems • e.g. Point location
Their Dual is a Voronoi diagram Therefore, some proximity queries can be solved efficiently
Why Delaunay Triangulations
It has been proven that they generate the best surface approximation independently of the z values - In terms of roughness Several efficient algorithms to calculate them exist!
Delaunay Triangulations
V
Voronoi Diagrams
Intuitively: given a set V of points, among all the triangulations that can be generated with the points of V, the Delaunay triangulation is the one in which triangles are as much equiangular as possible - Delaunay triangulations tend to avoid long and thin triangles
Does P lie inside t or on its boundary?
Given a set V of points in the plane, the Voronoi Diagram for V is the partition of the plane into polygons such that each polygon contains one point p of V and is composed of all points in the plane that are closer to p than to any other point of V
5
3/31/2015
Voronoi Diagrams
Voronoi Diagrams Example
Property
-The straight-line dual of the Voronoi diagram of V is a Delaunay triangulation of V
Dual
-Obtained by replacing each polygon with a point and each point with a polygon
Connect all pairs of points contained in Voronoi cells that share an edge
Voronoi Diagrams Example
Voronoi Diagrams Usage
Used as underlying structures to solve proximity problems:
- Nearest neighbor (what is the point of V nearest to P?)
- K-nearest neighbors (what are the k points of V nearest to P?)
Gridded Elevation Models
• A Gridded Elevation Model is defined on the basis of a domain partition into regular polygons
Regular Square Grids (RSGs)
• The most commonly used gridded elevation models are Regular Square Grids (RSGs) where each polygon in the domain partition is a square
• The function defined on each square can be a bilinear function interpolating all four elevation points corresponding to the vertices of the square
6
3/31/2015
RSG Example
RSG Stepped Model
Alternatively, a constant function can be associated with each square (i.e., a constant elevation value) This is called a stepped model - It presents discontinuity steps along the edges of the squares
Unidimensional profile of a stepped mode
TINsvs RSGs
Both models support automated terrain analysis operations
-RSGs are based on regular data distribution -TINs can be based both on regular and irregular data distribution
Irregular data distribution allows to adapt to the "variability" of the terrain relief
-More appropriate and flexible representation of the topographic surface
Digital Contours
• Given a sequence {vO,vn } of real values, a digital contour map of a mathematical terrain model (D, t\>) is an approximation of the set of contour lines
• {(x,y) e D, cfj(x,y) = vi}   i = 0, n
Plane z = v
Digital Contour Maps
V
Digital Contour Maps
Contours are usually available as sequences of points
A line interporlating points of a contour can be obtained in different ways
Typical examples -Polygonal chains
- Lines described by higher order equations
3/31/2015
Digital Contour Maps Properties
They are easily drawn on paper
They are very intuitive for humans
They are not good for complex automated terrain analysis
Problems with DEMs
Issues worth considering when
creating/using DTMs
-Quality of data used to generate DEM
-Interpolation technique
-Give rise to errors in surface such as:
-Sloping lakes and rivers flowing uphill
-Local minima
-Stepped appearance
-etc
Example Applications Visualisation
-Terrain and other 3D surfaces Visibility analysis
-Intervisibility matrices and viewsheds
Hydrological modelling
-Catchment modelling and flow models
Engineering
-Cut & fill, profiles, etc
V
Terrain Visualisation
Analytical hillshading Orthographic views
- Any azimuth, altitude, view distance/point
- Surface drapes (point, line and area data) Animated 'fly-through'
What if? modelling
- Photorealism
- Photomontage -CAD
8
3/31/2015
Hillshading Video
Fractal Terrain Generation
Terrain Map
Height Map: z = f(x, y)
- x and y are sampled on a 2D integer grid Real data
- Satellite, Elevation maps (previous lecture) Synthetic
-Texture map, Noise functions B^^H
Terrain Map
Connect samples into a mesh
0.0-i-1-1.0
Fake Terrain
V
Random Terrain
Generate the height-field Random process
- This can be control led
Reflects "realistic" terrain in some way
Simple:
- Terrain(x,y) = rand( MAX_HEIGHT )
- Results in random noise Next step:
- Smooth the terrain generated above
- Finite Impulse Response (FIR) filter:
1	1	
1	1	1
1	1	1
9
3/31/2015
Procedural Modeling With Fractals
Procedural Modeling -Compute geometry "on-the-fly" Fractals
-Model Natural Phenomena - Self Similarity
• Mountains, fire, clouds, etc, -Scales to infinity
• Add or "make up" natural looking details with mathematical tools
Fractals - A Definition
A geometrically complex object, the complexity of which arises through the repetition of some shape over a range of scales - Sufficient definition for describing terrains
A hybrid multifractal made from Worley's Voronoi distance basis
Fractals in Nature
Fractal Geometry
• Fractals are common in:
-Mountains, clouds, trees, turbulence, circulatory
systems in plants and animals
• Wide variety of other phenomena such as: -Noise in transistors -Fluctuations in river fluxes
Fractal geometry is very powerful! - But not sufficient for describing the complex forms found in Nature
Mathematics are simple -Based on the Euclidean geometry of lines, planes, spheres and cones
Fractal geometry has very little use in describing man-made objects
10
3/31/2015
Fractal Dimension
• Euclidean dimensions
- l, 2, 3, 4,...
• Fractal
- 1.2342, 2.7656
• Measure of detail or roughness of a fractal -D = (lnN)/(lnl/s)
Brownian Motion
• One of the most realistic ways of representing natural objects is the Brownian motion method because of the ability of creating peculiar curves
• Brownian method has been used to describe the chaotic and random manner in which a particle moves in a fluid -Application in terrains
Graph of a Brownian Sample Function A simulation of a Brownian Path in 2-Dimensions
Fractional Brownian motion (fBm)
• fBm consists of steps in a random direction and with a step-length that has some characteristic value
-Also known as the Random Walk Process - Hence the random walk process
• A key feature to fBm is that if you zoom in on any part of the function you will produce a similar random walk in the zoomed in part
Differences
• The main difference between fBm and regular Brownian motion is that while the increments in Brownian Motion are independent they are dependent in fBm -This dependence means that if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well
Code Example of fBm
total = O.Of;        //for each pixel, get the value
frequency = 1.0f/(float)hgrid; amplitude = gain;
for (i = 0; i < octaves; ++i)
{
total += noise((float)x * frequency,
(float)y*frequency) * amplitude; frequency *= lacunarity; amplitude *= gain;
}
map[x][y]=total; //now that we have the value, put it in
11
3/31/2015
Terrain using fBm Noise
Midpoint Displacement ID
A simple approach is to generate a heightmap using fBm noise - Results look ok but not very realistic • Since fBm is homogeneous and isotropic
Type of polygon subdivision algorithm, also a fractal function
Created to simulate tectonic uplift of mountain ranges
One of its main input parameters is the roughness constant r
Midpoint Displacement ID
Midpoint Displacement ID
Step 0
Displace the midpoint of the line
by some random value between (-d, d)
Step 3
Keep iterating until you get the required detail
Always remembering to reduce d After every step
Step 1
Step 2
Now reduce the range of your random function " depending on r by d* = pow( 2 , -r)
Again displace the midpoint
of all the line segments and reduce your
Random function's range
Nth step
How does r affect the outcome?
If r = 1
Your d will half each iteration
If r > 1 d increases faster
generates smooth terrain
If < 1 d increases slowly
generates chaotic terrain
Diamond - Square Algorithm
V
Diamond - Square Algorithm Example
Also called the cloud fractal, plasma fractal or random midpoint displacement The 2D version of the original Midpoint displacement algorithm
- Therefore it also has a roughness constant
The algorithm works best if it is run on square grids of width 2"n
- This ensures that the rectangle size will have an integer value at each iteration
12
3/31/2015
^^Diamond - Square Algorithm Methodology
Algorithm starts with a 2 x 2 grid
The heights at the corners can be set to either: - Zero, a random value or some predefined value
iamond - Square Algorithm Methodology.
The first step involves calculating the midpoint of the grid based on its corners and then adding the maximum displacement for the current iteration - Called the Diamond step Because if you render this terrain you will see four diamond shapes
E=(A+B+C+D)/4 + Rand(d)
Rand(d) can generate random values between -d and +d
9
V
First Step: Diamond Step
The first step involves:
- Calculating the midpoint of the grid based on its corners and then
- Adding the maximum displacement for the current iteration
E = (A+B+OD)/4 + Rand(d)
Rand(d) can generate random values between -d and +d
Second Step: Square Step
V
Calculate the midpoints of the edges between the corners
Since the first iteration is complete, now d is reduced by: - d *= pow(2, -r)
• where r is the roughness
wrapping
G = (A+B+E*E)/4 + rand(d)
H = (B+D+E+E)/4 + rand(d)
I = (D+C+E+E)/4 + rand(d)
F = (A+C+E+EJ/4 + rand(d)
Non-wrapping
G = (A+B+E)/3 +rand(d)
same for H.I.F
Third Step: Second Iteration
Start the second iteration
Again perform the diamond step
J = (A+G+F+E)/4 + rand(d) K = (G+B+E+HJ/4 + rand(d) L=(F+E+D+l)/4 + rand(d) M = (E+H+l+C)/4 + rand(d)
Remember this d is smaller than the one in the first iteration
Fourth Step: Square Step
®
Do the square step and continue subdividing
-Until you reach the desired level of detail
wrapping
O = (A+G+J+J)/4 + rand(d)
P = (J+G+K+EJ/4 + rand(d)
Q = (J+E+L+F)/4 + rand(d)
N = (A+F+J+J)/4 +rand(d)
Non-wrapping
O = (A+G*J)/3 + rand(d)
N= (A+F*J)/3 + rand(d)
13
3/31/2015
Diamond - Square Algorithm Summary
• While length of square sides > 0
- Pass through the whole array and apply the diamond step for each square
- Pass through the whole array and apply the square step for each diamond
- Reduce the range of the random displacement
Smoothness of Terrain
Lorentz and Gaussian distributions for the height array can control the smoothness of the terrain
height = y = (tandomnumbet) x--—^ ^ -
(x-x„) +(z-z„) +D2/8
— where (x0, z0) is the position of the peak and D2 is the length of the square
The value of the width affects the smoothness of the terrain
— By decreasing the width of the bell-shaped distribution the terrain becomes steeper
Outer Boundaries
Loop the Old Terrain
Three solutions can be considered: -Large terrain
• The user would never reach the outer boundary -Loop the old terrain
• When the user reaches a boundary the user reenters the map on the opposite side
-Infinite terrain
• When the user reaches a boundary an extension to the map is added
14
3/31/2015
Other Techniques
Cracked Terrains
Cracked terrains for dry lakes and riverbeds
Throw random points onto the plane Construct the Voronoi diagram and use the graph to etch into the terrain
Multifractals
Multifractal Code Example
A better choice would be to use Multifractals - Proposed by Kenton Musgrave These are fractals whose dimension/roughness varies with location
* Procedural raul tlfractal evaluated at 'point."
* Parameters:
* "H" determines tne highest fractal dlmensior
* "lacunarity" Is gap between successive frequencies
* "octaves" is the number of frequencies in the fBm
* "offset" is the aero offset, which determines multifractality
for Hit i-0; Koctaves; if+> 1
value *- Choi set point > + offset! * pow< la' point *- lacunarity:
Other Multifractals
V
Hybrid multifractals
-Called hybrid because they are both additive and multiplicative multifractals
Ridged multifractals
-Similar to Perlin's turbulence noise
-They calculate l-abs(noise) so that the
resulting "canyons" from abs(noise) become
high ridges
Other Multifractal Examples
Ridged multifractal terrains : taken from Texturing and Modeling: A Procedural Approach pg 518 (left) pg 480{right)
15
3/31/2015
More Controls
• To create plateaus, add a min function to flatten out high areas
More Controls..
• Multiply by a Guassian filter to limit the mountain range
More Controls.
• To create a plain, add a max function to flatten out low areas
More Controls...
• To create valleys, create a lower amplitude and rather smooth terrain to use as the max operator
— M	
	
16
3/31/2015
Ridges
• Quantize a terrain to create ridges - Use directly or as the min function -Can also be done as a transfer function that maps f(x)->g(x)
Fault Line Algorithm
• This algorithm is a very simple one, yet its results, although not the best, are pretty good
• The technique is not limited to planar height fields, being also applicable to spheres to generate artificial planets
-Can also approximate real world terrain features such as escarpments, mesas, and seaside cliffs
Fault Line Algorithm
Fault Line Algorithm
Start with a planar height field
- All points have zero height Then select a random line which divides the terrain in two parts
- The points to one side of the line will have their height displaced upwards
- The points on the other side will have their heights displaced downwa rds
The red points will have their height decreased, whereas the blue points will have their height increased
- So the terrain has two distinct heights
If we keep dividing the terrain can get something that has valleys, mountains and so on
First step in faulting process
1 iteration 100 iterations 400 iterations
^?5nerating Fault Lines in a Height Field v Grid
• Randomly pick two grid points p± and p2
• Calculate the line between them
• Go through all the points in the height field and add or subtract an offset value depending on what side of the line they are located
• Before the next fault is drawn reduce the range of the offset by some amount
dist = 0
I-«,
dist c 0__
17
3/31/2015
Filtering Height Fields
Variations to the Fault Line Algorithm
Height fields generated by this algorithm need to be filtered to look more realistic
-A low pass filter can be used
Particle Deposition
Simulates volcanic mountain ranges and island systems
- Drop random particles in a blank grid -Determine if the particle's neighboring cells are of a lower height
• If true, increment the height of the lowest cell
• Keep checking its surrounding cells for a set number of steps or until it is the lowest height among its surrounding cells
• If not, increment the height of the current cell
Particle Deposition
E
V
Generated after 5 series of 1000 iterations
Perlin Noise
V
Terrain Coloring
Using a sampling of 2D Perlin Noise provides smooth hills
Using a ID texture map based on the altitude can provide many useful mapping
18
3/31/2015
Terrain Coloring Striped ID texture map
Terrain Coloring
Using a 2D texture map provides richer detail, but is independent of the terrain
Terrain Coloring...
More advanced coloring is based on altitude and slope
Eft's
Rolling Hills
Scaling in one dimension gives smooth rolling hills
Height Fields Issues They cannot generate overhangs or caves
Height Fields Potential Solutions
• "Mushrooming" effects that involve the manipulation of vertex normals
-To render height field textures with overhangs
• The game Halo Wars implemented a new type of height field called a vector height field
-Stored a vector to displace a vertex instead of a height value
19
3/31/2015
Realistic Terrain Video 2
Erosion
Hydraulic Erosion
-Water (rain) depositing to settle in height field
Thermal Weathering -"Any material that knocks material loose,
which then falls down to pile up at the
bottom of an incline."
20
3/31/2015
Hydraulic Erosion
Hydraulic erosion refers to the natural process in which the motion of fluids, specifically water, produces mechanical weathering over the soil *
(Globe Tourism 2014|
Cairdshill Quarry (Burgess 2009|
Erosion Examples
	-
	
	
Erosion by Wind and Water
References
• https://www.goog! e.co.uk/#q=ordnance+survev
• http://ned.usgs.gov/
• http://en.wikipedia.org/wiki/Digital elevation model
• http://wiki.gis.com/wiki/index.php/Contour line
• http://wiki.gis.com/wiki/index.php/Digital Elevation Model
• http://www.gdcvault.com/plav/1277/HALO-WARS-The-Terrain-of
• http://davis.wpi.edu/~matt/courses/fractals/brownian.html
• http://code.google.eom/p/fractalterraingeneration/wiki/Fractional Brownian Motion
• http://www8.cs.umu.se/kurser/TDBD12/HT01/papers/MusgraveTerrain00.pdf
• http://www.lighthouse3d.com/opengl/terrain/index.php37fault
• http://www.decarpentier.nl/downloads/lnteractivelvSvnthesizingAndEditingVirtualOu tDoorTerrain report.pdf
• http://www.gameprogrammer.eom/fractal.html#midpoint
• Musgrave, et al. "The Synthesis and Rendering of Eroded Fractal Terrains", Siggraph 1989
• Ebert, David S., Musgrave, F. Kenton, Peachey, Darwyn, Perlin, Ken and Worley, Steve Texturing and Modeling: A Procedural Approach, 3rd edition. USA. Morgan Kaufman Publishers, 2003
Bibliography
• DeLoura, Mark. Game Programming Gems. Charles River Media, 2002.
• Martz, Paul. "Generating Random Fractal Terrain." Game Programmer. Publisher Robert C.
Questions
21