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PA 199 Advanced Game Design
Lecture 6 Terrain Generation
Dr. Fotis Liarokapis 25th March 2019
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Basic Terrain Generation
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 /4-dimensional surface
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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
A topographic surface or terrain can be mathematically modeled by the image of a real bi-variate function: z = (tp (x, y) Defined over a domain D such that D cSl2 The pair T=(D, (tp) is called a mathematical terrain model
Uni-dimensional profile of a mathematical terrain model
Digital Terrain Models Video
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Digital Elevation Models (DEMs)
DEM is set of regularly or irregularly spaced height values — Terrain elevation data No other information
DEM Video
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:
— 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
http://www.ordnancesurvev.co.uk/
Comparison
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
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Digital Terrain Models (DTMs)
• Digital terrain models represent an approximation of mathematical terrain models
Sampled model Digital terrain model
In general, a larger number of sampled points allows for a better representation: multiresolution terrain models
DTM Video 1 DTM Video 2
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DTM Types
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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
Triangulated Irregular Networks (TINs)
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Can be used for any type of sampled pointset
— Regularly and irregularly distributed Can adapt to the irregularity of terrains Represent continuous surfaces
• TINs are the most commonly used PTMs
— Each polygon of the domain partition is a triangle
TINs Example
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
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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
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?
Voronoi Diagrams
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
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Voronoi Diagrams
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Property
— The straight-line dual of the Voronoi diagram of Vis a Delaunay triangulation ofV
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 Example
Voronoi Diagrams Usage
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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
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RSG Example
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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 model
TINs vs 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
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Digital Contours
• Given a sequence {vO,vn } of real values, a digital contour map of a mathematical terrain model (D, (j)) is an approximation of the set of contour lines
• {(x,y) e D, (J)(x,y) = vi}   i = 0,n
Plane z = v
Digital Contour Maps
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
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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
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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 modeling -Catchment modeling and flow models
• Engineering
-Cut & fill, profiles, etc
Terrain Visualisation
• Analytical hillshading
• Orthographic views
- Any azimuth, altitude, view distance/point
- Surface drapes (point, line and area data)
• Animated 'fly-through'
• What if? modeling
- Photorealism
- Photomontage -CAD
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Hillshading Video
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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
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Terrain Map .
• Connect samples into a mesh
1.0
0.0-i-+1.0
Fake Terrain
Random Terrain
Generate the height-field Random process
- This can be controlled
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:
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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
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
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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
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Fractal Dimension
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Brownian Motion
Euclidean dimensions
- 1, 2, 3, 4,... Fractal
- 1.2342, 2.7656
Measure of detail or roughness of a fractal
- D = (In N)/(ln 1/s)
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
Brownian Motion Examples
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
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Terrain using fBm Noise
A simple approach is to generate a heightmap using fBm noise — Results look ok but not very realistic • Since fBm is homogeneous and isotropic
Midpoint Displacement ID
Step 0
Displace the midpoint of the line
by some random value between (-d, d)
Step 1
Now reduce the range of your random function L depending on r by d* = pow( 2 , -r)
Step 2
Again displace the midpoint
of all the line segments and reduce your
Random function's range
Midpoint Displacement ID
• 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
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Step 3
Keep iterating until you get the required detail
Always remembering to reduce d After every step
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 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
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ici^o^^jamonc| _ 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
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"cls,^TDiamoncl - 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
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E = (A+B+C+D)/4 + Rand(d)
Rand(d) can generate random values between -d and +d
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+C+D)/4 + Rand(d)
Rand(d) can generate random values between -d and +d
Second Step: Square Step
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 constant
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D
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+E)/4 + rand(d)
Mon-wrapping
G = (A+B+E)/3 +rand(d)
same tor H,I,F
Third Step: Second Iteration
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Start the second iteration
Again perform the diamond step
•-< F dh 4	i-• •K iE ti
IP 1 ♦L «i-«	' IP •M h-it
J = (A+G+F+E)/4 + rand(d) K = (G+B+E+H)/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
» m
it—it
ř—©x
E
wrapping
0 = (A+G+J+J)/4 + rand(d) P = (J+G+K+E)/4 + rand(d) Q = (J+E+L+FV4 + rand(d) N = (A+F+J+J)/4 +rand(d) Non-wrapping 0=(A+G+J)/3 + rand(d) N = (A+F+J)/3 + rand(d)
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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 - (randomnumber) x --^—^ )2-~2—
- 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 decreasingthe width of the bell-shaped distribution the terrain becomes steeper
Outer Boundaries
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
Loop the Old Terrain I-^
Upon reaching the right side of the landscape, a tile of ter in front of the player to provide the illusion of endles
Infinite Terrain
Upon reaching the right side of terrain grid A, the values of the far-right points are copied tothe far left points of terrain grid B
Terrain gridA
A
Terrain grid H
OpenGL Fractal Terrain Video
The other points of terrain grid B are then calculated via randomization and mid-point displacement, as done for grid A
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Other Techniques
• 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
• A better choice would be to use Multifractals - Proposed by Kenton Musgrave
• These are fractals whose dimension/roughness varies with location
Other Multifractals
• 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
Cracked Terrains
Multifractal Code Example
* Procedural multifractal evaluated at "point."
* Parameters:
* "H™ determines the highest fractal dimenslor
* "1 aeiinarlty" Is gap between successive frequencies
* "octaves" is the number of frequencies in trie fEm
* "offset" is the zero offset, which determines irwltifracta 1 itj */
double
multifractaU Vector point, double h. double lacunarity, int octaves, double offset )
■
double      value, NofseO: value - 1.0:
far (1nt i-u; ■Koctaves; {
value *- (Noise( point J + offset) * dow* lacunarlty. -K'i ); point *- 1acunarity:
:■
return value:
Other Multifractal Examples
Ridged multifractal terrains : taken from Texturing and Modeling: A Procedural Approach pg 518 (left) pg 480(right)
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More Controls
More Controls
Multiply by a Guassian filter to limit the mountain range
To create valleys, create a lower amplitude and rather smooth terrain to use as the max operator
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Ridges
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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
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 downwards
Fault Line Algorithm
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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
Fault Line Algorithm Examples
rierating Fault Lines in a Height Field Grid
100 iterations
400 iterations
Randomly pick two grid points px 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
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Filtering Height Fields
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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
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Variations to the Fault Line Algorithm
Cosine
		(	
disi <0			
Particle Deposition
Generated after 5 series of 1000 iterations
Perlin Noise
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
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Terrain Coloring
Rolling Hills
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More advanced coloring is based on altitude and slope
• 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
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Realistic Terrain Video 2 Erosion
Erosion by Water Erosion by Wind and Water
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Hydraulic Erosion Video
Case Study - Fractal Nature
Procedural Fractal Nature
Architecture
Simulate a selection of naturally occurring phenomena into a user friendly tool
Provide a set of techniques that can be used for creating realistic terrains for games
- i.e. flight simulators, action and strategy games
Emphasis on:
- Generating a realistic random terrain
- Implement hydraulic and thermal erosion
- Evaluate the tool
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Developed on Unity 3D game engine
- Can export the terrain into a file format Two visualisation modes:
- 'Flight Simulator1 and 'Free View'
GUI Overview Fractal Turbulence
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Terrain Filtering Thermal Erosion
Thermal erosion effects:
- Appearance of accentuated slopes
• Present natural-looking ridges
- Flat plateaus
• Where sediment tends to deposit
Implementation based on (Musgrave et al., 1989), (Jako and Toth, 2011)
Hydraulic Erosion
• Hydraulic erosion refers to the natural process in which the motion of fluids (specifically water) produces mechanical weathering over the soil - (Mei et al, 2007), (Anh et al., 2007), (Stava et al., 2008), (Jako and Toth, 2011)
■q=£>kTl^Pil« i .
;,i 1.1 ,.< wm a* ,
Virtual pipes representation (Mei et al, 2007)
Rain Simulation & Evaporation
The rainfall causes the water levels to rise up at random positions
Water movement by allowing water to escape
- Approximation of the sediment amount that is to be eroded
Water evaporation based on environmental heat
Video
Applications (V5-Games2015], IEEE D.   . iterSoc ?iy, Sk )vde, Sweden, 15-13 5eptember,B4-91, 2015.
Evaluation
• Qualitative evaluation with 12 expert users
- 6 academics with computer graphics/virtual reality background
- 6 industry professionals working in game companies
• Three stage evaluation:
- GUI exploration
- Simple flight simulator game
- Comparison with real landscapes
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Thermal erosion evaluation Hydraulic erosion evaluation
(a) Medium temperature values comparison (b) High temperature values comparison (a) Low density + heavy erosion comparison (b) Low density + high viscosity comparison
Academics Feedback
• Improve the visual realism of the terrain
- More accurate algorithms for hydraulic erosion
- Making the software open-source
- Compare these methods with Perlin noise generator
- Port the implementation (or part of it) on GPUs
- Implement fluid simulation for the representation of sea, rivers and lakes
- Stochastic smoothing algorithms to make the landscapes look more realistic
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Industry Feedback
• The level of realism quite satisfying and appropriate for the generation of interactive video games
- Strong similarities with real life landscapes
• All of them stated that the generated terrains cannot be used without manual editing into modern games
- Larger studios mentioned that they are using similar procedural approaches but then they perform a lot of editing
- Independent studios stated that they could use the generated terrain as they are
• More procedural elements such as different types of trees and plants
• Better texturing techniques
Conclusions
Real terrain data still used in games
— Very expensive
— Need a lot of manual editing Procedural are getting more and more attention
— More work is required to make them more realistic
References
https://www.eooele.co. uk/#q=ordnance+survey http://ned.uses.eov/
http://en.wikipedia.ore/wiki/Dieital elevation model http://wiki.eis.com/wiki/index.php/Contour line http://wiki.eis.com/wiki/index.php/Dieital Elevation Model http://www.edcvault.com/play/1277/HALO-WARS-The-Terrain-of http://davis.wpi.edu/~matt/courses/fractals/brownian.html
http://code.eooele.eom/p/fractalterraineeneration/wiki/Fractional Brownian Motion http://www8.cs.umu.se/kurser/TDBD12/HT01/pa pers/Muserave TerrainOO.pdf http://www.liehthouse3d.com/openel/terrain/index.php3Pfault http://www.decarpentier.nl/downloads/lnteractivelvSvnthesizineAndEditineVirtualOu tDoorTerrain report.pdf
http://www.eameproerammer.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.Texturingand Modeling: A Procedural Approach, 3rd edition. USA. Morgan Kaufman Publishers, 2003
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Bibliography Questions
DeLoura, Mark. Game Programming Gems. Charles River Media, 2002.
Martz, Paul. "Generating Random Fractal Terrain." Game Programmer. Publisher Robert C
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