Applications of
Computational Geometry
COSC 2126
Computational Geometry
Outline
 General categories of computational
geometry application domains.
 Triangulation and meshing
 Geocomputation
 Computational biology
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Application Domains
 Computer graphics
 2-D and 3-D intersections.
 Hidden surface elimination.
 Ray tracing.
 Virtual reality
 Collision detection (intersection).
http://www.linuxgraphic.org/section3d/article
s/raytracing/images/theiere.jpg
http://graphics.cs.unisb.de/Publications/2006/RTG/spheres.jpg
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Application Domains (2)
 Robotics
Motion planning, assembly
orderings, collision
detection, shortest path
finding
 Global information systems
(GIS)
 Large data sets  data
structure design.
 Overlays  Find points in
multiple layers.
 Interpolation  Find
additional points based on
values of known points.
 Voronoi diagrams of points.

http://mathworld.wolfram.com/VoronoiDiagram.html
http://skagit.meas.ncsu.edu/~helena/classwork/topics/F1a.gif
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Spatial elevation model
Application Domains (3)
 Computer aided design and manufacturing (CAD /
CAM)
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Design and manipulate 3-D objects.
 Possible manipulations: merge (union), separate,
move.
“Design for assembly”
 CAD/CAM provides a test on objects for ease of
assembly, maintenance, etc.
 Computational biology
 Determine how proteins combine together based on
folds in structure.

Surface modeling, path finding, intersection.
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Triangulation and Meshing
 Used to generate surfaces and solids from
unstructured data (point clouds).
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Surfaces  triangles
Solids  tetrahedra
 Important in most sciences:
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Medical imaging.
Engineering – finite element modeling.
Art.
Computer games.
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Delaunay Triangulation
 Delaunay triangulation for a set
P of points in the plane is a
triangulation DT(P) s.t. no point
in P is inside the circumcircle of
any triangle in DT(P).
 The Delaunay triangulation of a
discrete point set P corresponds
to the dual graph of the Voronoi
tessellation for P.
 For a set P of points in ddimensional Euclidean space,
DT(P) is s.t. no point in P is inside
the circum-hypersphere of any
simplex in DT(P).
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Finite Element Method
Stress distributions on the
foot.
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http://www.grc.nasa.gov/WWW/RT/2003/7000/7740morales.html
FEM (2)
 Truck crash simulation.
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http://en.wikipedia.org/wiki/Finite_element_method
Photorealism in Computer Graphics
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Meshing in Game Graphics
http://www.math.tu-berlin.de/geometrie/gallery/vr/bilder/FarCry0001.jpg
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http://graphics.ethz.ch/~mattmuel/projects/project.htm
http://www.gamingtarget.com/images/media/Specials/Essential_Tech_Terminology_For_Gamers/page/p002.jpg
Meshing in Game Graphics (2)
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Finding Next Gen – CryEngine 2, Martin Mittring14, Crytek GmbH
Surface Reconstruction With
Radial Basis Functions
Scanning a bone
section with a laser
scanner.
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Surface Reconstruction With
Radial Basis Functions (2)
Point cloud
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Surface Reconstruction With
Radial Basis Functions (3)
Final surface
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Scattered Point Interpolation with
Radial Basis Functions
Interpolate
scattered
points
Original point cloud from segmented
contours in CT volume.
Enhanced point cloud
Radial basis
interpolation
Surface normals
Courtesy: Derek Cool, Robarts Research Imaging Laboratories
Final RBF model
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Geocomputation
 Geocomputation – a new paradigm for
multidisciplinary/interdisciplinary research that enables
the exploration of extremely complex and previously
unsolvable problems in geography.
 Used to study spatial data:
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Population distributions.
Movement patterns of migratory animals.
Locations of natural resources.
Epidemiology.
Source and extent of environmental pollution and
contamination.
Extent of natural disasters.
Many other applications.
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Geocomputation (2)
 Geocomputation depends on the
contributions of many fields of study:
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Computational geometry.
Interactive exploratory data analysis.
Data mining.
Numerical methods.
Graphics and visualization.
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Geographic Information Systems
 Also known as geomatics – the application of
computational methods and systems to geographical
problems.
 Computational geometry provides useful tools and
algorithms for GIS, including:
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Data correction (after data acquisition and input).
Data retrieval (through queries).
Data analysis (e.g map overlay and geostatistics).
Data visualization (for maps and animations).
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Global Positioning System (GPS)
 Global positioning system (GPS) – A
specialized, dedicated distributed system for
determining geographical position anywhere
on Earth.
 Satellite-based system launched in 1978.
 Initially for military applications, but extended
for civilian use (traffic navigation), and other
tracking uses.
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GPS (2)
 29 satellites, each circulating in
an orbit at height  20,000 km,
and having up to four regularly
calibrated atomic clocks.
 Each satellite (i) continuously
broadcasts its position (xi, yi,
zi), and timestamps each
message.
 This allows every receiver on
Earth to accurately compute its
own position using three
satellites.
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Location Calculation
 For the GPS receiver to locate itself, two data are
needed:
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The location of at least 3 reference satellites.
The distance between the receiver and each of those satellites.
 The receiver obtains both of these by analyzing high-
frequency, low-power radio waves from the GPS
satellites.
 Because radio waves travel at the speed of light,
receivers can calculate the distance the wave traveled
by the amount of time it took to travel.
 Each GPS receiver contains a database of the
locations of each satellites at a given time.
 Using this information, the receiver uses trilateration to
find the exact spot on earth.
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GPS (3)
 Trilateration – a method for determining the intersections of
three sphere surfaces given the centers and radii of the three
spheres.
(Altitude)
(Earth’s surface at sea level)
Computing a position in a two-dimensional space.
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Time Calculation
 Each satellite tracks time by an atomic clock.
  They are all synchronized.
 Upon receiving the signal from the satellites,
the receiver can calculate the time delay of
each, providing the travel time.
 By multiplying the travel time by the speed of
light, the distances of the satellites are
obtained.
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GPS (4)
 Principle of intersecting circles can be re-
formulated to 3D.
 Three (3) satellites are needed to compute
the longitude, latitude, and altitude of a
receiver on Earth.
 Real world facts that complicate GPS:
Some time elapses before data on a
satellite’s position reaches the receiver.
2. The receiver’s clock is generally not
synchronized to the satellite.
1.
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Computing Position Using GPS
 Dr: Deviation of receiver’s clock from the actual time.
 Ti: Timestamp received from satellite i.
 di: Real distance between the receiver and satellite i.
However,
di  cTnow  Ti  D r   ( xi  xr ) 2  ( yi  yr ) 2  ( zi  zr ) 2
4 equations (3 satellites + time difference) are
needed to solve for four unknowns, xr, yr, zr, and Dr.
 GPS can also be used for synchronization.
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Limitations of CG w.r.t. GIS
 Computational geometry algorithms are often very
complex, and require a large effort to implement.
 Efficiency analysis, which is based on worst-case
inputs to the algorithm, is often performed.
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The theoretical worst-case data sets may be rather
artificial, and never appear in real-world applications.
 Another problem lies in the abstraction of the original
problem, in which several criteria to be met “at least
to some extent” simultaneously.
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This leads to vague problem statements, but CG
generally considers well-defined, simple-to-state
problems.
This problem will be more difficult than the first two.
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Bioinformatics – Protein Folding
 Proteins are large 3D molecules with
complicated geometries and topologies.
 Basic idea – Create “designer proteins” that
can be used to treat a variety of disease
conditions.
 Lock-and-key mechanism – proteins have
binding sites where other ions or molecules
form chemical bonds.
 Proteins can therefore bind to harmful
pathogens (e.g. viruses), rendering them
harmless.
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Protein Binding to a Pathogen
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www.physorg.com/news138885789.html
Geometric Representation of Proteins
– Primary Structure
 The primary structure of a protein is its sequence of
amino acids, which determines what the protein does,
how it interacts with other proteins, and how it folds.
Sequence of amino acids and peptide
bonds.
3-D curve
{vi}, i = 1…n
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Geometric Representation of Proteins –
Secondary Structure
 Secondary structure refers to the way a single
protein (macromolecule) folds together.
 Secondary structure consists of helix (helices),
strand(s), and random coil(s).
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http://mcl1.ncifcrf.gov/integrase/asv_secstr.html
Geometric Representation of Proteins –
Tertiary Structure
 Tertiary structure refers to the protein’s 3D shape.
 It is determined by the protein’s primary structure.
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Geometric Representation of Proteins –
Quaternary Structure
 Quaternary structure refers to the arrangement of
multiple folded protein molecules in a multi-subunit
complex.
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http://www.cryst.bbk.ac.uk/PPS2/course/section12/haemogl2.html
Protein Folding
 The “grand challenge” in bioinformatics and
proteomics.
 Allows the transition from sequence to
structure.
 Currently, relatively simple computational
folding models have proven to be NP
complete even in the 2D case!
 Example.
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Other Non-Traditional Applications
 Spatial databases.
 Radiation therapy planning.
 Computational topology.
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Use of geometry and topology to study
complex and massive data sets.
Applications range from medical, GIS,
CAD/CAM, and crystallography to financial
and economic models, music, and quantum
computing.
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End
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