Applications of Computational Geometry (presentation by Dr. Mark
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Transcript Applications of Computational Geometry (presentation by Dr. Mark
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)
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).
Surfaces triangles
Solids tetrahedra
Important in most sciences:
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:
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:
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:
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:
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 cTnow 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.
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.
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.
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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