Applications of Computational Geometry

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Applications of Computational Geometry

• COSC 2126
• Computational Geometry

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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/articles/ray
tracing/images/theiere.jpg http//graphics.cs.uni
-sb.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 h
ttp//skagit.meas.ncsu.edu/helena/classwork/topic
s/F1a.gif
Spatial elevation model
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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 d-dimensional 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.
http//www.grc.nasa.gov/WWW/RT/2003/7000/7740moral
es.html
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FEM (2)

• Truck crash simulation.

http//en.wikipedia.org/wiki/Finite_element_method
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Photorealism in Computer Graphics
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Meshing in Game Graphics
http//www.math.tu-berlin.de/geometrie/gallery/vr/
bilder/FarCry0001.jpg
http//graphics.ethz.ch/mattmuel/projects/project
.htm
http//www.gamingtarget.com/images/media/Specials/
Essential_Tech_Terminology_For_Gamers/page/p002.jp
g
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Meshing in Game Graphics (2)
Finding Next Gen CryEngine 2, Martin
Mittring14, Crytek GmbH
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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
Final RBF model
Courtesy Derek Cool, Robarts Research Imaging
Laboratories
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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)
(Earths 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 satellites
  position reaches the receiver.
• The receivers clock is generally not
  synchronized to the satellite.

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Computing Position Using GPS

• Dr Deviation of receivers clock from the
  actual time.
• Ti Timestamp received from satellite i.
• di Real distance between the receiver and
  satellite i.

However,
?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
www.physorg.com/news138885789.html
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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 1n
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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).

http//mcl1.ncifcrf.gov/integrase/asv_secstr.html
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Geometric Representation of Proteins Tertiary
Structure

• Tertiary structure refers to the proteins 3D
  shape.
• It is determined by the proteins 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.

http//www.cryst.bbk.ac.uk/PPS2/course/section12/h
aemogl2.html
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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