INTERSECTIONS -- TOPOLOGY, ACCURACY, & NUMERICS FOR GEOMETRIC OBJECTS. I-TANGO III. NSF/DARPA. Intellectual Integration of Project Team . New conceptual model (Stewart - UConn) Intersection improvements (Sakkalis – MIT) Polynomial evaluation (Hoffmann – Purdue)
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Neil Stewart, Université de Montréal
with L.-E. Andersson and M. Zidani
Thanks: T. J. Peters and J. Bisceglio
Well-conditionedWell- and ill-conditioned problems
Well-conditionedStable numerical methods
I-TANGO: Intersections-Topology, Accuracy and Numerics for Geometric Objects
N. M. Patrikalakis
K. H. Ko
Massachusetts Institute of Technology
Transversal intersection of rational parametric surfaces
Self intersection of a bi-cubic surface
Intersection of a hyperbolic surface with a plane
Tangential intersections of parametric surfaces
Validated ODE solver can correctly trace the intersection curve segment even through closely spaced features, where standard methods fail.
Result from a validated interval scheme
Christoph Hoffmann, Purdue University
Neil Stewart, University of Montreal
Gahyun Park, Purdue
J.-R. Simard, Montreal
Method converges to the true solution even when the residuals are inaccurately computed
Block floating-point arithmetic used in Wilkinson’s proof
Accurate evaluation may require more when there are multiple roots, and the accurate inner product is essential in those situations.
p1: Horner = 0, others = 10-32, enc=3ulp
p2: Horner = 8·10-14, others = 10-48, enc=1ulp
p3: Horner = 1.02·10-12, others = 10-64, enc=1ulp
p4: Horner within 1 ulp of accurate value
p5 – p7: Horner within 2 ulp of accurate value
NACA Airfoils (~1915) Equations
Roy Liming Equations would boast, with pardonable hyperbole, that the Britain-based Mustangs could fly to Berlin and back because their surface contours did not deviate from the mathematical ideal.
Geometric variation presents a new Equations challenge for geometry systems and intersections: Smooth Morphing
For a approximately 1, the approximation fluctuates between a constant spline and a spline with one interior knot, i.e., changing the parameter a may cause the model to change discontinuously.
Approximating f(x) = ax2
A Final Thought: We are seeing a fundamental change in the use of geometry
Previously, geometry was used solely to document what was built. Geometry had to match the product.
Now, it’s the product that has to match the geometry: Liming’s goal.
1. Invited tutorial: Effective Computational Geometry
2. Talk at ICIAM (Australia, 2003)
a. Three papers, Italy, Solids & Shapes
b. Dagstuhl Seminar