This document contains a listing of some of my publications.

- Envelopes of Nonlinear Geometry
- Smooth paths in a polygonal channel
- Tight linear envelopes for splines
- Computing linear envelopes for uniform B--spline curves
- Sharp, quantitative bounds on the distance between a polynomial piece and its Bezier control polygon

Envelopes of Nonlinear Geometry
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Ph.D. Thesis | ||

David Lutterkort |
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Paper
| 1999-11-09
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A general framework for comparing objects commonly used to represent
nonlinear geometry with simpler, related objects, most notably their
control polygon, is provided. The framework enables the efficient
computation of bounds on the distance between the nonlinear geometry
and the simpler objects and the computation of envelopes of nonlinear
geometry.
The framework is used to compute envelopes for univariate
splines, the four point subdivision scheme, tensor product
polynomials and Bezier triangles.
The envelopes are used to approximate solutions to continuously
constrained optimization problems.
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Available Formats: | PDF | gzipped Postscript |

Smooth paths in a polygonal channel
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Extended Abstract for SCG '99 in Miami | ||

David Lutterkort and Jörg Peters |
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Paper
| 1998-11-20
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Appeared in Proceedings of the Fifteenth Annual
Symposium on Computational Geometry, Miami Beach, Florida, 1999
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We show how to efficiently smooth a polygon with an
approximating spline that stays to one side of the polygon. We also
show how to find a smooth spline path between two polygons that form a
channel. Problems of this type arise in many physical motion planning
tasks where not only forbidden regions have to be avoided but also a
smooth traversal of the motion path is required. Both algorithms are
based on a new tight and efficiently computable bound on the distance
of a spline from its control polygon and employ only standard linear
and quadratic programming techniques.
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Available Formats: | gzipped Postscript |

Tight linear envelopes for splines
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David Lutterkort and Jörg Peters |
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Paper
| 1998-11-20
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To appear in Numerische Mathematik
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A sharp bound on the
distance between a spline and its B-spline control polygon is
derived. The bound yields a piecewise linear envelope enclosing spline
and polygon. This envelope is particularly simple for uniform splines and
splines in Bernstein-B\'ezier form and shrinks by a factor of~4 for each
uniform subdivision step. The envelope can be easily and efficiently
implemented due to its explicit and constructive nature.
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Available Formats: | gzipped Postscript |

Computing linear envelopes for uniform B--spline curves
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Submission for SIAM Student Paper Prize | ||

David Lutterkort |
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Paper
| 1999-02-03
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Submitted to
Proceedings St. Malo
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We derive a new, efficiently computable bound on the distance
between a uniform spline and its B--Spline control polygon in terms of
the second differences of the control points. The bound is piecewise
linear and sharp for quadratic and cubic splines and decreases by a
factor of 4 under uniform refinement. Using this bound, we describe a
simple algorithm for enveloping parametric curves.
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Available Formats: | Extended Abstract (ps) | Full paper(ps) |

Sharp, quantitative bounds on the distance between a polynomial
piece and its Bezier control polygon
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D. Nairn and J. Peters and D. Lutterkort |
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Paper
| 1998-09-01
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Appeared in
Computer Aided Geometric Design 16 (1999) pp. 613-631
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The maximal distance between a Bezier segment and its control
polygon is bounded in terms of the differences of the control point
sequence and a constant that depends only on the degree of the
polynomial. The constants derived here for various norms and orders of
differences are the smallest possible.
In particular, the bound in terms of the maximal absolute second
difference of the control points is a sharp bound for the Hausdorff
distance between the control polygon and the curve segment. It provides
a straightforward proof of the quadratic convergence of the sequence of
control polygons to the Bezier segment under subdivision or degree-fold
degree-raising, and establishes the explicit convergence constants, and
allows analyzing the optimal choice of the subdivision parameter for
adaptive refinement of quadratic and cubic segments and yields
efficient bounding regions.
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Available Formats: | gzipped Postscript |