12,843 research outputs found
Symmetry Detection of Rational Space Curves from their Curvature and Torsion
We present a novel, deterministic, and efficient method to detect whether a
given rational space curve is symmetric. By using well-known differential
invariants of space curves, namely the curvature and torsion, the method is
significantly faster, simpler, and more general than an earlier method
addressing a similar problem. To support this claim, we present an analysis of
the arithmetic complexity of the algorithm and timings from an implementation
in Sage.Comment: 25 page
Smooth quasi-developable surfaces bounded by smooth curves
Computing a quasi-developable strip surface bounded by design curves finds
wide industrial applications. Existing methods compute discrete surfaces
composed of developable lines connecting sampling points on input curves which
are not adequate for generating smooth quasi-developable surfaces. We propose
the first method which is capable of exploring the full solution space of
continuous input curves to compute a smooth quasi-developable ruled surface
with as large developability as possible. The resulting surface is exactly
bounded by the input smooth curves and is guaranteed to have no
self-intersections. The main contribution is a variational approach to compute
a continuous mapping of parameters of input curves by minimizing a function
evaluating surface developability. Moreover, we also present an algorithm to
represent a resulting surface as a B-spline surface when input curves are
B-spline curves.Comment: 18 page
Detecting Similarity of Rational Plane Curves
A novel and deterministic algorithm is presented to detect whether two given
rational plane curves are related by means of a similarity, which is a central
question in Pattern Recognition. As a by-product it finds all such
similarities, and the particular case of equal curves yields all symmetries. A
complete theoretical description of the method is provided, and the method has
been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
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