1,209 research outputs found
B\'ezier curves that are close to elastica
We study the problem of identifying those cubic B\'ezier curves that are
close in the L2 norm to planar elastic curves. The problem arises in design
situations where the manufacturing process produces elastic curves; these are
difficult to work with in a digital environment. We seek a sub-class of special
B\'ezier curves as a proxy. We identify an easily computable quantity, which we
call the lambda-residual, that accurately predicts a small L2 distance. We then
identify geometric criteria on the control polygon that guarantee that a
B\'ezier curve has lambda-residual below 0.4, which effectively implies that
the curve is within 1 percent of its arc-length to an elastic curve in the L2
norm. Finally we give two projection algorithms that take an input B\'ezier
curve and adjust its length and shape, whilst keeping the end-points and
end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure
A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve
We derive a variational model to fit a composite B\'ezier curve to a set of
data points on a Riemannian manifold. The resulting curve is obtained in such a
way that its mean squared acceleration is minimal in addition to remaining
close the data points. We approximate the acceleration by discretizing the
squared second order derivative along the curve. We derive a closed-form,
numerically stable and efficient algorithm to compute the gradient of a
B\'ezier curve on manifolds with respect to its control points, expressed as a
concatenation of so-called adjoint Jacobi fields. Several examples illustrate
the capabilites and validity of this approach both for interpolation and
approximation. The examples also illustrate that the approach outperforms
previous works tackling this problem
High-order adaptive methods for computing invariant manifolds of maps
The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
Preserving Derivative Information while Transforming Neuronal Curves
The international neuroscience community is building the first comprehensive
atlases of brain cell types to understand how the brain functions from a higher
resolution, and more integrated perspective than ever before. In order to build
these atlases, subsets of neurons (e.g. serotonergic neurons, prefrontal
cortical neurons etc.) are traced in individual brain samples by placing points
along dendrites and axons. Then, the traces are mapped to common coordinate
systems by transforming the positions of their points, which neglects how the
transformation bends the line segments in between. In this work, we apply the
theory of jets to describe how to preserve derivatives of neuron traces up to
any order. We provide a framework to compute possible error introduced by
standard mapping methods, which involves the Jacobian of the mapping
transformation. We show how our first order method improves mapping accuracy in
both simulated and real neuron traces under random diffeomorphisms. Our method
is freely available in our open-source Python package brainlit
Constructing G1 quadratic Bezier curves with arbitrary endpoint tangent vectors
International audienceQuadratic BĂ©zier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic BĂ©zier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing G1 quadratic BĂ©zier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the G1 quadratic BĂ©zier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs
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