1,911 research outputs found
Coaxing a planar curve to comply
AbstractA long-standing problem in computer graphics is to find a planar curve that is shaped the way you want it to be shaped. A selection of various methods for achieving this goal is presented. The focus is on mathematical conditions that we can use to control curves while still allowing the curves some freedom. We start with methods invented by Newton (1643–1727) and Lagrange (1736–1813) and proceed to recent methods that are the subject of current research. We illustrate almost all the methods discussed with diagrams. Three methods of control that are of special interest are interpolation methods, global minimization methods (such as least squares), and (Bézier) control points. We concentrate on the first of these, interpolation methods
Instabilities of Spiral Shocks I: Onset of Wiggle Instability and its Mechanism
We found that loosely wound spiral shocks in an isothermal gas disk caused by
a non-axisymmetric potential are hydrodynamically unstable, if the shocks are
strong enough. High resolution, global hydrodynamical simulations using three
different numerical schemes, i.e. AUSM, CIP, and SPH, show similarly that
trailing spiral shocks with the pitch angle of larger than ~10 deg wiggle, and
clumps are developed in the shock-compressed layer. The numerical simulations
also show clear wave crests that are associated with ripples of the spiral
shocks. The spiral shocks tend to be more unstable in a rigidly rotating disk
than in a flat rotation. This instability could be an origin of the secondary
structures of spiral arms, i.e. the spurs/fins, observed in spiral galaxies. In
spite of this local instability, the global spiral morphology of the gas is
maintained over many rotational periods. The Kelvin-Helmholtz (K-H) instability
in a shear layer behind the shock is a possible mechanism for the wiggle
instability. The Richardson criterion for the K-H stability is expressed as a
function of the Mach number, the pitch angle, and strength of the background
spiral potential. The criterion suggests that spiral shocks with smaller pitch
angles and smaller Mach numbers would be more stable, and this is consistent
with the numerical results.Comment: 11 pages, 14 figures, to be published in MNRAS, high quality figures
can be downloaded from http://th.nao.ac.jp/~wada/paperlist.htm
Fast and accurate clothoid fitting
An effective solution to the problem of Hermite interpolation with a
clothoid curve is provided. At the beginning the problem is naturally
formulated as a system of nonlinear equations with multiple solutions that is
generally difficult to solve numerically. All the solutions of this nonlinear
system are reduced to the computation of the zeros of a single nonlinear
equation. A simple strategy, together with the use of a good and simple guess
function, permits to solve the single nonlinear equation with a few iterations
of the Newton--Raphson method.
The computation of the clothoid curve requires the computation of Fresnel and
Fresnel related integrals. Such integrals need asymptotic expansions near
critical values to avoid loss of precision. This is necessary when, for
example, the solution of interpolation problem is close to a straight line or
an arc of circle. Moreover, some special recurrences are deduced for the
efficient computation of asymptotic expansion.
The reduction of the problem to a single nonlinear function in one variable
and the use of asymptotic expansions make the solution algorithm fast and
robust.Comment: 14 pages, 3 figures, 9 Algorithm Table
Applying inversion to construct rational spiral curves
A method is proposed to construct spiral curves by inversion of a spiral arc
of parabola. The resulting curve is rational of 4-th order. Proper selection of
the parabolic arc and parameters of inversion allows to match a wide range of
boundary conditions, namely, tangents and curvatures at the endpoints,
including those, assuming inflection.Comment: 18 pages, 11 figure
NGC 1300 Dynamics: I. The gravitational potential as a tool for detailed stellar dynamics
In a series of papers we study the stellar dynamics of the grand design
barred-spiral galaxy NGC~1300. In the first paper of this series we estimate
the gravitational potential and we give it in a form suitable to be used in
dynamical studies. The estimation is done directly from near-infrared
observations. Since the 3D distribution of the luminous matter is unknown, we
construct three different general models for the potential corresponding to
three different assumptions for the geometry of the system, representing
limiting cases. A pure 2D disc, a cylindrical geometry (thick disc) and a third
case, where a spherical geometry is assumed to apply for the major part of the
bar. For the potential of the disc component on the galactic plane a Fourier
decomposition method is used, that allows us to express it as a sum of
trigonometric terms. Both even and odd components are considered, so that the
estimated potential accounts also for the observed asymmetries in the
morphology. For the amplitudes of the trigonometric terms a smoothed cubic
interpolation scheme is used. The total potential in each model may include two
additional terms (Plummer spheres) representing a central mass concentration
and a dark halo component, respectively. In all examined models, the relative
force perturbation points to a strongly nonlinear gravitational field, which
ranges from 0.45 to 0.8 of the axisymmetric background with the pure 2D being
the most nonlinear one. We present the topological distributions of the stable
and unstable Lagrangian points as a function of the pattern speed .
In all three models there is a range of values, where we find
multiple stationary points whose stability affects the overall dynamics of the
system.Comment: 14 pages, 11 figures, published in MNRA
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