Tomasz Popiel

Bézier curves and C2 interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bezier curves are C^~ curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bezier curve of arbitrary degree and use the formulae to express the curves control points in terms of these quantities. These results allow generalised Bezier curves to be pieced together into C^2 splines, and thereby allow C^2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C^2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.

C2 spherical Bézier splines

The classical de Casteljau algorithm for constructing Bezier curves can be generalised to a sphere of arbitrary dimension by replacing line segments with shortest great circle arcs. The resulting spherical Bezier curves are C^~ and interpolate the endpoints of their control polygons. In the present paper, we address the problem of piecing these curves together into C^2 splines. For this purpose, we compute the endpoint velocities and accelerations of a spherical Bezier curve of arbitrary degree and use the formulae to define control points that give the curve a desired initial velocity and acceleration. In addition, for uniform splines we establish a simple relationship between the control points of neighbouring curve segments that is necessary and sufficient for C^2 continuity. As illustration, we solve an interpolation problem involving sparse data using both the present method and a normalised polynomial interpolant. The normalised spline exhibits large variations in speed and magnitude of acceleration, whilst the spherical Bezier spline is far better behaved. These considerations are important in applications where velocities and accelerations need to moderated or estimated, notably computer animation and rigid body trajectory planning, where interpolation in the 3-sphere is a fundamental task.

Geometry for robot path planning

There have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out. Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical constructions that imitate well-known constructions from classical approximation theory. Part III reviews a class of methods where classical criteria are extended to curves in Riemannian manifolds, including several recent mathematical results that have not yet found their way into the literature.

ELASTICA IN SO(3)

In a Riemannian manifold M, elastica are solutions of the Euler�Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3 -space. For compact G, we show that elastica extend to the whole real line. For G=SO(3) , we solve the Euler�Lagrange equation by quadratures.

Quadratures and cubics in SO(3) and SO(1,2)

Then we shall solve (1) by a quadrature for x(t) in terms of V (t). Differential equations of the form (2) are called Lax equations, and their solutions (Z , V ) Lax pairs. Lax equations are fundamental in the study of integrable systems since if a differential equation can be put in the Lax pair form (2), the spectrum of Z is preserved by the flow. In particular, the coefficients of the characteristic polynomial of Z are conserved quantities. Lax equations appear in many classical mechanical systems; a well-known example is Euler’s equation describing geodesics in a Lie group equipped with a left-invariant Riemannian metric. They are also central to the application of inverse scattering theory to the Korteweg–de Vries and non-linear Schrödinger equations. For more background, see Babelon et al. (2003) or Hitchin et al. (1999). In the present paper, solutions of Lax equations (2) are used to solve associated equations of the form (1). As illustration, we look at applications to variational curves in SO(3) and SO(1,2) equipped with bi-invariant semi-Riemannian metrics. In Section 3, we consider • Riemannian cubics, studied in Camarinha (1996), Camarinha et al. (1995, 2001), Crouch & Silva Leite (1995), Giambo et al. (2002), Noakes (2003, 2004a,b) and Noakes et al. (1989),

Abundant p-singular elements in finite classical groups

Abstract In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of | G | distinct from the characteristic, the proportion of p-singular elements in G (elements with order divisible by p) is at least a constant multiple of ( 1 − 1 / p ) / e , where e is the order of q modulo p. Motivated by algorithmic applications, we define a subfamily of p-singular elements, called p-abundant elements, which leave invariant certain ‘large’ subspaces of the natural G-module. We find explicit upper and lower bounds for the proportion of p-abundant elements in G, and prove that it approaches a (positive) limiting value as the dimension of G tends to infinity. It turns out that the limiting proportion of p-abundant elements is at least a constant multiple of the Isaacs–Kantor–Spaltenstein lower bound for the proportion of all p-singular elements.

On semiregular permutations of a finite set

In this paper we establish upper and lower bounds for the proportion of permutations in symmetric groups which power up to semiregular permutations (permutations all of whose cycles have the same length). Provided that an integer n has a divisor at most d, we show that the proportion of such elements in Sn is at least cn−1+1/2d for some constant c depending only on d whereas the proportion of semiregular elements in Sn is less than 2n−1.

On parametric smoothness of generalised B-spline curves

In a Riemannian manifold, generalised B-spline curves are piecewise C^~ curves defined by a generalisation of the classical Cox-de Boor algorithm, in which line segments are replaced by minimal geodesics. Their applications include rigid body motion planning and computer graphics. We prove that, like classical B-spline curves, they are C^1 at knots of multiplicity at most m-1, where m is the degree. We then compute the difference between their left and right (covariant) accelerations at knots of multiplicity at most m-2. Unlike classical B-spline curves, generalised B-spline curves are not in general C^2 at such knots

Point-primitive, line-transitive generalised quadrangles of holomorph type

Abstract Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set 𝒫

ALGORITHMS TO IDENTIFY ABUNDANT p-SINGULAR ELEMENTS IN FINITE CLASSICAL GROUPS

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