Ueber eine Stelle in Jacobis Aufsatz "Observatiunculae ad theoriam aequationum pertinentes"

L. Kronecker
1891 Journal für die Reine und Angewandte Mathematik  
Acta Technologica Agriculturae 1/2016 Dušan Páleš et al. The most effective way for determination of curves for practical use is to use a set of control points. These control points can be accompanied by other restriction for the curve, for example boundary conditions or conditions for curve continuity (Sederberg, 2012) . When a smooth curve runs only through some control points, we refer to curve approximation. The B-spline curve is one of such approximation curves and is addressed in this
more » ... dressed in this contribution. A special case of the B-spline curve is the Bézier curve Rédl et al., 2014) . The B-spline curve is applied to a set of control points in a space, which were obtained by measurement of real vehicle movement on a slope (Rédl, 2007 (Rédl, , 2008 . Data were processed into the resulting trajectory (Rédl, 2012; Rédl and Kučera, 2008) . Except for this, the movement of the vehicle was simulated using motion equations (Rédl, 2003; Rédl and Kročko, 2007) . B-spline basis functions Bézier basis functions known as Bernstein polynomials are used in a formula as a weighting function for parametric representation of the curve (Shene, 2014). B-spline basis functions are applied similarly, although they are more complicated. They have two different properties in comparison with Bézier basis functions and these are: 1) solitary curve is divided by knots, 2) basis functions are not nonzero on the whole area. Every B-spline basis function is nonzero only on several neighbouring subintervals and thereby it is changed only locally, so the change of one control point influences only the near region around it and not the whole curve. These numbers are called knots, the set U is called the knot vector, and the half-opened interval 〈u i , u i + 1 ) is the i-th knot span. Seeing that knots u i may be equal, some knot spans may not exist, thus they are zero. If the knot u i appears p times, hence u i = u i + 1 = ... = u i + p -1 , where p >1, u i is a multiple knot of multiplicity p, written as u i (p). If u i is only a solitary knot, it is also called a simple knot. If the knots are equally spaced, i.e. (u i + 1 -u i ) = constant, for every 0 ≤ i ≤ (m -1), the knot vector or knot sequence is said uniform, otherwise it is non-uniform. Knots can be considered as division points that subdivide the interval 〈u 0 , u m 〉 into knot spans. All B-spline basis functions are supposed to have their domain on 〈u 0 , u m 〉. We will use u 0 = 0 and u m = 1. To define B-spline basis functions, we need one more parameter k, which gives the degree of these basis functions. Recursive formula is defined as follows: This definition is usually referred to as the Cox-de Boor recursion formula. If the degree is zero, i.e. k = 0, these basis functions are all step functions that follows from Eq. (1). N i, 0 (u) = 1 is only in the i-th knot span 〈u i , u i + 1 ). For example, if we have four knots u 0 = 0, u 1 = 1, u 2 = 2 and u 3 = 3, knot spans 0, 1 and 2 are 〈0, 1), 〈1, 2) and 〈2, 3), and the basis functions of degree 0 are N 0, 0 (u) = 1 on interval 〈0, 1) Acta In this contribution, we present the description of a B-spline curve. We deal with creation of its basis function as well as with creation of the curve itself from entered control points. Following the literature, we formed an algorithm for B-spline modelling and we used it for the planar and spatial curve. The planar curve is made of chosen points. The spatial curve approximates the trajectory of a real vehicle, which trajectory was obtained by the set of measured points. The modelled curve very exactly describes the polygon created from the linked control points. With the lowering degree of the curve, this one is more clamping to the given polygon and for the extreme case it is transformed to the polygon itself. The advantage of the B-spline curve use is, for example in comparison with a Bézier curve, high adaptability, which is expressed in its parameters -besides entered control points, these are knots generated on the curve and degree of the curve.
doi:10.1515/crll.1891.107.349 fatcat:2lglmrs37zgdbe6hycly33yqny