polynomipolku

Polynomipolku is a term used in Finnish mathematical literature to describe a continuous path whose coordinates are polynomials in a parameter t, typically taken in the interval [0, 1]. In this sense, a polynomipolku P(t) maps a parameter value to a point in Euclidean space, with P(0) representing the start point and P(1) the end point.

Definition and construction: A polynomipolku in R^n is expressed as P(t) = (x1(t), x2(t), ..., xn(t)), where each xi(t) is a real polynomial. The degree of the path is the maximum degree among the coordinate polynomials. By choosing appropriate polynomials, one can enforce boundary conditions such as P(0) = p0 and P(1) = p1, and, if desired, prescribed derivatives at the endpoints. A common construction uses polynomial basis representations, for example when using Bezier-type forms where the coordinates are linear combinations of polynomials in t with fixed control points.

Properties: Polynomipolku curves are infinitely differentiable, since polynomials are smooth. Their exact shape depends on the chosen coefficients, which can be tuned to meet constraints like tangents or curvature at endpoints. Reparameterization by a monotonic function φ(t) yields another polynomipolku if φ is chosen so that the resulting coordinates remain polynomials.

Applications: The concept serves as a simple model for polynomial trajectories in computer graphics, animation, and robotics motion planning, as well as for interpolation in numerical analysis and geometric design.

Relation to other concepts: Polynomipolku is closely related to Bezier curves and polynomial splines, which also rely on polynomial coordinate functions to describe smooth paths.

See also: Bezier curve, polynomial interpolation, spline, motion planning.