In this paper, we introduce a new algebraic method to characterize rational PH plane curves. And using this method, we study the algebraic characterization of generic strongly regular rational plane ...PH curves expressed in the complex formalism which is introduced by R. T. Farouki. We prove that generic strongly semi-regular rational PH plane curves are completely characterized by solving a simple functional equation whereh is a complex polynomial and is a bi-linear operator defined by for complex polynomialsf, g.
We study the (plane polynomial) Pythagorean hodograph curves from the viewpoint of their roots. The loci of root-related parameters of PH curves show us very interesting geometric properties. They ...include regular 2
n + 1-
gon and isosceles triangles with the ratio of sides
n : 1 :
n.
The envelope of the one-parameter set of congruent spheres which are centered at the points of a curve
S
is a pipe surface with
S
as spine curve. We prove that pipe surfaces with rational spine curve ...always admit a rational parameterization and propose an algorithm for its computation.
Many guidance laws for UAVs focus on path following. And trajectory tracking problem is enough when it satisfies path and time constrains. One more constraint will be considered in this research, ...that is, approaching direction to a terminal position. Two methods will be introduced; they are based on Pythagorean-Hodograph curve, and the Lyapunov vector field, respectively. The first algorithm consists of two steps. The first step is trajectory-planning using PH curve, and the second step is tracking algorithm for that curve. The second algorithm is based on vector field including a variable parameter. Using this parameter, contraction vectors toward the origin are tuned, and a desired terminal approaching condition is achieved.
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