Curvas alabeadas con hodógrafo pitagórico

Abstract

In this work, we investigate the properties of polynomial space curves r(t) = (x(t), y(t), z(t)) whose hodographs, r 0 (t) = (x 0 (t), y0 (t), z0 (t)), satisfy x 02 (t) + y 02 (t) + z 02 (t) ≡ σ 2 (t) for some real polynomial σ(t). We give a characterization for all cubic Pythagoreanhodograph space curve and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility. We focus on the curves of degree 5, that admits a constructive representation. Pythagorean-hodograph space curves have two geometrically attractive attributes: (i) the arc length of any segment can be determined exactly without numerical aproximation; and (ii) the canal surfaces based on such curves as spines have precise rational parametrizations.