Cubic Helices in Minkowski Space

Jiří Kosinka
2010 Sitzungsberichte und Anzeiger der mathematisch-naturwissenschaftlichen Klasse  
We discuss space-like and light-like polynomial cubic curves in Minkowski (or pseudo-Euclidean) space R 2;1 with the property that the Minkowski-length of the first derivative vector (or hodograph) is the square of a polynomial. These curves, which are called Minkowski Pythagorean hodograph (MPH) curves, generalize a similar notion from the Euclidean space (see FAROUKI [3] ). They can be used to represent the medial axis transform (MAT) of planar domains, where they lead to domains whose
more » ... ies are rational curves. We show that any MPH cubic (including the case of light-like tangents) is a cubic helix in Minkowski space. Based on this result and on certain properties of tangent indicatrices of MPH curves, we classify the system of planar and spatial MPH cubics. Cubic helices for other slopes are obtained by a uniform scaling of the z-coordinate. According to WAGNER and RAVANI [11] , cubic helices are the only cubics which are equipped with a rational Frenet-Serret motion. More precisely, the unit tangent, normal and binormal of the curve can be described by rational functions. Pythagorean hodograph (PH) curves in Euclidean space were introduced by FAROUKI and SAKKALIS [4] . While the only planar PH cubic is the so-called Tschirnhausen cubic, FAROUKI and SAKKALIS [5] proved later that spatial PH cubics are helices, i.e. curves of constant slope. A classification of PH cubics in Euclidean space can be obtained by combining these results: Any PH cubic can be constructed as a helix with any given slope "over" the Tschirnhausen cubic. Later, this notion was generalized to Minkowski (pseudo-Euclidean) space. As observed by MOON [10] and by CHOI et al. [1], Minkowski Pythagorean hodograph (MPH) curves are very well suited for representing the so-called medial axis transforms (MAT) of planar domains. Recall that the MAT of a planar domain is the closure of the set containing all points ðx; y; rÞ, where the circle with center ðx; yÞ and radius r touches the boundary in at least two points and is fully contained within the domain. When the MAT is an MPH curve, the boundary curves of the associated planar domain admit rational parameterizations. Moreover, rational parameterizations of their offsets exist too, since the offsetting operations correspond to a translation in the direction of the time axis, which clearly preserves the MPH property. These observations served to motivate constructions for MPH curves. Interpolation by MPH quartics was studied by KIM and AHN [6]. Recently, it was shown that any space-like MAT can approximately be converted into a G 1 cubic MPH spline curve (KOSINKA and J € U UTTLER [7] ). This paper analyzes the geometric properties of MPH cubics. As the main result, it is shown that these curves are again helices and can be classified, similarly to the Euclidean case. The remainder of this paper is organized as follows. Section 2 summarizes some basic notions and facts concerning three-dimensional Minkowski geometry, MPH curves, and the differential geometry of curves in Minkowski space. Section 3 recalls some properties of helices in Euclidean space and it discusses helices in Minkowski space. Section 5 presents a classification of planar MPH cubics. Based on these results and using the so-called tangent indicatrix of a space-like curve we give a complete classification of spatial MPH cubics. Finally, we conclude the paper. 14 J. Kosinka and B. Jüttler
doi:10.1553/sunda2006ssbii-13 fatcat:o3efh6xs4zhvblahyo2hplkxie