G² blends of linear segments with cubics and Pythagorean-hodograph quintics

Abstract

Fillets, also known as blend arcs, are used in CNC machining to round corners. Fillets are normally circular arcs, which have G ¹ contact with the straight line segments to which they are joined. Recent advances in machining technology allow NURBS, including Pythagorean-hodograph (PH) curve segments, to be incorporated in CNC tool paths. This article examines the use of cubic and PH quintic Bézier curve segments that have a single curvature extremum, and which have G ² contact with the straight line segments to which they are joined, as fillets. It is shown how the extreme circle of curvature can be determined. The point of curvature extremum and the corresponding value of the curvature can be changed by adjusting the joining points of the blending curve with the neighbouring straight lines. These blending curves can also be incorporated in computer-aided design packages for curve or surface design.

Keywords:

• linear blends
• cubic
• Pythagorean-hodograph quintic

2000 AMS Subject Classification :

• 65D17
• 68U07

CCS Category :

• I.3.5
• J.6

Acknowledgements

The authors acknowledge the support of The Natural Sciences and Engineering Research Council of Canada. The first author also acknowledges, with thanks, the hospitality of The Institute of Biodiagnostics of the National Research Council in Winnipeg, Canada. The authors also acknowledge, with thanks, the suggestions of the anonymous reviewer that helped to improve the manuscript.