Real-time CNC interpolators for Pythagorean-hodograph curves
Computer Aided Geometric Design
Contour machining of free-form surfaces with real-time PH curve CNC interpolators
Computer Aided Geometric Design
Differential geometry of intersection curves of two surfaces
Computer Aided Geometric Design
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Curvature formulas for implicit curves and surfaces
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Computer Graphics in China: Convergence analysis for B-spline geometric interpolation
Computers and Graphics
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We use the canonical equations (CE) of differential geometry, a local Taylor series representation of any smooth curve with parameter the arc length, as a unifying framework for the development of new CNC algorithms, capable of interpolating 2D and 3D curves, represented parametrically, implicitly or as surface intersections, with accurate feedrate control. We use a truncated form of the CE to compute a preliminary point, at an arc distance from the last interpolation point selected to achieve a desired feedrate profile. The next interpolation point is derived by projecting the preliminary point on the curve. The coefficients in the CE involve the curve's curvature, torsion and their arc length derivatives. We provide computing procedures for them for common Cartesian representations, demonstrating the generality of the proposed method. In addition, our algorithms admit corrections, which render them more accurate in terms of the programmed feedrate, compared to existing parametric algorithms of the same order.