3 ms·

This is understandable, since each control point affects the curve globally it is very difficult to get all the points in the correct position at once, particul

by starky 11y ago

This is understandable, since each control point affects the curve globally it is very difficult to get all the points in the correct position at once, particularly when the curve becomes higher order. Using NURBS is much more intuitive as using a particular knot will allow for specific local control.

jacobolus 11y ago
In a piecewise Bézier curve, as used in vector graphics tools like Adobe Illustrator or Inkscape, or as the main curve primitive in PostScript, PDF, SVG, font formats, etc., each control point affects only a particular segment. Every shape tends to be made of numerous tiny segments.
- delhanty 11y ago
  NURBS curve can always be converted to piecewise Bézier by repeated knot-insertion. Having said that I too prefer spirals. Designing with piecewise Bézier curves is just a bad idea. Piecewise Bézier is a fine representation format once the curve is immutably defined. Bézier curves and surfaces, and the Bernstein Polynomial basis have "excellent numerical stability properties" http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf which is just what you need for downstream processing. For example, intersection algorithms in CAD systems. Edit: I just realised that we are talking about polynomial Bézier curves here (rather than rational Bézier), so what I suggested for converting from NURBS to piecewise Bézier isn't going to work in general.
  - jacobolus 11y ago
    Farouki’s paper is nice, but the advantages of the Bernstein basis are oversold IMO. In my opinion Chebyshev basis would be a better core representation for nearly every purpose (including rendering efficiency, degree raising / lowering, intersections, computing approximate arclengths, approximation of arbitrary parametric curves, etc.). I’ve been meaning to email Pomax about this for a while, but it’s going to take a somewhat long and involved email, so I keep putting it off.

by starky 11y ago

jacobolus 11y ago

In a piecewise Bézier curve, as used in vector graphics tools like Adobe Illustrator or Inkscape, or as the main curve primitive in PostScript, PDF, SVG, font formats, etc., each control point affects only a particular segment. Every shape tends to be made of numerous tiny segments.

delhanty 11y ago
NURBS curve can always be converted to piecewise Bézier by repeated knot-insertion. Having said that I too prefer spirals. Designing with piecewise Bézier curves is just a bad idea. Piecewise Bézier is a fine representation format once the curve is immutably defined. Bézier curves and surfaces, and the Bernstein Polynomial basis have "excellent numerical stability properties" http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf which is just what you need for downstream processing. For example, intersection algorithms in CAD systems. Edit: I just realised that we are talking about polynomial Bézier curves here (rather than rational Bézier), so what I suggested for converting from NURBS to piecewise Bézier isn't going to work in general.
- jacobolus 11y ago
  Farouki’s paper is nice, but the advantages of the Bernstein basis are oversold IMO. In my opinion Chebyshev basis would be a better core representation for nearly every purpose (including rendering efficiency, degree raising / lowering, intersections, computing approximate arclengths, approximation of arbitrary parametric curves, etc.). I’ve been meaning to email Pomax about this for a while, but it’s going to take a somewhat long and involved email, so I keep putting it off.