Improvements to the offsetting code

[jonschz]

Dear Pomax,

first of all, thank you so much for your work! My project would not have been possible without it. I gave the due credits in multiple places.

My project involved implementing a variant of your offsetting algorithm in LaTeX (you can check it out at https://github.com/jonschz/tikz-nfold if you are interested). While implementing the code I discovered some improvements which I explain here, p. 6 ff in detail. A quick summary:

• BezierInfo-2/docs/js/graphics-element/lib/bezierjs/bezier.js

  Line 1748 in 40607a4

  return abs$1(acos$1(s)) < pi$1 / 3;

  This check is equivalent to s > 1/2; furthermore, the abs() is redundant because acos() never returns a negative value.

• BezierInfo-2/docs/js/graphics-element/lib/bezierjs/bezier.js

  Line 1820 in 40607a4

  // TODO: add special handling for degenerate (=linear) curves.

  It is possible to get around computing the origin altogether, as I explained here, pp. 7–8. Essentially, using a bit of geometry and trigonometry one can find a closed formula for the intersection points that does not have a singularity when the origin approaches infinity (i.e. when the control points approach a straight line). The clockwise/anticlockwise checks are also no longer needed, and most (or all) trigonometric function calls can be removed.

• BezierInfo-2/docs/js/graphics-element/lib/bezierjs/bezier.js

  Line 1715 in 40607a4

  const nv = this.normal(0),

  BezierInfo-2/docs/js/graphics-element/lib/bezierjs/bezier.js

  Lines 1557 to 1561 in 40607a4

  	__normal2(t) {
  	const d = this.derivative(t);
  	const q = sqrt$1(d.x * d.x + d.y * d.y);
  	return { x: -d.y / q, y: d.x / q };
  	}

  As far as I can tell, computing the normal in t=0 and t=1 will not work if the first two or the last two control points are equal. I computed the tangent (and hence the normal) in t=0 as follows (pseudocode):

  if (points[0] != points[1]) { tangent = normalize(points[1]-points[0]); }
  else if (points[0] != points[2]) { tangent = normalize(points[2]-points[0]); }
  else if (points[0] != points[3]) { tangent = normalize(points[3]-points[0]); }
  else { error("Curve is degenerate to a point"); tangent = (1,0[,0]); }

Don't feel obligated to implement any of these improvements, I merely wanted to share what I found out.