You can reduce the number of points using the Ramer–Douglas–Peucker algorithm there is a C# implementation here. I gave this a try using WPFs PolyQuadraticBezierSegment and it showed a small amount of improvement depending on the tolerance. After a bit of searching sources (1,2) seem to indicate that using the curve fitting algorithm from Graphic … Read more
Maybe someone is searching for an answer, like me for two days now: // Draw a path and hide it: var root = paper.path(‘M0 50L30 50Q100 100 50 50’).hide(); var length = root.getTotalLength(); // Setup your animation (in my case jQuery): element.animate({ ‘to’: 1 }, { duration: 500, step: function(pos, fx) { var offset = … Read more
I hope you found my math paper useful Quadratic bezier offsetting with selective subdivision https://microbians.com/mathcode
I have similar problem and I have found “An algorithm for automatically fitting digitized curves” from Graphics Gems (1990) about Bezier curve fitting. Additionally to that I have found source code for that article. Unfortunately it is written in C which I don’t know very well. Also, the algorithm is quite hard to understand (at … Read more
Let P0, P1, P2 be the control points, and Pc be your fixed point you want the curve to pass through. Then the Bezier curve is defined by P(t) = P0*t^2 + P1*2*t*(1-t) + P2*(1-t)^2 …where t goes from zero to 1. There are an infinite number of answers to your question, since it might … Read more
The gradient is a mathematical operator that may help you. If you can turn your interpolation into a differentiable function, the gradient of the height will always point in the direction of steepest ascent. All curves of equal height are perpendicular to the gradient of height evaluated at that point. Your idea about starting from … Read more
Intersection of Bezier curves is done by the (very cool) Asymptote vector graphics language: look for intersect() here. Although they don’t explain the algorithm they actually use there, except to say that it’s from p. 137 of “The Metafont Book”, it appears that the key to it is two important properties of Bezier curves (which … Read more
As you’ve discovered, both Quadratic curves and Cubic Bezier curves just connect 2 points with a curve. Since the Cubic curve has more control points, it is more flexible in the path it takes between those 2 points. For example, let’s say you want to draw this letter “R”: Start drawing with the “non-curvey” parts … Read more
The standard way to represent 2D/3D transformations nowadays is by using homogeneous coordinates. [x,y,w] for 2D, and [x,y,z,w] for 3D. Since you have three axes in 3D as well as translation, that information fits perfectly in a 4×4 transformation matrix. I will use column-major matrix notation in this explanation. All matrices are 4×4 unless noted … Read more
From https://fontforge.org/docs/techref/bezier.html#converting-truetype-to-postscript: Any quadratic spline can be expressed as a cubic (where the cubic term is zero). The end points of the cubic will be the same as the quadratic’s. CP0 = QP0 CP3 = QP2 The two control points for the cubic are: CP1 = QP0 + 2/3 *(QP1-QP0) CP2 = QP2 + 2/3 … Read more