If I have a set of points from a black-white image I scanned.
It looks like a curve and I want to simulate a smooth curve using cubic Bezier curve from those points.
How can I figure out the start point, 2 control points and the end point?
From the image, there are several cubic Bezier curves that can be simulated but I don't know how to figure out the control points.
I have found the solution for my problems and I want to share it.
First, I write a python program to trace the curve and tell me all x,y of the points on the line.
Now, I have one curve.
I then create a line from start to end point and compare the distance between the perpendicular points on those 2 lines. If the distance is not low enough I will move the control point up, down, left, right and find which direction has the least distance. Do this multiple time and I will get a curve that is similar to the curve I have.
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I'm struggling to fit an elliptical arc to some points. The points are either from line segments or all part of the same polyline or they are generated from a bezier curve. I basically have two issues:
I would like to fit an elliptical arc to the points if it is reasonable and i straight line if that is a better fit. I'm thinking about starting at the first point and then constructing a line and a ellip_arc to the next points. The one with the lowest error wins. Or something like that at least. The problem is that an ellipse has many free parameters as opposed to a straight line. So how could I fit an ellipse arc to the points? It also has to start and stop at the points.
Fitting seems to be easiest when the ellipse arc is parameterized. Even though I find it hard to define where to start and stop the arc. But, as output, I need to have the ellipse described like in SVG format (Center Point, Minor and Major Axis, rotation to x-axis etc. https://www.w3.org/TR/SVG2/paths.html#PathDataEllipticalArcCommands). I'm not sure how to convert to this representation, or maybe I can fit it like that?
see:
Circular approximation of polygon (or its part)
If you compute this for your curve samples from local change of radius and center you could group and estimate parts of curve that belongs to the same ellipse and also its eccentricity and or a,b semiaxises sizes , center and orientation to ease up your fitting... Even if not get the precise value it will be a start point and range for ellipse parameter fitting hugely improving speed and stability of fitting
Some of the ellipse parameters might be obtained directly if your data has enough large chunks of ellipses see:
Algorithms: Ellipse matching
Fitting SVG like parameters fully will be very slow easier would be to fit just the ellipses first (center,a,b,rotation) and then convert/fit to SVG form elliptic arc. See:
Converting an svg arc to lines
Express SVG arc as series of curves
And finally you can use any fitting algorithm my favorite is this one:
approximation search
I do not code in C# so I have no idea about any existing packages for tasks like this.
I am aiming to create the following (a directed arrow that connects two nodes) :
At the moment I have this (a quadratic bezier curve drawn from the center point of one node to the center of another):
(Note I have drawn the bezier above the nodes to show where it begins and ends)
I need a method - heuristic or otherwise - to calculate the point of intersection (circled in red, above) between the bezier curve and the node's (ellipse) circumference.
With this, I could calculate the angle between the node center and the point of intersection to draw the arrow head lines at the correct location and angle.
As a last resort, I could use the quadratic Bézier formula to generate a list of points that lie along the curve and also generate a list of points that lie on the circumference of the circle and use one of the two coordinates that have the least euclidian distance between each other as my intersection point. I'm hoping any answers can leverage geometry or whatever else to better solve it.
The general problem is uneasy as the intersection equation is quartic ((X(t)-Xc)² + (Y(t)-Yc)²=R²), where X and Y are quadratic polynomials). If you have a quartic solver handy you can use it but you'll have to select the right root.
A more reasonable approach is just to intersect the circle with the line segment between the control points. This is approximate but probably unnoticeable if the circle radius is small.
If you want more accuracy, perform one or two Newton's iterations from this point.
I want to fit a bezier curve with known end points (p0 and p3) to noisy 2d data. This seems like an easier problem than traditional 4-point bezier curve fitting but still too hard for me to figure out.
Can someone point me to existing code or an algorithm to find the best values for the control points p1 and p2?
edit: The points that I'm trying to fit with a bezier curve comes from curves drawn with a mouse (imagine drawing something with a brush in Paint, there could be hundreds of recorded points in one long stroke). The anchor points p0 and p3 are created in advance but the control points p1 and p2 should be calculated so that the bezier fits the shape of the curve sketched out with the mouse.
I stumbled on a paper called "Approximation of data using cubic Bezier curve least square fitting" by "M.Khan" which describes an algorithm to calculate the exact thing I'm looking for.
Implementation in javascript was easy. It works quite good and is fast but the resulting bezier curves are not perfect. Could be a bug in my code but I suspect that better curves could be obtained by iteratively adjusting the matching points on the bezier curve to better fit the data .
edit: It turns out you can use newton-raphson to optimize each individual t-value for the bezier curve. After doing that the curve fits great, atleast for curves with only few points that don't self intersect but I have to do some more testing.
The operation mentioned in the title is common in many Computer Aided Design (CAD) softwares such as AutoCAD, where it is called fillet. However, I found it is really difficult to implement this function in my own program.
The method I thought of is to use the condition that the distances of the arc center to the tangent lines of the curves are equal to the specified radius. Considering that actual curves are defined with piece-wise nonlinear functions, and the contact points could be anywhere on the curves, it is not easy to get the solution. Anyone good ideas?
Given that you don't describe in enough details the characteristics of the curves, it's hard to come with a specific/specified algo, but let's try a descriptive approach:
take a circle of the given radius and roll it on one curve until the circle touches the other one.
I assume you can parametrize you curves.
To "roll the circle" along the curve you need the tangent (or better said the normal, which of course is normal to the tangent) in the point "rolling track curve"-to-circle tangent point. You have this normal, you know the radius, you can compute your circle. You have the circle, you can see if/where it intersects the other curve.
The idea of "rolling" is to bracket your solution (parameter of the tangent-point on one curve) between a point when the circle does not intersect the other curve and another point where it intersects (possible in more than 1 point).
Once you have the bracket, go with a bisection method (binary search) between the two positions until your circle becomes "tangent enough" to the other curve (i.e the intersection points with the other curve are so close that they fall below your acceptable epsilon).
You will now have two points (one on each curves) and the circle that realizes the solution: just keep the arc on this circle corresponding what make sense (based on the convergence or divergence of the two tangents).
To find the arc center you need two robust and strategic algorithms:
Curve offset
Curve intersection
What AutoCAD does to find the arc center is to offset the two curves of the arc radius distance and intersect them. Depending on the curve offset direction you can easily switch between all possible solutions to the problem.
At this point, trimming the curves at tangent points will be trivial.
Given a few sample points on a bézier curve, is it possible to work out the set of possible curves these points may lie on?
In my specific application there is a limited set of endpoints the curve may have, so I want to generate the set of possible curves, enumerate all of them and pick out all the ones which may end on a valid end point.
Some people have asked for more detail.
I have a set of points which I know are on a quadratic bezier curve, I want to calculate the formula of the curve and be able to extrapolate new points on the curve.
Bezier curves will always go through starting and ending control points.
If the degree of the curve is equal to the number of sample points then there will be only one curve that will pass through all your points (in a normal case, where all points are different and they don't lie on a bezier curve of a lesser degree).
If the degree of a curve is less then the number of the sample points then, in general case, you will not be able to make the curve go through all the points (in a normal case).
If the degree of a curve is higher then the number of the sample points then, in general case, you will be able to draw infinite number of curves.
In the wiki article you will find references to control points only, but still I believe that I remember the above properties correctly and that they hold for the points on the curves as well.
I think you need to redefine your question and exactly define what type of curves (and of which degree) do you need. Also as Georg pointed out you might be looking for paths - a combination of curves.
EDIT:
First a correction - curve is defined with degree plus one number of control points points (quadratic need three). Control points are not the same as points on the curve - and for three points on the curve and quadratic curve you could have infinite number of solutions (see this for quadratic curve and four points)
As for the solution (but still under assumption that you are looking at a single curve):
For an equation for single quadratic curve you have
B(t) = (1-t)^2*P0 + 2*(1-t)*t*P1 + t^2*P2
Capital letters above are vectors, and P0 corresponds to starting control point (first point), P2 corresponds to ending control point (last point), so you still need to find P1. The variable t is scalar that ranges from 0 to 1.
If working with 2D curves the above vector equation gives two scalar equations for each point on the curve.
Still there is t as an unknown, so you should take 2 more points (4 in total) which will give you 4 unknowns (t for first point, t for second point, x and y of the P1, middle control point) and 4 equation to solve (2 from each sample point).
Solve that with your favourite numerical method and you will get the original curve on which the points came from.
If you still think that you can get more curves and that you will have to choose something then you are not working with bezier curves, but with bezier splines (in a sense of multiple curves joined together). Still the same principle applies and if you work out a way to solve a single curve from the above equations (and if you have enough points) then you can divide the problem into n-segments of actual bezier curves and solve each as outlined above.
If it turns out that you don't have enough points then look at the linked article again - you are probably looking for the smoothest curve and there are some suggestions in the article on how to get there as looking for the exact solution (shortest curve/smoothest curve) seems to be rather complex.