Bezier curve is a parametric curve, meaning that there is a paramater t at which one can evaluate the polynomials in order to find out the positions of points laying on the curve.
Polynomials for some common cases can be found at en.wikipedia.org/wiki/B%C3%A9zier_curve#Specific_cases
To draw a Bezier curve on screen, one could evaluate the polynomials from 0 to 1 at ever increasing t by tiny little steps. However, that would very wasteful because, in general, the parameter "space" does not correspond to screen "space", i.e. several little steps may fall onto just one pixel.
My question is: how to find the smallest step which increases Cartesian distance from previous point at least by 1 pixel?
To put it in other way: I would like to draw a Bezier curve on screen. How to choose the (uniform) step by which t should grow so that I never draw at one pixel more the once? I don't mind the "holes" when the t grows too quickly, I just don't want to redraw already drawn pixels.
Edit
By "how to find" I mean O(1). Yes, I could use De Casteljau's algorithm but I was hoping there is a way to "guess" the optimal t step quickly.
The comment above (jozxyqk) gives you a hint. I would give it a try with a recursive binary division of the spline drawing.
Lets say you start with a coarse resolution of the parameter space (delta_t = 0.1), that gives you 11 points on the spline curve s , s(t=0), s(t=0.1), ..., s(t=0.9), s(t=1).
Calculate the distance between s(t_i) and s(t_i+1). If it is >1 than make a binary subdivision between those two points. And so on...
But honestly, I guess it is faster to calculate all points a a higher resolution without any recursive loops or subdivisions. Especially if you are using multithread programming.
Related
I am dealing with a reverse-engineering problem regarding road geometry and estimation of design conditions.
Suppose you have a set of points obtained from the measurement of positions of a road. This road has straight sections as well as curve sections. Straight sections are, of course, represented by lines, and curves are represented by circles of unknown center and radius. There are, as well, transition sections, which may be clothoids / Euler spirals or any other usual track transition curve. A representation of the track may look like this:
We know in advance that the road / track was designed taking this transition + circle + transition principle into account for every curve, yet we only have the measurement points, and the goal is to find the parameters describing every curve on the track, this is, the transition parameters as well as the circle's center and radius.
I have written some code using a nonlinear optimization algorithm, where a user can select start and end points and fit a circle that to the arc section between them, as it shows in next figure:
However, I don't find a suitable way to take the transition into account. After giving it some thought I came to think that this s because, given a set of discrete points -with their measurement error- representing a full curve, it is not entirely clear where to consider it "begins" and where it "ends" and, moreover, it is less clear where to consider the transition, the proper circle and the exit transition "begin" and "end".
Is there any work on this subject which I may have missed? is there a proper way to fit the whole transition + curve + transition structure into the set of points?
As far as I know, there's no method to fit a sequence clothoid1-circle-clothoid2 into a given set of points.
Basic facts are that two points define a straight, and three points define a unique circle.
The clothoid is far more complex, because you need: The parameter A, the final radius Rf, an initial point px,py, the radius Ri at that point, and the tangent T (angle with X-axis) at that point.
These are 5 data you may use to find the solution.
Due to clothoid coords are calculated by expanded Fresnel integrals (see https://math.stackexchange.com/a/3359006/688039 a little explanation), and then apply a translation & rotation, there's no an easy way to fit this spiral into a set of given points.
When I've had to deal with your issue, what I've done is:
Calculate the radius for triplets of consecutive points: p1p2p3, p2p3p4, p3p4p5, etc
Observe the sequence of radius. Similar values mean a circle, increasing/decreasing values mean a clothoid; Big values would mean a straight.
For each basic element (line, circle) find the most probably characteristics (angles, vertices, radius) by hand or by some regression method. Many times the common sense is the best.
For a spiral you may start with aproximated values, taken from the adjacent elements. These values may very well be the initial angle and point, and the initial and final radius. Then you need to iterate, playing with Fresnel and 'space change' until you find a "good" parameter A. Then repeat with small differences in the other values, those you took from adjacents.
Make the changes you consider as good. For example, many values (A, radius) use to be integers, without decimals, just because it was easier for the designer to type.
If you can make a small applet to do these steps then it's enough. Using a typical roads software helps, but doesn't avoid you the iteration process.
If the points are dense compared to the effective radii of curvature, estimate the local curvature by least square fitting of a circle on a small number of points, taking into account that the curvature is most of the time zero.
You will obtain a plot with constant values and ramps that connect them. You can use an estimate of the slope at the inflection points to figure out the transition points.
I'm using a polygonal chain to approximate a curve. I want to approximate the average of a function of curvature of all points that lie on the curve. One function of curvature that I need is, for example, the square of curvature.
I can get near that by choosing some points on the chain, calculating the curvature in those points, applying the function on it (for example squaring it), and then averaging the calculated values.
I need both accuracy and speed. I appreciate both — fast, but approximate; as well as accurate, but slow solutions. I'm working in Java, but the answer doesn't need to be written in Java — it doesn't even need to contain any code at all.
Polygonal chain with uniform segment length
If the polygonal chain's segments all have equal length, I can just calculate the curvature in the vertices and then average that. I see two ways to get the curvature in a vertex.
One way is to get the circle that goes through the selected vertex, the vertex before it, and the one after it. The curvature is then 1/radius of the circle.
The other way is to calculate the external angle (in radians) of the two segments connected at the selected vertex and then divide its absolute value by the length of a segment. In the following image, φ marks the external angle:
I am not sure if this method is correct, as I haven't mathematically derived it, but I've noticed through experimentation that it gives similar results to the above method.
Polygonal chain with non-uniform segment length
Unfortunately, though, there's no guarantee that the segments have uniform length.
If I try using the first of the above methods, vertices connected to longer segments give lower curvatures, even if they are visibly sharper. I tried substituting previous and next vertices with a point x units before the selected vertex and a point x units after it. I don't know what to set the x constant to, to get accurate results. All the values I've tried seemed to give inaccurate results.
If I try using the second method, I don't know what length to divide the angle by. If I don't divide by anything at all, I actually get pretty good results for comparing two curves and determining which one is curvier, but I need to be able to determine the actual curvature in a point.
With both of these methods there's also the problem that parts with shorter segments (where points are denser) will affect the average more.
Another possible solution would be to ignore the vertices and instead use an array of points on the chain that are evenly spaced, treat them as a new polygonal chain (connect the points with straight lines), and then calculate curvatures on this new chain instead, using one of the methods I mentioned under the header titled "Polygonal chain with uniform segment length".
Finding such an array of points is not trivial, though, because I have to choose a segment length, and only after producing the points, I can see if the length of the resulting chain is divisible by the chosen segment length.
If you aren't short on space, the last solution you mentioned would be the best, because the "sphere" approximation, as you've perhaps realized, would give awful results in more extreme cases, especially if the curvature is large or changes sign quickly.
There are many ways to do interpolations, the simplest being quadratic and cubic splines. However if you have more pre-processing time, Lagrange polynomials produce very good results: https://en.wikipedia.org/wiki/Lagrange_polynomial.
Side note on your angle division method, consider this diagram:
(From simple geometry the inside angle there is also theta)
For a << l. So the curvature:
So your approximation is in fact correct for small curvatures.
An alternative is to use a local parabola approximation to estimate the curvature. Basically, to estimate the curvature at point P(i), you take P(i-1), P(i) and P(i+1) and construct a parabola from these 3 points. Then, you compute the curvature at P(i) from the parabola. Remember to use chord-length (or centripetal) parametrization when constructing the parabola.
The curve is in fact the trajectory of a bus, the curve is represented by many (up to a few thousand) discrete points on the curve (the points were recorded by a GPS device installed on the bus).
Input a point P, I need to find the closest point on the curve to the point P. The point P is usually no more than 30m away from the trajectory of the bus. Note, the closest point isn't necessary a point recorded by the GPS device, it could be a point somewhere between two recorded points.
First I need an algorithm to recover the trajectory from those recorded points. It would be great if the interpolated curve could show sharp turns made by the bus. Which curve is best for such task ? Is Bezier curve good enough ? And finally I have to calculate the closest point on the curve, of course the algorithm completely depends on the kind of curve chosen.
I'm doing some research, and don't have much knowledge in curve interpolation, so any suggestions are welcome.
For computing the trajectory from recorded points, I recommended using the centripetal or chord-length Catmull-Rom splines. See link for more details. Catmll-Rom splines are in fact special cubic Hermite curves, which can be easily converted into cubic Bezier curves. Please note that the result from Catmull-Rom spline is a G1 curve only in general. If you want the trajectory to be with higher continuity (such as C2), you can go with natural cubic splines or general B-spline interpolation. Whatever approach you take, it is advised to keep the spline's degree no higher than 5. Degree 3 is a popular choice.
Once you have the mathematical representation for the trajectory, you can compute the minimum distance between a given point P and the trajectory. In general, the squared distance between point P and a curve C(t) is represented as D(t) = |P-C(t)|^2. The minimum of D will happen at where its first derivative is zero, which means we have to find the root for the following equation:
dD/dt = 2*(P-C(t)).C'(t) =0
When C(t) is of degree 3, dD/dt will be of degree 5. This is the reason why it is recommended to use a low degree curve earlier.
There are many literatures or online materials talking about how to find the root of a polynomial (of any degree) efficiently and robustly. Here is another SO post that might be useful.
here is what I want to do (preferably with Matlab):
Basically I have several traces of cars driving on an intersection. Each one is noisy, so I want to take the mean over all measurements to get a better approximation of the real route. In other words, I am looking for a way to approximate the Curve, which has the smallest distence to all of the meassured traces (in a least-square sense).
At the first glance, this is quite similar what can be achieved with spap2 of the CurveFitting Toolbox (good example in section Least-Squares Approximation here).
But this algorithm has some major drawback: It assumes a function (with exactly one y(x) for every x), but what I want is a curve in 2d (which may have several y(x) for one x). This leads to problems when cars turn right or left with more then 90 degrees.
Futhermore it takes the vertical offsets and not the perpendicular offsets (according to the definition on wolfram).
Has anybody an idea how to solve this problem? I thought of using a B-Spline and change the number of knots and the degree until I reached a certain fitting quality, but I can't find a way to solve this problem analytically or with the functions provided by the CurveFitting Toolbox. Is there a way to solve this without numerical optimization?
mbeckish is right. In order to get sufficient flexibility in the curve shape, you must use a parametric curve representation (x(t), y(t)) instead of an explicit representation y(x). See Parametric equation.
Given n successive points on the curve, assign them their true time if you know it or just integers 0..n-1 if you don't. Then call spap2 twice with vectors T, X and T, Y instead of X, Y. Now for arbitrary t you get a point (x, y) on the curve.
This won't give you a true least squares solution, but should be good enough for your needs.
I'm trying to find/make an algorithm to compute the intersection (a new filled object) of two arbitrary filled 2D objects. The objects are defined using either lines or cubic beziers and may have holes or self-intersect. I'm aware of several existing algorithms doing the same with polygons, listed here. However, I'd like to support beziers without subdividing them into polygons, and the output should have roughly the same control points as the input in areas where there are no intersections.
This is for an interactive program to do some CSG but the clipping doesn't need to be real-time. I've searched for a while but haven't found good starting points.
I found the following publication to be the best of information regarding Bezier Clipping:
T. W. Sederberg, BYU, Computer Aided Geometric Design Course Notes
Chapter 7 that talks about Curve Intersection is available online. It outlines 4 different approaches to find intersections and describes Bezier Clipping in detail:
https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1000&context=facpub
I know I'm at risk of being redundant, but I was investigating the same issue and found a solution that I'd read in academic papers but hadn't found a working solution for.
You can rewrite the bezier curves as a set of two bi-variate cubic equations like this:
∆x = ax₁*t₁^3 + bx₁*t₁^2 + cx₁*t₁ + dx₁ - ax₂*t₂^3 - bx₂*t₂^2 - cx₂*t₂ - dx₂
∆y = ay₁*t₁^3 + by₁*t₁^2 + cy₁*t₁ + dy₁ - ay₂*t₂^3 - by₂*t₂^2 - cy₂*t₂ - dy₂
Obviously, the curves intersect for values of (t₁,t₂) where ∆x = ∆y = 0. Unfortunately, it's complicated by the fact that it is difficult to find roots in two dimensions, and approximate approaches tend to (as another writer put it) blow up.
But if you're using integers or rational numbers for your control points, then you can use Groebner bases to rewrite your bi-variate order-3 polynomials into a (possibly-up-to-order-9-thus-your-nine-possible-intersections) monovariate polynomial. After that you just need to find your roots (for, say t₂) in one dimension, and plug your results back into one of your original equations to find the other dimension.
Burchburger has a layman-friendly introduction to Groebner Bases called "Gröbner Bases: A Short Introduction for Systems Theorists" that was very helpful for me. Google it. The other paper that was helpful was one called "Fast, precise flattening of cubic Bézier path and offset curves" by TF Hain, which has lots of utility equations for bezier curves, including how to find the polynomial coefficients for the x and y equations.
As for whether the Bezier clipping will help with this particular method, I doubt it, but bezier clipping is a method for narrowing down where intersections might be, not for finding a final (though possibly approximate) answer of where it is. A lot of time with this method will be spent in finding the mono-variate equation, and that task doesn't get any easier with clipping. Finding the roots is by comparison trivial.
However, one of the advantages of this method is that it doesn't depend on recursively subdividing the curve, and the problem becomes a simple one-dimensional root-finding problem, which is not easy, but well documented. The major disadvantage is that computing Groebner bases is costly and becomes very unwieldy if you're dealing with floating point polynomials or using higher order Bezier curves.
If you want some finished code in Haskell to find the intersections, let me know.
I wrote code to do this a long, long time ago. The project I was working on defined 2D objects using piecewise Bezier boundaries that were generated as PostScipt paths.
The approach I used was:
Let curves p, q, be defined by Bezier control points. Do they intersect?
Compute the bounding boxes of the control points. If they don't overlap, then the two curves don't intersect. Otherwise:
p.x(t), p.y(t), q.x(u), q.y(u) are cubic polynomials on 0 <= t,u <= 1.0.
The distance squared (p.x - q.x) ** 2 + (p.y - q.y) ** 2 is a polynomial on (t,u).
Use Newton-Raphson to try and solve that for zero. Discard any solutions outside 0 <= t,u <= 1.0
N-R may or may not converge. The curves might not intersect, or N-R can just blow up when the two curves are nearly parallel. So cut off N-R if it's not converging after after some arbitrary number of iterations. This can be a fairly small number.
If N-R doesn't converge on a solution, split one curve (say, p) into two curves pa, pb at t = 0.5. This is easy, it's just computing midpoints, as the linked article shows. Then recursively test (q, pa) and (q, pb) for intersections. (Note that in the next layer of recursion that q has become p, so that p and q are alternately split on each ply of the recursion.)
Most of the recursive calls return quickly because the bounding boxes are non-overlapping.
You will have to cut off the recursion at some arbitrary depth, to handle weird cases where the two curves are parallel and don't quite touch, but the distance between them is arbitrarily small -- perhaps only 1 ULP of difference.
When you do find an intersection, you're not done, because cubic curves can have multiple crossings. So you have to split each curve at the intersecting point, and recursively check for more interections between (pa, qa), (pa, qb), (pb, qa), (pb, qb).
There are a number of academic research papers on doing bezier clipping:
http://www.andrew.cmu.edu/user/sowen/abstracts/Se306.html
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.6669
http://www.dm.unibo.it/~casciola/html/research_rr.html
I recommend the interval methods because as you describe, you don't have to divide down to polygons, and you can get guaranteed results as well as define your own arbitrary precision for the resultset. For more information on interval rendering, you may also refer to http://www.sunfishstudio.com