I don't think there's a good answer to this, but I'd like to find out if there's a better way to do this.
I need to plot a mathematical function, which is nearly flat at one end of the display, and nearly vertical at the other end. The bottom left quadrant of a circle would be a good model. I can auto-generate as many points as required.
The problem is, I can't do this without all sorts of artefacts.
I haven't tried Bezier fitting; I don't think this would be even close. My understanding is that Bezier is for one-off manually-constructed pretty graphics, and not for real curve-fitting.
That leaves polylines. There are only 2 things I can do with polylines - I can select the line length (in other words, the number of points I auto-generate), and I can disable anti-aliasing (setAttributeNS(null, "shape-rendering", "crisp-edges").
If I generate lots of points, then I get jaggies everywhere, and the result is unusable. It can also look very much like it's oscillating, which makes it appear that I've incorrectly calculated the function. The anti-aliasing doesn't make any difference, since it doesn't operate across point boundaries.
The only solution I've got is to draw fewer points, so that it's obvious that I'm drawing segments. It's no longer smooth, but at least there are no jaggies or oscillation. I draw this with the default anti-aliasing.
Any ideas?
Edit:
It seems like the only answer to this is actually Bezier curve fitting. You have to preprocess to find the parameters of the required segments, and then plot the results. Google comes up with a number of hits on curve fitting with Beziers.
You have the mathematical function, and can therefore generate as many points as you need.
I assume the problem is that because you do not know the output resolution (SVG is device independent) you do not know how many points to generate. Otherwise you could just create a polyline where each line is approximately 1 pixel long.
Fitting your mathematical function to a bezier curve is (probably) not going to get a perfect match - just like a circle cannot be matched perfectly by a cubic bezier curve. And I think the task of fitting your function to a bezier curve would not be trivial (I've never done this).
Could you rather output your mathematical function to a canvas element? Then you could write some javascript code to plot your mathematical function dependant on the output resolution. Similar to how a graphics system renders a Bezier curve.
Do you know how graphics systems render Bezier curves? They approximate the bezier curve with a polyline, and then measure the error difference between the polyline and the bezier curve. If the difference is greater than a certain tolerance - where the tolerance is determined by the output resolution - the bezier is subdivided and the process repeated for each bezier curve. When the difference between beziers and polylines is below the tolerance, the polylines are drawn. http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Computer_graphics
I suppose you want to draw y=f(x) over a certain interval [a,b]
A classical solution is to take N points uniformly distributed over [a,b], to compute f over these points and draw lines (or polynoms).
It of course doesn't work in your case, since y is nearly vertical in certain area. But why don't you take more points in these areas (and less points where the function is nearly horizontal) ?
You can compute the derivative of your function (or approximate this derivative with (f(x+h)-f(x))/h and h small) and determine the step between two successive points with this derivative
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A part of the project I am working on requires an interface where a user draws a curve, and I want to output an approximate mathematical equation of that curve to perform a variety of tests on it.
I have thought of 2 approaches so far and was wondering if they are feasible or if there exists a better way of approaching the problem that I am missing/ cool mathematical trick that allows me to pull this off.
Approach 1:
Instead of having users draw a curve, give them the option of inserting bezier curve points and tweaking them to make their curve. Since bezier curves have a parametric equation that describes them, I could directly get the exact equation.
Cons:
-It's more cumbersome for users to tweak and make a bezier than simply draw a curve
Approach 2:
Get the curve drawn and extract 'n' points. (I don't think this should be hard to do).
Somehow go from these 'n' points to an equation of a curve passing through them. Naturally 'n' would be pretty large, say 100.
Is there a neat way to get an equation for the "simplest continuous function" passing through 'n' points?
Approach 1 -- insert Bezier points and tweak control points -- is used in a lot of vector drawing programs, especially CAD-type applications where precision is important and points are laid out on a grid, etc. It is a slower way to draw, but it's used in these situations, because it's really difficult to draw a freehand curve carefully.
Approach 2 -- just draw and get an smooth approximation -- is also pretty popular when precision isn't required. I'm not sure what methods are usually used for this, or even if there is a "usually", but I don't think it's particularly difficult to do a pretty good job. I would:
Connect all the points with line segments in the order in which they were drawn.
Apply the Ramer-Douglas-Peucker algorithm to reduce the number of points.
Interpolate the points with cubic spline interpolation. Be careful looking this up-there are many flavors of cubic spline interpolation and their names overlap. I would specifically prefer the versions that preserve 2nd derivative continuity.
Consider the following polygon (an agricultural plot)
From this polygon, I would like to extract the "headlands" of the plot, being the consecutive lines (sides) of the polygon (Wikipedia) used for turning on the field. While often only the rows running perpendicular to the lay of the field are considered, I need all sides of the polygon.
Here, a consecutive line means any set of coordinates, where the angle between any two coordinates of the set is not larger than a value X (e.g 30 degrees).
For the given example, the resulting headlands should look like the following:
I wrote a small algorithm trying to accomplish this, basically checking the angle between two coordinates and either pushing the given coordinate to the existing lineString if the angle is below X degrees or creating a new lineString (headland) if not.
Check out the following Gist
However, in some cases corners of a field are rounded, therefore may consist of many coordinates within small distances of each other. The relative angles then may be less than the value X, even though the corner is too sharp to actually be cultivated without turning.
In order to overcome that issue, I added an index that increases whenever a coordinate is too close for comparison, so that the next coordinate will be checked against the initial coordinate. Check out the following Gist.
This works for simple plots like the one in the example, however I am struggling with more complex ones as the following.
Here, the bottom headland is recognised as one lineString together with the headland on the right, even though optically a sharp corner is given. Also, two coordinates in the upper right corner were found to be a separate headland even though they should be connected to the right headland. The result should therefore yield in the following:
What I would like to know is if there is an approach that efficiently decomposes any polygon into it's headlands, given a specific turning angle. I set up a repo for the code here, and an online testing page with many examples here if that helps.
I have a large svg path which has quadratice bezier curves.
The path data is used for drawing maps.
How to reduce the number of points of the bezier curve without distorting the overall shape?
You don't say whether you want to do this offline (pre-prepared paths) or online (on the fly). If offline is fine, use a tool like Inkscape.
If you want to calculate the simplified curve yourself, then the typical algorithm used to do this is also the same one that has been used for drawing bezier curves. The algorithm is called "flattening".
Basically the idea is to convert the bezier curves to a series of straight line segments. Because you don't want the the flatness of the line segments to be visible, you have to take into account the scale of the drawing and how curvy the bezier is. If the bezier is very curvy you have to use more line segments than if it is fairly straight.
What you typically do is divide each bezier into two using De Casteljau's algorithm. Then look at each of the two half bezier curves. If a bezier is straight enough to meet a flatness limit you decide on, then stop dividing. Otherwise, divide in half and try again.
At the end of the process you should get a polyline that is indistinguishable from the bezier version. Or if you use a "flatness test" that is a bit courser than that, you will get a rougher approximation of the shape. In your case a map.
If you google bezier flattening you can find a number of papers on the technique. And a few pages which describe how to do it in a more friendly accessible way. For example this one, which is about generating offset curves, but starts out by describing how to flatten a curve:
https://seant23.wordpress.com/2010/11/12/offset-bezier-curves/
Here's what I am trying to solve. I have bezier curves that contain 3 points (x1, y1), (x2, y2), (x3, y3) (that are in a 2-dimensional plane). What I am trying to figure out is if a fourth point is clicked by a user on the bezier curves whether the click point lies between points 1 and 2 or if lies between points 2 and 3. The click point is only recorded when the line is directly clicked so it must lie between either points 1 and 2 or points 2 and 3.
The lines are randomly created at compile time and can start and end from any (x, y) position.
The 3 sets of points that make up the line are the start point, curve point and the end point. These three points are the control points of the line. A line object is then created from the control points. The control points are randomly created during each run of the program, making the spline different every time.
Is there any specific algorithm that should be followed for this problem. I am coding this in javascript but any c++ or java like pseudo code is fine. Thanks for the help.
Build a LUT (lookup table) for your curve, so that when the user clicks on it, you can resolve the (x,y) coordinate they clicked to the curve's t value (or whatever you called your control variable, of course). Rather than evaluating based on (x,y) coordinates, which is virtually impossible, resolve all four coordinates to t values, and it becomes really simple:
With (x1,y1) being t=0, (x2,y2) being some t=T and (x3,y3) being t=1, if a user clicks anywhere on the curve, we get a new t value. If that value is less than T, the point lies between points 1 and 2, and if it's greater than T, it lies between points 2 and 3.
Build the lookup table should be a one-time operation per curve, run the first time you draw the curve, since that's when you're already mapping t values to (x,y) coordinates, so you can build the reverse mapping "for free". If you don't control the draw code, then you'll have to run your own when you create the curve.
That has one problem: the curve you've given here, defined by three on-curve points, is not the usual way to define a Bezier curve. For Bezier curves, the control points define the curve "hull"; for a quadratic curve (with three points), that means points 1 and 3 are on the curve, but point 2 is very much not. To find the true Bezier curve based on these three points (i.e. the curve that goes through those three points) you'd need to run the algorithm that turns three points into a true curve.
(Full code to tell you how to do that is pretty much beyond the scope of this answer, but I explain it in a long article on Bezier curves, culminating in http://pomax.github.io/bezierinfo/#pointcurves for forming true curves based on three points)
I am still working on my "javascript 3d engine" (link inside stackoverflow).
at First, all my polygons were faces of cubes, so sorting them by average Z was working fine.
but now I've "evolved" and I want to draw my polygons (which may contain more than 4 vertices)
in the right order, namely, those who are close to the camera will be drawn last.
basically,
I know how to rotate them and "perspective"-ize them into 2D,
but don't know how to draw them in the right order.
just to clarify:
//my 3d shape = array of polygons
//polygon = array of vertices
//vertex = point with x,y,z
//rotation is around (0,0,0) and my view point is (0,0,something) I guess.
can anyone help?
p.s: some "catch phrases" I came up with, looking for the solution: z-buffering, ray casting (?!), plane equations, view vector, and so on - guess I need a simple to understand answer so that's why I asked this one. thanks.
p.s2: i don't mind too much about overlapping or intersecting polygons... so maybe the painter's algorthm indeed might be good. but: what is it exactly? how do I decide the distance of a polygon?? a polygon has many points.
The approach of sorting polygons and then drawing them bottom-to-top is called the "Painter's algorithm". Unfortunately the sorting step is in general an unsolvable problem, because it's possible for 3 polygons to overlap each other:
Thus there is not necessarily any polygon that is "on top". Alternate approaches such as using a Z buffer or BSP tree (which involves splitting polygons) don't suffer from this problem.
how do I decide the distance of a polygon?? a polygon has many points.
Painter's algorithm is the simplest to implement, but it works only in very simple cases because it assumes that there is only a single "distance" or z-value for each polygon (which you could approximate to be the average of z-values of all points in the polygon). Of course, this will produce wrong results if two polygons intersect each other.
In reality, there isn't a single distance value for a polygon -- each point on the surface of a polygon can be at a different distance from the viewer, so each point has its own "distance" or depth.
You already mentioned Z-buffering, and that is one way of doing this. I don't think you can implement this efficiently on a HTML canvas, but here's the general idea:
You need to maintain an additional canvas, the "z-buffer", where each pixel's colour represents the z-depth of the corresponding pixel on the main canvas.
To draw a polygon, you go through each point on its surface and draw only those points which are closer to the viewer than any previous objects, as indicated by the z-buffer.
I think you will have some ideas by investigating BSP tree ( binary spaces partition tree ), even if the algo will require to split some of your polygon in two.
Some example could be find here http://www.devmaster.net/articles/bsp-trees/ or by google for BSP tree. Posting some code as a reply is, in my opinion, not serious since is a complex topic.