Given a list of Vector3s in no order, I would like to create an ordering of the elements such that when I draw a Shape from those points, I will be outlining the shape without ever going across its face.
How would you do this? I think the first thing is to always pick the point closest to you, but what if more than one point have exactly the same distance from a point?
Let's call the point we're on x, and a potential "neighbor" (point whose distance to x is minimal) y where there can be multiple ys.
I've thought about these two approaches:
Find the center of mass by averaging out the positions of the points, then make sure that going from x to y never goes through the center. This approach has many problems like 1. it's not guaranteed to cross the CoM or 2. imagine a fidget spinner with just two circles, and one much smaller than the other. When tracing the smaller bit, the center of mass is never crossed (or come close to), but we might still ran into problems
Randomly pick any other three points, and make sure that y is not within the triangle created. But there are cases where y is the correct choice, but still falls within the triangle (imagine a shape one of whose edge is created by two tangent circles).
Any help would be much appreciated!
Related
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've been looking for a way to find the the vertices within a certain radius from a given point. One way to this is brute force. After selection of a point (raypicking), loop over all vertices, check whether it is within a set radius and voila. However, this tends to get quite slow for models with lots of vertices.
What I would want to do is use raypicking to select a point on the model. This would give me the face this point is on. Then from that face I can get the vertices belonging to that face. These vertices can be "shared" over faces. This might allow me to forward search from this point, flagging visited vertices and stop whenever the distance reaches the set maximum (radius). However, from what I can see from a dump of the geometry I can get the vertices belonging to a face directly, but there's no way to get the faces that a vertex belongs to. That is without preprocessing. Am I right here, or did I miss something?
Ok, so... here's what I'm trying to do. In HTML/JS. With a math knowledge that would embarrass a twelve-year old.
With, say, 20 checkboxes on a page, the user picks.. five of them. A circle then appears on the page (I'd anticipate using one of the canvas libraries like Raphael) with their five answers arranged around it. So that's the first thing I don't know how to do; split a circle into x equal segments. Strange, foreign terms like "cos" and "sin" are looming on the horizon.
But then it gets even more fun: the user can click a point on the circle. And that point is translated into some sort of percentage value for each of the segments. So if the user checks Happy, Grumpy, Sneezy, Dopey and Bashful, and clicks the circle, I can tell them that they're 37% Happy, 42% Dopey and 21% Sneezy.
The best analogy is probably a colour-picker wheel, but I can't find any JS ones that I could repurpose. A pie chart is close - and there's a nice Raphael demo - but I've got a feeling that the fixed boundaries of a pie chart segment is going to take me down the wrong route for estimating percentage positions within the circle as a whole.
So, given my vague and poorly thought out request, and the noticeable absence of any "here's what I've tried so far" code - because I don't have the foggiest where to even start, apart from high school trigonometry books, can anyone suggest any code libraries or snippets that might get me at least pointing in the right direction?
Thanks :)
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Edit :
Wow, thanks for all the answers. I'll try to refine the question/s a bit:
I'm going to need to draw a circle with x number of points arranged equally around its circumference. The number of points will correlate to the number of checkboxes that the user checked.
That's the first bit I'm stuck on: I reckon that I can draw a circle, with dimensions and at a position on the page of my own choosing, (using Raphael or similar canvas library) but how can I calculate what the x/y pixel coordinates of those points should be on the circle circumference?
The second bit: the user then clicks anywhere in the circle. I guess what I'd do is calculate how far each of the circumference points are from that user click point - I'm not sure how to do that, apart from a vague suspicion it involves imaginary triangles - and then how much of a total distance each of those distances are. That last bit, at least, I can manage.
Actually, this is starting to make sense. I'm still not sure how the trig stuff works but it's amazing what typing your problem out so that strangers will understand it can do to help your own understanding...
You need to revise your question to explain more clearly what you are trying to do. In the meantime, I can give you the following information that may help you get started.
First of all, you'll need to know the locations of the five checkboxes that the user selects. In order to do this, the jQuery library offers some convenient functions such as $.position and $.offset.
Your question does not make clear exactly how the circle you want to draw is positioned in relation to the five checkboxes that the user clicks. As for actually drawing the circle, you may want to use something like the HTML5 canvas element. I've not yet used it myself, so I can't tell you much about it.
You might want to try asking another question on StackOverflow about how to draw a circle on your web page once you've computed the center and radius of the circle.
As for doing the math about the circle, you need to know that a circle can be parameterized on an x-y plane by the following equations:
x = x0 + r cos(theta)
y = y0 + r sin(theta)
where (x0,y0) is the center of the circle, r is the radius of the circle, and theta ranges over 0 to 2*Pi radians (0 to 360 degrees)
Let us know more about what you're doing and we can give you some more specific information.
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
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.