I have a set of points which form some curve, see picture. This curve consists of straight and curved sections and I do not know exactly where they start or end. In the presented picture I know at least five pieces: two straight sections, one section with a constant radius, two other types of curves. I want to be able to recognise those sections, to separate them from each other and to work with them as with separate curves.
Update.
In my opinion splines would not work in the way I want, and not surely Besier. I was thinking about Non-linear Least Squares but was not sure if it suits this case. If I am able to separate the parts, then I can use Linear Least Squares for straights and some non-linear for other parts. Otherwise, I need some universal method which will work for all types of curves: 1, 2 and 3 power.
Please share your thoughts.
Thank you.
For a bent tube I would suggest multidimensional spline fitting. Your tube does not look like a composition of of straight and curved sections, but curves smoothly all the way.
http://en.wikipedia.org/wiki/Spline_interpolation
This is the starting point if you want to read further.
Taking a guess I suppose you are not experienced with splines, so if you could put out the dataset in table form me or some other reader could interpolate the given data with a 3D spline curve.
Related
I am trying to create an algorithm that can take a set of objects and organize them in a given area such that a box bounding all of the shapes is optimized (either by area used, or by maximizing the span along one of the dimensions, etc.). All of the shapes are closed and bounded.
The purpose of this is to try and minimize material waste from using a laser cutter. The shapes are generated in CAD and can read into this algorithm. The algorithm will then take arguments for the working area (effective laser cutting area) as well as the minimum separation between any two objects, then attempt to organize the objects within the specified dimensions while trying to minimize the area usage. Alternatively, the algorithm can also try to maximize the object locations along one axis while minimizing the span along the other dimension. This would be akin to cutting off a smaller workpiece to cut from.
Ideally, the algorithm would be able to make translations AND rotations, but rotations aren't necessary.
For example, this Picture depicts the required transformation.
It should work with an arbitrary, but small (<25) number of objects.
Lastly, I don't expect anyone to solve this for me, but I would appreciate help toward either finding an algorithm that can do this, or developing my own. Thank you.
I dont know to what extent you want to create said algorithm or how you want to implement it, But i know of a program called OptiNest that can do what you ask. It organizes geometric shapes to optimize the layout and minimize waste on a plane, i think in an autocad format.
Currently I'm working on a little project just for a bit of fun. It is a C++, WinAPI application using OpenGL.
I hope it will turn into a RTS Game played on a hexagon grid and when I get the basic game engine done, I have plans to expand it further.
At the moment my application consists of a VBO that holds vertex and heightmap information. The heightmap is generated using a midpoint displacement algorithm (diamond-square).
In order to implement a hexagon grid I went with the idea explained here. It shifts down odd rows of a normal grid to allow relatively easy rendering of hexagons without too many further complications (I hope).
After a few days it is beginning to come together and I've added mouse picking, which is implemented by rendering each hex in the grid in a unique colour, and then sampling a given mouse position within this FBO to identify the ID of the selected cell (visible in the top right of the screenshot below).
In the next stage of my project I would like to look at generating more 'playable' terrains. To me this means that the shape of each hexagon should be more regular than those seen in the image above.
So finally coming to my point, is there:
A way of smoothing or adjusting the vertices in my current method
that would bring all point of a hexagon onto one plane (coplanar).
EDIT:
For anyone looking for information on how to make points coplanar here is a great explination.
A better approach to procedural terrain generation that would allow
for better control of this sort of thing.
A way to represent my vertex information in a different way that allows for this.
To be clear, I am not trying to achieve a flat hex grid with raised edges or platforms (as seen below).
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I would like all the geometry to join and lead into the next bit.
I'm hope to achieve something similar to what I have now (relatively nice undulating hills & terrain) but with more controllable plateaus. This gives me the flexibility of cording off areas (unplayable tiles) later on, where I can add higher detail meshes if needed.
Any feedback is welcome, I'm using this as a learning exercise so please - all comments welcome!
It depends on what you actually want and what you mean by "more controlled".
Do you want to be able to say "there will be a mountain on coordinates [11, -127] with radius 20"? Complexity of this this depends on how far you want to go. If you want just mountains, then radial gradients are enough (just add the gradient values to the noise values). But if you want some more complex shapes, you are in for a treat.
I explore this idea to great depth in my project (please consider that the published version is just a prototype, which is currently undergoing major redesign, it is completely usable a map generator though).
Another way is to make the generation much more procedural - you just specify a sequence of mathematical functions, which you apply on the terrain. Even a simple value transformation can get you very far.
All of these methods should work just fine for hex grid. If artefacts occur because of the odd-row shift, then you could interpolate the odd rows instead (just calculate the height value for the vertex from the two vertices between which it is located with simple linear interpolation formula).
Consider a function, which maps the purple line into the blue curve - it emphasizes lower located heights as well as very high located heights, but makes the transition between them steeper (this example is just a cosine function, making the curve less smooth would make the transformation more prominent).
You could also only use bottom half of the curve, making peaks sharper and lower located areas flatter (thus more playable).
"sharpness" of the curve can be easily modulated with power (making the effect much more dramatic) or square root (decreasing the effect).
Implementation of this is actually extremely simple (especially if you use the cosine function) - just apply the function on each pixel in the map. If the function isn't so mathematically trivial, lookup tables work just fine (with cubic interpolation between the table values, linear interpolation creates artefacts).
Several more simple methods of "gamification" of random noise terrain can be found in this paper: "Realtime Synthesis of Eroded Fractal Terrain for Use in Computer Games".
Good luck with your project
Im trying to transform a path along an arc.
My project is running on osX 10.8.2 and the painting is done via CoreAnimation in CALayers.
There is a waveform in my project which will be painted by a path. There are about 200 sample points which are mirrored to the bottom side. These are painted 60 times per second and updated to a song postion.
Please ignore the white line, it is just a rotation indicator.
What i am trying to achieve is drawing a waveform along an arc. "Up" should point to the middle. It does not need to go all the way around. The waveform should be painted along the green circle. Please take a look at the sketch provided below.
Im not sure how to achieve this in a performant manner. There are many points per second that need coordinate correction.
I tried coming up with some ideas of my own:
1) There is the possibility to add linear transformations to paths, which, i think, will not help me here. The only thing i can think of is adding a point, rotating the path with a transformation, adding another point, rotating and so on. But this would be very slow i think
2) Drawing the path into an image and bending it would surely lead to image-artifacts.
3) Maybe the best idea would be to precompute sample points on an arc, then save save a vector to the center. Taking the y-coordinates of the waveform, placing them on the sample points and moving them along the vector to the center.
But maybe i am just not seeing some kind of easy solution to this problem. Help is really appreciated and fresh ideas very welcome. Thank you in advance!
IMHO, the most efficient way to go (in terms of CPU usage) would be to use some form of pre-computed approach that would take into account the resolution of the display.
Cleverly precomputed values
I would go for the mathematical transformation (from linear to polar) and combine two facts:
There is no need to perform expansive mathematical computation
There is no need to render two points that are too close from each other
I have no ready-made algorithm for you, but you could use a pre-computed sin or cos table, and match the data range to the display size in order to work with integers.
For instance imagine we have some data ranging from 0 to 1E6 and we need to display the sin value of each point in a 100 pix height rectangle. We can use a pre-computed sin table and work with integers. This way displaying the sin value of a point would be much quicker. This concept can be refined to get a nicer result.
Also, there are some ways to retain only significant points of a curve so that the displayed curve actually looks like the original (see the Ramer–Douglas–Peucker algorithm on wikipedia). But I found it to be inefficient for quickly displaying ever-changing data.
Using multicore rendering
You could compute different areas of the curve using multiple cores (can be tricky)
Or you could use pre-computing using several cores, and one core to do finish the job.
I'm trying to develop an app which allows you to walk around, and where you walked will be drawn on a map. I have this all working fine, but I'm finding that even with a reasonably accurate GPS location the points still jump around a bit. When drawn on a map this has the effect of creating a squiggly or zig-zag line.
I'm looking for suggestions/strategies on how to smooth the data, so that the line drawn on the map is more of a smooth best fit, rather than an accurate point to point drawing.
There are many different types of smoothing algorithms you could apply to the data (for a few starting points, see this Wikipedia article). The only way to know for sure which is/are suitable for your application is to implement and test them.
Simple or weighted moving averages are fairly common (taking the last n samples and averaging them), but have the problem of lagging behind the data. A common one for filtering signal noise is a high-pass filter, which attenuates small (noisy) movements while passing through larger ones. Apple has some code for this in their AccelerometerGraph sample.
I'd suggest trying those out first as they're easy to implement, before looking at the move complex ones.
If I have a graph of a reasonable size (e.g. ~100 nodes, ~40 edges coming out of each node) and I want to represent it in R^3 (i.e. map each node to a point in R^3 and draw a straight line between any two nodes which are connected in the original graph) in a way which would make it easy to understand its structure, what do you think would make a good drawing criterion?
I know this question is ill-posed; it's not objective. The idea behind it is easier to understand with an extreme case. Suppose you have a connected graph in which each node connects to two and only two other nodes, except for two nodes which only connect to one other node. It's not difficult to see that this graph, when drawn in R^3, can be drawn as a straight line (with nodes sprinkled over the line). Nevertheless, it is possible to draw it in a way which makes it almost impossible to see its very simple structure, e.g. by "twisting" it as much as possible around some fixed point in R^3. So, for this simple case, it's clear that a simple 3D representation is that of a straight line. However, it is not clear what this simplicity property is in the general case.
So, the question is: how would you define this simplicity property?
I'm happy with any kind of answer, be it a definition of "simplicity" computable for graphs, or a greedy approximated algorithm which transforms graphs and that converges to "simpler" 3D representations.
Thanks!
EDITED
In the mean time I've put force-based graph drawing ideas suggested in the answer into practice and wrote an OCaml/openGL program to simulate how imposing an electrical repulsive force between nodes (Coulomb's Law) and a spring-like behaviour on edges (Hooke's law) would turn out. I've posted the video on youtube. The video starts with an initial graph of 100 nodes each with approximately 1-2 outgoing edges and places the nodes randomly in 3D space. Then all the forces I mentioned are put into place and the system is left to move around subject to those forces. In the beginning, the graph is a mess and it's very difficult to see the structure. Closer to the end, it is clear that the graph is almost linear. I've also experience with larger-sized graphs but sometimes the geometry of the graph is just a mess and no matter how you plot it, you won't be able to visualise anything. And here is an even more extreme example with 500 nodes.
One simple approach is described, e.g., at http://en.wikipedia.org/wiki/Force-based_algorithms_%28graph_drawing%29 . The underlying notion of "simplicity" is something like "minimal potential energy", which doesn't really correspond to simplicity in any useful sense but might be good enough in practice.
(If you have 100 nodes of degree 40, I have some doubt as to whether any way of drawing them is going to reveal much in the way of human-accessible structure. That's a lot of edges. Still, good luck!)