I have a quadratic curve in canvas (defined by start, end and control point). What I need is to get more of the curve's coordinates (eg. the middle, quarters etc.). With that, I'd be able (after a few iterations of finding coordinates) to animate an object based on this curve (I know it can be done easily with SVG, but that's not an option).
image taken from http://www.html5canvastutorials.com/tutorials/html5-canvas-quadratic-curves/
Is there an easy way to do this? When I checked for bisecting curve on google, the only thing I found was some complicated Math including drawing, which is out of the question.
These functions may be of interest to you:
Kinetic.Path.getPointOnCubicBezier = function(pct, P1x, P1y, P2x, P2y, P3x, P3y, P4x, P4y)
Kinetic.Path.getPointOnQuadraticBezier = function(pct, P1x, P1y, P2x, P2y, P3x, P3y)
I am seeking a similar solution for bezier curves. How do I get a list of the coordinates on the curve?
To solve your problem, perhaps you could plugin the x-coordinate (or y-coordinate) into the quadratic equation:
y = a*x^2 + b*x + c
I can't seem to find a function that returns the coordinates of the curve. It would be nice to get a list of the coordinates for any curve / shape.
Related
Explanation of the problem:
I have points with (x,y,z) coordinates at two+ distinct times. For convenience, they can be imagined as irregularly spaced points along the surface of an inverted paraboloid.
There is some minimal thickness to the paraboloid. The paraboloid changes shape slightly as time proceeds (like a balloon inflating) and when it does so, all of the points move.
By substracting the coordinates at time2 - time1, I can get the displacement vectors at each point.
It is important to note (and I suspect this might be the source of the problem) that at the first time point, the x and y coordinates range from 0 to 2000, and the z coordinates are all within a narrower range - say 350 to 450. During the deformation, each point has an x component of displacement, y component, and z component.
The x and y components are small (~50 at most), while the z component is the largest (goes up to 400 near the center, much less near the edges).
Using weighted moving least squares at the location of each point, I am trying to fit the components of displacements to a second degree polynomial surface in terms of the original x,y,z coordinates of the point: eg.
x component of
displacement = ax^2 + bxy + cx + dy^2.. + hz^2 + iz + j
I use the lsqr function in MATLAB,like so, looping through each point for each time interval:
Ux = displacements{k,1}(:,1);
Cx = lsqr((adjust_B_matrix'*W*adjust_B_matrix),(adjust_B_matrix'*W*Ux),1e-7,10000);
W is the weight matrix, and adjust_B_matrix is the matrix of all (x,y,z) coordinates at time 1, shifted so that they're all centered around the point at which I'm trying to fit the function.
What is going wrong?
It's just not working -- once I have the functions, they're re-centered around the actual coordinates of the points.
But once I plot the resulting points (initial pointx + displacementx, initial pointy + displacementy, initial pointz + displacementz) by plugging in the coordinates at time 1 into the now-discovered functions, it just spits out a surface that looks just like the surface at time 1.
What might be going wrong? Things I have tried:
It's not an issue with the code itself- I generated 'fake' data using a grid of points and it worked perfectly. The predicted locations were superimposed with the actual coordinates and I was able to get back the function I started with. But in my trial example, I used x,y,z from 0 to 5, evenly spaced.
Global fitting works (but I need local fitting...).
I tried MATLAB's curve fitting toolbox and just tried to fit one of the displacements to only x and y coordinates, globally. It worked perfectly.
I think I shouldn't have a singular matrix issue because I use a large radius (around 75-80) points in the calculations, somewhat dispersed in 3D space.
Suspicions:
I think it has to do with the uneven distribution of initial (x,y,z) coordinates, but I don't know why or how to fix the issue, or even what method I can use.
If you read this far, thank you so much. Any advice would be greatly appreciated.
Figure for reference:
green = predicted points at time 2. Overlapping mostly with red, the actual coordinates of the points at time 1.
blue is the correct coordinates of points at time 2 (this is where the green ones should be if things were working).
image
Updated link for files:
http://a.tmp.ninja/eWfkNmFZyTFk.zip
Contents - code, sample data (please load the .mat files).
I can't actually access the code you posted, so here's some general suggestions.
It does look like the curve fitting toolbox has tools that do exactly what you are looking for, checkout the bottom of this page: https://www.mathworks.com/help/curvefit/polynomial.html#bt9ykh.
It looks like for whatever your learned function for the displacement is just very small or zero everywhere. I suspect the issue is just a minor typo/error on your part somewhere in your pipeline, possibly translating what you have to work with the fit function will reveal the issue.
This really shouldn't be the issue, but in the future if you had much more unbalanced data you could normalize it all before fitting (x_norm = (x - x_mu)/x_std).
Also, I don't think this is your problem either, but you can check if your matrix is close to singular by checking the condition number using the cord() function. So you could check cond(adjust_B_matrix'Wadjust_B_matrix). Second, If you check the documentation for lsqr there is an option to get a debug return flag, that is worth checking too.
I have a multiple plants in a single binary image. How would I identify each leaf in the image assuming that each leaf is approximately elliptical?
example input: http://i.imgur.com/BwhLVmd.png
I was thinking a good place to start would be finding the tip of each leaf and then getting the center of each plant. Then I could fit the curves starting from the tip and then going to the center. I've been looking online and saw something involving a watershed method, but I do not know where to begin with that idea.
You should be aware that these things are tricky to get working robustly - there will always be a failure case.
This said, I think your idea is not bad.
You could start as follows:
Identify the boundary curve of each plant (i.e. pixels with both foreground and background in their neighbourhood).
Compute the centroid of each plant.
Convert each plant boundary to a polar coordinate system, with the centroid as the origin. This amounts to setting up a coordinate system with the distance of each boundary curve point on the Y axis and the angle on the X axis.
In this representation of the boundary curve, try to identify maxima; these are the tips of the leaves. You will probably need to do some smoothing. Use the parts of the curve before and after the maxima the start fitting your ellipses or some other shape.
Generally, a polar coordinate system is always useful for analysing stuff thats roughly circular.
To fit you ellipses, once you have a rough initial position, I would probably try an EM-style approach.
I would do something like this (I is your binary image)
I=bwmorph(bwmorph(I, 'bridge'), 'clean');
SK=bwmorph(I, 'skel', Inf);
endpts = bwmorph(SK,'endpoints');
props=regionprops(I, 'All');
And then connect every segment from the centroids listed in props.centroid to the elements of endpts that should give you your leaves (petals?).
A bit of filtering is probably necessary, bwmorph is your friend. Have fun!
I have a convex polygon in 3D. For simplicity, let it be a square with vertices, (0,0,0),(1,1,0),(1,1,1),(0,0,1).. I need to arrange these vertices in counter clockwise order. I found a solution here. It is suggested to determine the angle at the center of the polygon and sort them. I am not clear how is that going to work. Does anyone have a solution? I need a solution which is robust and even works when the vertices get very close.
A sample MATLAB code would be much appreciated!
This is actually quite a tedious problem so instead of actually doing it I am just going to explain how I would do it. First find the equation of the plane (you only need to use 3 points for this) and then find your rotation matrix. Then find your vectors in your new rotated space. After that is all said and done find which quadrant your point is in and if n > 1 in a particular quadrant then you must find the angle of each point (theta = arctan(y/x)). Then simply sort each quadrant by their angle (arguably you can just do separation by pi instead of quadrants (sort the points into when the y-component (post-rotation) is greater than zero).
Sorry I don't have time to actually test this but give it a go and feel free to post your code and I can help debug it if you like.
Luckily you have a convex polygon, so you can use the angle trick: find a point in the interior (e.g., find the midpoint of two non-adjacent points), and draw vectors to all the vertices. Choose one vector as a base, calculate the angles to the other vectors and order them. You can calculate the angles using the dot product: A · B = A B cos θ = |A||B| cos θ.
Below are the steps I followed.
The 3D planar polygon can be rotated to 2D plane using the known formulas. Use the one under the section Rotation matrix from axis and angle.
Then as indicated by #Glenn, an internal points needs to be calculated to find the angles. I take that internal point as the mean of the vertex locations.
Using the x-axis as the reference axis, the angle, on a 0 to 2pi scale, for each vertex can be calculated using atan2 function as explained here.
The non-negative angle measured counterclockwise from vector a to vector b, in the range [0,2pi], if a = [x1,y1] and b = [x2,y2], is given by:
angle = mod(atan2(y2-y1,x2-x1),2*pi);
Finally, sort the angles, [~,XI] = sort(angle);.
It's a long time since I used this, so I might be wrong, but I believe the command convhull does what you need - it returns the convex hull of a set of points (which, since you say your points are a convex set, should be the set of points themselves), arranged in counter-clockwise order.
Note that MathWorks recently delivered a new class DelaunayTri which is intended to superseded the functionality of convhull and other older computational geometry stuff. I believe it's more accurate, especially when the points get very close together. However I haven't tried it.
Hope that helps!
So here's another answer if you want to use convhull. Easily project your polygon into an axes plane by setting one coordinate zero. For example, in (0,0,0),(1,1,0),(1,1,1),(0,0,1) set y=0 to get (0,0),(1,0),(1,1),(0,1). Now your problem is 2D.
You might have to do some work to pick the right coordinate if your polygon's plane is orthogonal to some axis, if it is, pick that axis. The criterion is to make sure that your projected points don't end up on a line.
Anyone have any starting tips for me? I want to learn from this (ie Don't want to be lazy and have someone answer this for me).
I would like to develop my understanding of mathematical 3D surfaces. My own personal project is to produce a 3D surface/graph of the concourse structure in MATLAB.
I found a link with good pictures of its geometry here. I am not expecting to get it 100% perfectly but I'd like to come close!
At the end of this exercise I would like to have a mathematical definition of the geometry as well as a visual representation of the surface. This can involve cartesian equations, parametric equations, matrices, etc.
Any help would be very much appreciated!
To give some specific advice for MATLAB:
I would load in the 'section' image from the web page you have linked, and display this in a MATLAB figure window. You can then try plotting lines over the top until you find one that fits nicely. So you might do something like:
A = imread('~/Desktop/1314019872-1244-n364-1000x707.jpg');
imshow(A)
hold on
axis on
%# my guess at the function - obviously not a good fit
x = [550:900];
plot(x, 0.0001*x.^2 + 300)
Of course, you might want to move the position of the origin or crop the picture and so on.
As an arguably better alternative to this trial-and-error method, you could trace the outline of the section (e.g by clicking points with something like ginput), and then use one of MATLAB's curve-fitting tools (e.g. fit) to fit a function to the data.
The final 3D shape looks to me (at a casual glance) to be a 3D revolution of the section shape around a central axis. Use of a cylindrical coordinate system could therefore be a good idea.
The final plotting of your 3D shape could be done with a function such as surf or mesh.
I would start by defining a function that defines for each x, y coordinate whether there is a point z, and if so with which altitude.
The shape reminds me a bit of a log or a square root.
like what i want is if i move my finger fast on the iphone screen , then i want like something that it make a proper curve using quartz 2d or opengl es whatever.
i want to draw a path in curve style......
i had seen that GLPaint(OpenglES) example ,but it will not help me alot , considering if your finger movement is fast.....
something like making a smooth curve.....
any one have some kind of example please tell me
thanks
Edit: Moved from answer below:
thanks to all.......
but i had tried the bezier curve algo with two control points but problem is first how to calculate the control points whether there is no predefined points....
as i mentioned my movement of finger is fast...... so most of the time i got straight line instead of curve, due to getting less number of touch points.......
now as mark said piecewise fashion, ihad tried it like considering first four touch points and render them on screen , then remove the first point then again go for next four points ex. step 1: 1,2,3,4 step 2: 2,3,4,5 like that where as in that approach i got an overlap , which is not the issue actually , but didn't get smooth curve........
but for fast movement of finger i have to find something else?????
Depending on the number of sample points you are looking at, there are two approaches that I would recommend:
Simple Interpolation
You can simply sample the finger location at set intervals and then interpolate the sample points using something like a Catmull-Rom spline. This is easier than it sounds since you can easily convert a Catmull-Rom spline into a series of cubic Bezier curves.
Here's how. Say you have four consecutive sample points P0, P1, P2 and P3, the cubic Bezier curve that connects P1 to P2 is defined by the following control points:
B0 = P1
B1 = P1 + (P2 - P0)/6
B3 = P2 + (P1 - P3)/6
B4 = P2
This should work well as long as your sample points aren't too dense and it's super easy. The only problem might be at the beginning and end of your samples since the first and last sample point aren't interpolated in an open curve. One common work-around is to double-up your first and last sample point so that you have enough points for the curve to pass through each of the original samples.
To get an idea of how Catmull-Rom curves look, you can try out this Java applet demonstrating Catmull-Rom splines.
Fit a curve to your samples
A more advance (and more difficult) approach would be to do a Least Squares approximation to your sample points. If you want try this, the procedure looks something like the following:
Collect sample points
Define a NURBS curve (including its knot vector)
Set up a system of linear equations for the samples & curve
Solve the system in the Least Squares sense
Assuming you can pick a reasonable NURBS knot vector, this will give you a NURBS curve that closely approximates your sample points, minimizing the squared distance between the samples and your curve. The NURBS curve can even be decomposed into a series of Bezier curves if needed.
If you decide to explore this approach, then the book "Curves and Surfaces for CAGD" by Gerald Farin, or a similar reference, would be very helpful. In the 5th edition of Farin's book, section 9.2 deals specifically with this problem. Section 7.8 shows how to do this with a Bezier curve, but you'd probably need a high-degree curve to get a good fit.
Naaff gives a great overview of the NURBS technique. Unfortunately, I think generating a smooth bezier on-the-fly might be too much for the iPhone. I write drawing apps, and getting a large number of touchesMoved events per second is quite a challenge to begin with. You really need to optimize your drawing code just to get good performance while recording individual points - much less constructing a bezier path.
If you end up going with a bezier or NURBS curve representation - you'll probably have to wait until the user has finished touching the screen to compute the smoothed path. Doing the math continuously as the user moves their finger (and then redrawing the entire recomputed path using Quartz) is not going to give you a high enough data collection rate to do anything useful...
Good luck!
Do something like Shadow suggested. Get the position of the touch with some frequency and then make a Bézier curve out of it. This is how paths are drawn with a mouse (or tablet) in programs like Illustrator.