I would like to produce a realistic 3D demonstration of a ball rolling down a Conical Helix path. The reference that has helped me get close to a solution can be found here. [I am creating my solution in Actionscript 3, using Stage3D, but would be happy to have any suggested coding solutions in other languages, using other 3D frameworks, with which you may be more familiar.]
As I entered the title for my posting, the system pointed me to a wealth of "Questions that may already have your answer", and that was helpful, and I did check each of them out. Without wanting to hijack an existing thread, I should say that this oneincludes a good deal of very helpful commentary about the general subject, but does not get to the specific challenges I have been unable to resolve.
Using the cited reference, I am happy with this code snippet that traces the path I would like the ball to follow. [N.B. My reference, and most other math-based references, treat Z as being up-down; my usage, however, is the more usual 3D graphics of Y for up-down.]
This code is executed for each frame.
ft += 0.01; // Where ft is a global Number.
var n:Number = Math.pow (0.5, (0.15 * ft));
// Where s is a constant used to scale the overall path.
obj.moveTo (
(s * n * Math.cos (2.0 * ft)),
(s * n),
(s * n * Math.sin (2.0 * ft))
);
The ball follows a nice path, and owing to the lighting and other shader code, a very decent effect is viewed in the scene.
What is not good about my current implementation is that the ball does not appear to be rolling along that path as it moves from point to point. I am not using any physics engine, and am not seeking any solution dealing with collisions, but I would like the ball to correctly demonstrate what would be happening if the movement were due to the ball rolling down a track.
So, to make a little more clear the challenge, let's say that the ball is a billiard ball with the stripe and label for #15. In that case, the visual result should be that the number 15 should be turning head over heals, but, as you can probably surmise from the name of my obj.moveTo() function, that only results in changes in position of the 3D object, not its orientation.
That, finally, brings me to the specific question/request. I have been unable to discover what rotation changes must be synchronized with each positional change in order to correctly demonstrate the way the billiard ball would appear if it rolled from point 1 from point 2 along the path.
Part of the solution appears to be:
obj.setRotation ((Math.atan2 (Math.sin (ft), Math.cos (ft))), Vector3D.Y_AXIS);
but that is still not correct. I hope there is some well-known formula that I can add to my render code.
I want to know certain method so that i can tell if a pixel is inside a 4-point polygon or quadrilateral figure (not necessarily to be rectangle) given the 4 co-ordinates of that polygon.
I tried several methods, but none of them worked really well.
Thanx and Regards
Uday Gupta
A simple method is to use areas : You first decompose your polygon into two triangles ABC and CDA, and check whether the point is in either triangle.
For that, assuming the triangle ABC for example and your point to test is M, you can test whether the area of the triangle ABC is equal to the sum of the areas of ABM + BCM + CAM.
Computing the area of a triangle is done using half the norm of the cross product.
Another solution that directly uses cross products can be found here:
http://www.blackpawn.com/texts/pointinpoly/default.html
I have to do some image processing but I don't know where to start. My problem is as follows :-
I have a 2D fiber image (attached with this post), in which the fiber edges are denoted by white color and the inside of the fiber is black. I want to choose any black pixel inside the fiber, and travel from it along the length of the fiber. This will involve comparing the contrast with the surrounding pixels and then travelling in the desired direction. My main aim is to find the length of the fiber
So can someone please tell me atleast where to start? I have made a rough algorithm in my mind on how to approach my problem but I don't know even which software/library to use.
Regards
Adi
EDIT1 - Instead of OpenCV, I started using MATLAB since I found it much easier. I applied the Hough Transform and then Houghpeaks function with max no. of peaks = 100 so that all fibers are included. After that I got the following image. How do I find the length now?
EDIT2 - I found a research article on how to calculate length using Hough Transform but I'm not able to implement it in MATLAB. Someone please help
If your images are all as clean as the one you posted, it's quite an easy problem.
The very first technique I'd try is using a Hough Transform to estimate the line parameters, and there is a good implementation of the algorithm in OpenCV. After you have them, you can estimate their length any way you want, based on whatever other constraints you have.
Problem is two-fold as I see it:
1) locate start and end point from your starting position.
2) decide length between start and end points
Since I don't know your input data I assume it's pixel data with a 0..1 data on each pixel representing it's "whiteness".
In order to find end points I would do some kind of WALKER/AI that tries to walk in different locations, knowing original pos and last traversed direction then continuing along that route until "forward arc" is all white. This assumes fiber is somewhat straight (is it?).
Once you got start and end points you can input these into a a* path finding algorithm and give black pixels a low value and white very high. Then find shortest distance between start and end point, that is the length of the fiber.
Kinda hard to give more detail since I have no idea what techniques you gonna use and some example input data.
Assumptions:
-This image can be considered a binary image where there are only 0s(black) and 1s(white).
-all the fibers are straight and their starting and ending points are on borders.
-we can come up with a limit for thickness in fiber(thickness of white lines).
Under these assumptions:
start scanning the image border(start from wherever you want in whichever direction you want...just be consistent) until you encounter with the first white pixel.At this point your program will understand that this is definitely a starting point. By knowing this, you will gather all the white pixels until you reach a certain limit(or a threshold). The idea here is, if there is a fiber,you will get the angle between the fiber and the border the starting point is on...of course the more pixels you get(the inner you get)the surer you will be in the end. This is the trickiest part. after somehow ending up with a line...you need to calculate the angle(basic trigonometry). Since you know the starting point, the width/height of the image and the angle(or cos/sin of those) you will have the exact coordinate of the end point. Be advised...the exactness here is not really what you might have understood because we may(the thing is we will) have calculation errors in cos/sin values. So you need to hold the threshold as long as possible. So your end point will not be a point actually but rather an area indicating possibility that the ending point is somewhere inside that area. The rest is just simple maths.
Obviously you can put too much detail in this method like checking the both white lines that makes the fiber and deciding which one is longer or you can allow some margin for error since those lines will not be straight properly...this is where a conceptual thickness comes to the stage etc.
Programming:
C# has nice stuff and easy for you to use...I'll put some code here...
newBitmap = new Bitmap(openFileDialog1.FileName);
for (int x = 0; x < newBitmap.Width; x++)
{
for (int y = 0; y < newBitmap.Height; y++)
{
Color originalColor = newBitmap.GetPixel(x, y);//gets the pixel value...
//things go here...
}
}
you'll get the image from a openfiledialog and bitmap the image. inside the nested for loop this code scans the image left-to-right however you can change this...
Since you know C++ and C, I would recommend OpenCV
. It is open-source so if you don't trust anyone like me, you won't have a problem ;). Also if you want to use C# like #VictorS. Mentioned I would use EmguCV which is the C# equivilant of OpenCV. Tutorials for OpenCV are included and for EmguCV can be found on their website. Hope this helps!
Download and install the latest version of 3Dslicer,
Load your data and go the the package>EM segmenter without Atlas>
Choose your anatomical tree in 2 different labels, the back one which is your purpose, the white edges.
The choose the whole 2D image as your ROI and click on segment.
Here is the result, I labeled the edges in green and the black area in white
You can modify your tree and change the structures you define.
You can give more samples to your segmentation to make it more accurate.
I am using the Path class and a BezierSegment in my Silverlight application.
I am drawing different arches and lines using it.
What I'd like to do is get the middle of the my arch / line.
I noticed that the BezierSegment doesn't go through the second point, but merely comes near to it.
How can I get the real curve point (its coordinates) ?
The BezierSegment class uses cubic Bézier curves, and this Wikipedia page gives the equation of such curves. You can take this equation and put t = 0.5 into it to get a point somewhere on the curve.
Note that the point at t = 0.5 isn't guaranteed to be exactly half-way along the curve from the start to the end. Getting the exact midpoint of the curve is more complicated, and if you really need that, you'll be best off asking how to calculate it at https://math.stackexchange.com/.
I do not have any experience with programming fractals. Of course I've seen the famous Mandelbrot images and such.
Can you provide me with simple algorithms for fractals.
Programming language doesn't matter really, but I'm most familiar with actionscript, C#, Java.
I know that if I google fractals, I get a lot of (complicated) information but I would like to start with a simple algorithm and play with it.
Suggestions to improve on the basic algorithm are also welcome, like how to make them in those lovely colors and such.
Programming the Mandelbrot is easy.
My quick-n-dirty code is below (not guaranteed to be bug-free, but a good outline).
Here's the outline:
The Mandelbrot-set lies in the Complex-grid completely within a circle with radius 2.
So, start by scanning every point in that rectangular area.
Each point represents a Complex number (x + yi).
Iterate that complex number:
[new value] = [old-value]^2 + [original-value] while keeping track of two things:
1.) the number of iterations
2.) the distance of [new-value] from the origin.
If you reach the Maximum number of iterations, you're done.
If the distance from the origin is greater than 2, you're done.
When done, color the original pixel depending on the number of iterations you've done.
Then move on to the next pixel.
public void MBrot()
{
float epsilon = 0.0001; // The step size across the X and Y axis
float x;
float y;
int maxIterations = 10; // increasing this will give you a more detailed fractal
int maxColors = 256; // Change as appropriate for your display.
Complex Z;
Complex C;
int iterations;
for(x=-2; x<=2; x+= epsilon)
{
for(y=-2; y<=2; y+= epsilon)
{
iterations = 0;
C = new Complex(x, y);
Z = new Complex(0,0);
while(Complex.Abs(Z) < 2 && iterations < maxIterations)
{
Z = Z*Z + C;
iterations++;
}
Screen.Plot(x,y, iterations % maxColors); //depending on the number of iterations, color a pixel.
}
}
}
Some details left out are:
1.) Learn exactly what the Square of a Complex number is and how to calculate it.
2.) Figure out how to translate the (-2,2) rectangular region to screen coordinates.
You should indeed start with the Mandelbrot set, and understand what it really is.
The idea behind it is relatively simple. You start with a function of complex variable
f(z) = z2 + C
where z is a complex variable and C is a complex constant. Now you iterate it starting from z = 0, i.e. you compute z1 = f(0), z2 = f(z1), z3 = f(z2) and so on. The set of those constants C for which the sequence z1, z2, z3, ... is bounded, i.e. it does not go to infinity, is the Mandelbrot set (the black set in the figure on the Wikipedia page).
In practice, to draw the Mandelbrot set you should:
Choose a rectangle in the complex plane (say, from point -2-2i to point 2+2i).
Cover the rectangle with a suitable rectangular grid of points (say, 400x400 points), which will be mapped to pixels on your monitor.
For each point/pixel, let C be that point, compute, say, 20 terms of the corresponding iterated sequence z1, z2, z3, ... and check whether it "goes to infinity". In practice you can check, while iterating, if the absolute value of one of the 20 terms is greater than 2 (if one of the terms does, the subsequent terms are guaranteed to be unbounded). If some z_k does, the sequence "goes to infinity"; otherwise, you can consider it as bounded.
If the sequence corresponding to a certain point C is bounded, draw the corresponding pixel on the picture in black (for it belongs to the Mandelbrot set). Otherwise, draw it in another color. If you want to have fun and produce pretty plots, draw it in different colors depending on the magnitude of abs(20th term).
The astounding fact about fractals is how we can obtain a tremendously complex set (in particular, the frontier of the Mandelbrot set) from easy and apparently innocuous requirements.
Enjoy!
If complex numbers give you a headache, there is a broad range of fractals that can be formulated using an L-system. This requires a couple of layers interacting, but each is interesting in it own right.
First you need a turtle. Forward, Back, Left, Right, Pen-up, Pen-down. There are lots of fun shapes to be made with turtle graphics using turtle geometry even without an L-system driving it. Search for "LOGO graphics" or "Turtle graphics". A full LOGO system is in fact a Lisp programming environment using an unparenthesized Cambridge Polish syntax. But you don't have to go nearly that far to get some pretty pictures using the turtle concept.
Then you need a layer to execute an L-system. L-systems are related to Post-systems and Semi-Thue systems, and like virii, they straddle the border of Turing Completeness. The concept is string-rewriting. It can be implemented as a macro-expansion or a procedure set with extra controls to bound the recursion. If using macro-expansion (as in the example below), you will still need a procedure set to map symbols to turtle commands and a procedure to iterate through the string or array to run the encoded turtle program. For a bounded-recursion procedure set (eg.), you embed the turtle commands in the procedures and either add recursion-level checks to each procedure or factor it out to a handler function.
Here's an example of a Pythagoras' Tree in postscript using macro-expansion and a very abbreviated set of turtle commands. For some examples in python and mathematica, see my code golf challenge.
There is a great book called Chaos and Fractals that has simple example code at the end of each chapter that implements some fractal or other example. A long time ago when I read that book, I converted each sample program (in some Basic dialect) into a Java applet that runs on a web page. The applets are here: http://hewgill.com/chaos-and-fractals/
One of the samples is a simple Mandelbrot implementation.
Another excellent fractal to learn is the Sierpinski Triangle Fractal.
Basically, draw three corners of a triangle (an equilateral is preferred, but any triangle will work), then start a point P at one of those corners. Move P halfway to any of the 3 corners at random, and draw a point there. Again move P halfway towards any random corner, draw, and repeat.
You'd think the random motion would create a random result, but it really doesn't.
Reference: http://en.wikipedia.org/wiki/Sierpinski_triangle
The Sierpinski triangle and the Koch curve are special types of flame fractals. Flame fractals are a very generalized type of Iterated function system, since it uses non-linear functions.
An algorithm for IFS:es are as follows:
Start with a random point.
Repeat the following many times (a million at least, depending on final image size):
Apply one of N predefined transformations (matrix transformations or similar) to the point. An example would be that multiply each coordinate with 0.5.
Plot the new point on the screen.
If the point is outside the screen, choose randomly a new one inside the screen instead.
If you want nice colors, let the color depend on the last used transformation.
I would start with something simple, like a Koch Snowflake. It's a simple process of taking a line and transforming it, then repeating the process recursively until it looks neat-o.
Something super simple like taking 2 points (a line) and adding a 3rd point (making a corner), then repeating on each new section that's created.
fractal(p0, p1){
Pmid = midpoint(p0,p1) + moved some distance perpendicular to p0 or p1;
fractal(p0,Pmid);
fractal(Pmid, p1);
}
I think you might not see fractals as an algorithm or something to program. Fractals is a concept! It is a mathematical concept of detailed pattern repeating itself.
Therefore you can create a fractal in many ways, using different approaches, as shown in the image below.
Choose an approach and then investigate how to implement it. These four examples were implemented using Marvin Framework. The source codes are available here
Here is a codepen that I wrote for the Mandelbrot fractal using plain javascript and HTML.
Hopefully it is easy to understand the code.
The most complicated part is scale and translate the coordinate systems. Also complicated is making the rainbow palette.
function mandel(x,y) {
var a=0; var b=0;
for (i = 0; i<250; ++i) {
// Complex z = z^2 + c
var t = a*a - b*b;
b = 2*a*b;
a = t;
a = a + x;
b = b + y;
var m = a*a + b*b;
if (m > 10) return i;
}
return 250;
}
The mandelbrot set is generated by repeatedly evaluating a function until it overflows (some defined limit), then checking how long it took you to overflow.
Pseudocode:
MAX_COUNT = 64 // if we haven't escaped to infinity after 64 iterations,
// then we're inside the mandelbrot set!!!
foreach (x-pixel)
foreach (y-pixel)
calculate x,y as mathematical coordinates from your pixel coordinates
value = (x, y)
count = 0
while value.absolutevalue < 1 billion and count < MAX_COUNT
value = value * value + (x, y)
count = count + 1
// the following should really be one statement, but I split it for clarity
if count == MAX_COUNT
pixel_at (x-pixel, y-pixel) = BLACK
else
pixel_at (x-pixel, y-pixel) = colors[count] // some color map.
Notes:
value is a complex number. a complex number (a+bi) is squared to give (aa-b*b+2*abi). You'll have to use a complex type, or include that calculation in your loop.
Sometimes I program fractals for fun and as a challenge. You can find them here. The code is written in Javascript using the P5.js library and can be read directly from the HTML source code.
For those I have seen the algorithms are quite simple, just find the core element and then repeat it over and over. I do it with recursive functions, but can be done differently.
People above are using finding midpoints for sierpinski and Koch, I'd much more recommend copying shapes, scaling them, and then translating them to achieve the "fractal" effect.
Pseudo-code in Java for sierpinski would look something like this:
public ShapeObject transform(ShapeObject originalCurve)
{
Make a copy of the original curve
Scale x and y to half of the original
make a copy of the copied shape, and translate it to the right so it touches the first copied shape
make a third shape that is a copy of the first copy, and translate it halfway between the first and second shape,and translate it up
Group the 3 new shapes into one
return the new shape
}