Draw a circle with a specific number of pixels

I'm working on a project that requires me to accurately control the number of pixels that are used to draw (roughly) circular stimuli, and although Bresenham's algorithms are great, they don't draw circles of an arbitrary area (to my knowledge). I've tried scripts that开发者_C百科 interrupt Bresenham's algorithm when the desired area has been plotted, but the results are decidedly hit-or-miss. Does anyone know of a way to plot the "best" circle (somewhat subjective, I know) using a given number of pixels? Many thanks!

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A rough way of doing it, for example:

The radius of a circle of area 1000 sq px is sqrt(1000/pi) = 17.8... That circle should then fit in a 35x35 matrix. If you make "indices" for that matrix where the central pixel is (0,0), you can check easily if the pixel falls in the circle or not by substitution into the equation of a circle x^2 + y^2 = r^2. Or you can use the alternative equation for a circle centered at (a,b). If it evaluates to TRUE, it does, if not, it's outside the circle.

As a pseudocode/example, in Python I would do an optimized version of:

import numpy, math

target_area = 1000.0

r = (target_area / math.pi) ** 0.5
m = numpy.zeros((2*r+2,2*r+2))

a, b = r, r

for row in range(0, m.shape[0]):
    for col in range(0, m.shape[1]):
        if (col-a)**2 + (row-b)**2 <= r**2:
            m[row,col] = 1

numpy.sum(m)
#>>> 999

Here is the result when the target area is 100,000 pixels (the actual circle generated is 99988.0):

You could also write a routine to find which areas can be matched more closely than others with this algorithm, and select those values to ensure conformity.

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The area of a circle is A=Pi*r². You're starting from the area and (apparently) want the radius, so we divide both sides by Pi to get: r²=A/pi. Taking the square root of both sides then gives us: r=sqrt(A/pi). Once you have the radius, drawing with most of the normal algorithms should be straightforward.

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A simple (but somewhat naive approach) would be to simply count the number of pixels drawn by Bresenham's algorithm for a given radius, and then use binary search to find the radius that produces the desired number of pixels.

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My first thought is to use an algorithm with sub-pixel precision. Consider what happens if you're center has irrational coordinates and you gradually increase the radius. This would fill seemingly random pixels around the perimeter as they became included in the circle. You want to avoid symmetry that causes the 4 quadrants of the circle adding pixels at the same time so you get closer to single pixels getting added. How something like this could be implemented I haven't a clue.

I had to solve a single instance of the 3d version once. I needed to a set of lattice points inside a sphere to be less-than or equal to 255. IIRC if r*r = 15 there are 240 points inside the sphere. I was not concerned with getting 255 exactly though.

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Supposedly you have 2000 pixels in total that should make up your complete circle. By complete I mean there should be no breakage in pixels and must be connected to each other. Since 2Pi*R = circumference, the running length of diameter of the circle, this is the total amount of pixels you have. Now simply write R = 2000/2*Pi and this will give you the radius. Now you should be able to draw a circle the comprised of 2000 pixels. I hope this is what you wanted.

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Let's forget about pixels for a second and let's work through the basic math/geometry.

We all know that

Area of a Circle = Pi * Radius ^2

which is the same as saying

Area of a Circle = Pi * (Diameter / 2) ^2

We all know that

Area of the Square Enclosing the Circle (i.e. each side of the square is tangent to the circle) = Diameter * Diameter

Thus

Ratio of the Circle Area to the Square Area = Circle Area / Square Area = (Pi * (Diameter / 2) ^2) / (Diameter * Diameter) = Pi / 4

Now let's assume that we have a circle and square with a pixel count large enough so that we don't have to worry about the troublesome edge cases around the border of the circle. In fact let's assume for a second that we have a very large diameter (maybe 10,000 or maybe even infinite). With this assumption the following holds:

Number of Pixels in the Circle = (Number of Pixels in the Square) * (Ratio of the Circle Area to the Square Area)

In other words for a sufficiently large number of pixels, the ratio of the areas of a perfectly drawn circle to a perfectly drawn square will approximate the ratio of the number of pixels in a pixelated circle to the number of pixels in the enclosing pixelated square.

Now in a pixelated square, the number of pixels in that square is the number of pixels across times the number of pixels high. Or in other words it is the square's diameter (in pixels) squared. Let's call the square's pixel diameter d. So substituting with the formulas above we have:

Number of Pixels in the Circle = (d * d) * (Pi /4)

So now let's solve for d

d = Sqrt(4 * (Num of Pixels in the Circle) / Pi)

Well we said earlier that d was the diameter of the square. Well it also happens to be the diameter of the circle. So when you want to draw a circle with a certain number of pixels, you draw a circle with the diameter being:

Diameter of Circle = Sqrt(4 * (Desired Number of Pixels in Circle Area) / Pi)

Now obviously you have to make some choices about rounding and so forth (there is no such thing as a fractional pixel), but you get the point. Also, this formula is more accurate as the desired number of pixels for the area of the circle goes up. For small amounts of pixels the roundoff error may not give you exactly the right number of pixels.