You're in effect drawing a straight line in a discontinous space, the coordinates being time and color value. The standard way to do this efficiently is the Bresenham algorithm. The idea is to incrementally compute the current error between the next integer color value (y coordinate) and the next "ideal" color value, and when this error becomes too large to update the current color value accordingly. Here is the code for the inner loop (excerpt from the wikipedia page) :

for x from x0 to x1
plot(x,y)
error := error + deltaerr
if error ≥ 0.5 then
y := y + 1
error := error - 1.0

Note that there is no integer division in the loop, hence no room for rounding errors. And it's efficient: never more than a comparison and two additions at each iteration.

@DrFPGA: If I were using the controller to do something else at the same time, then I would look at using a timer and interrupts as you suggest ... but in this case all I'm doing is controlling the LEDs, so I'm just using the basic Arduine delay() function.

If you think of doing this in the time domaine does it become easier? Every so many milliseconds you generate an interrupt to process colors. Check where you are and where you want to be and take a step in the direction you need with a fixed sized step. Would that work ok?

@Alvie: Sometimes using Fixed-Point arithmetic can help with these issues...

It all sounds so simple the way you say it LOL

Actually, you are correct in that I had completely neglected to consider using something like a 10.6 fixed-point representation. For some tasks this would be very appropriate -- but for my current application I think I'll stick with the solution from Javi (Garcia).

Sometimes using Fixed-Point arithmetic can help with these issues.

The basics behind FixedPoint are quite simple, and easy to implement in most architectures. Let's have a look at your concrete problem (255/256) and see how we could get a somewhat accurate output from it.

The easiest way to implement this would be to use a 10.6 fixed point, where 10 bits are used for the real part (signed number range would be between +511 and -512), and 6 bits for the "fractionary" part - since 6 bits can represent 64 different values, each bit can represent 1/64, 0.01625. This is your resolution in terms of the fractionary part.

The beauty about fixed-point is that most of the arithmetic is done with the integer representation. This includes addition/subtraction and multiplication.

In order to add two 10.6 fixed-point values, represented by a 16-bit integer, just... add them. The result will also be a 10.6 fixed point value. Same for subtraction. For multiplication, the result will not be 10.6, but rather 20.12 - 32 bits. In order to convert this multiplication back to 10.6, just shift it by 6 bits, and truncate the result (note: if you're dealing with signed numbers, you must do this in a slightly different way, in order to preserve the highest bit signal).

So, let's take a look at your case: 255/256.

First, let's represent them as fixed-point, 10.6 numbers, in binary :

255 => 0011111111.000000

256 => 0100000000.000000

and:

1 => 0000000001.000000

In the following explanations, I'll prefix the numbers with "f" if they are representations of fixed-point values, and no prefix for their integer representations.

Now, let's invert f256. This is done by computing (f1/f256), also in fixed-point format. In order to perform this division, we use the integer representations to perform an integer division. But, similarly to the multiplication (which doubles the number of fractionary bits), the division will also change the fractionary bits - not only that, it will indeed remove them.

Let's take a look at a simple (f1/f1) division. This ought to result in f1, of course. If we take a clear look at the f1 representation above, and compute it's integer value, we get 2^6 == 64. If we divide the values using their integer representation, we get 64/64, which gives us 1. But this integer "1" does not represent "f1", it represents "0000000000.000001". So, we just lost a lot of precision (6 bits actually) by performing the division. To fix this, we shift the result by 6 bits to the left, and get the correct value: "0000000001.000000".

Now, another division. Let's divide f3 by f2, the result should be f1.5. The representation of f3 is "0000000011.000000", and f2 is "0000000010.000000". The integer representations are 192 and 128, respectively. If we perform the integer division, we get 1. If we shift this by 6-bits as in previous example, we get "0000000001.000000", which is f1 - not f1.5. So, we just lost all precision !!!

There are a few solutions to overcome this. One of them is to get a few more resolution by extending the numbers to the next representation, which in this case would be 32-bit. Basically, what we will be doing is to transform the equation:

x = a / b;

into

x = ( (a * N) / b ) / N

which is equivalent. The "N" value would be ((2^6)-1) in this case - which matches the 6-bit shift we have to perform to correct the division result - if we multiply the first argument by the same "shift amount" we have to perform at the end, we don't need to perform the final shift. We can also use a shift to perform the computation of the first operand.

So, for the case of f3/f2, we will get:

a*N (f3 times N): 0000000011.000000[000000] (integer representation: 12288)

And the result is 96 (12288 / 128). In binary form (and fixed point form) it's : [000000]0000000001.100000.

Integer part is 1, and fractionary part is '100000' in binary, 32 in decimal. If we multiply 32 by the fractionary resolution we computed in first place (0.01625), we get... 0.52, close enough to the "0.5" we need to add to the integer part.

Doing the same for your 255/256 values, we get a nicely looking value (result of (255<<6)/256):

0000000000.111111

We are very close to f1, and we can also truncate the result. But that would not be wise : the result is way more close to f1 than to f0.

So, we want to do rounding before converting that to an integer (which means shifting by 6 to the right). In order to do so, we add a '1' in the MSB of the to-be-discarded part. We are discarding 6 bits, so we have to add '100000' to the result:

0000000000.111111

+0000000000.100000

= 0000000001.011111.

Shift by 6 to the right.... gives "1".

So, here's your answer, and a quick, dirty and confuse explanation of fixed point arithmetic :)

@Garcia: The more I look at your solution the more I realize just how elegant it is. The part I especially like is the fact that the color value automatically ends up at 0 or 255 without any checking or tweaking (capping). I cannot wait to go home and try your solution thsi evening...

Of course, I'm handling all 64 LEDs each with three color channels, but the way I'm doing it is by means of a plan so cunning we could pin a tail on it and call it a weasel :-)

David Patterson, known for his pioneering research that led to RAID, clusters and more, is part of a team at UC Berkeley that recently made its RISC-V processor architecture an open source hardware offering. We talk with Patterson and one of his colleagues behind the effort about the opportunities they see, what new kinds of designs they hope to enable and what it means for today’s commercial processor giants such as Intel, ARM and Imagination Technologies.

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