I'm curious about infinite numbers in computing, in particular pi.
For a computer to render a circle it would have to understand pi. But how can it if it is infinite?
Am I looking too much into this? Would it just use a rounded value?
+6
A:
An approximation is generally sufficient. To "render" a circle, the computer only needs to understand pi well enough to render accurately at whatever resolution (finite) is required.
Edit: as others have pointed out, you don't even need pi to render a circle. Still, the gist of the question was "how do computers deal with numbers like pi?" They use approximations, and whoever is using those approximations must decide whether they are precise enough for the given purpose.
Donnie DeBoer
2009-07-27 22:06:48
To render a circle, you usually don't use pi at all.
Nosredna
2009-07-27 22:42:32
PI does NOT have repeating decimal patterns as it is transcendental, which means that not only is it irrational (cannot be expressed as a fraction of two integers) it is also not algebraic (it isn't the solution to any rational polynomial equation).
spatz
2009-07-27 22:27:40
Not repeating. It has an infinite number of *nonzero* decimal digits, but they don't repeat. (See the comments to the question for the debate about what this is actually called)
David Zaslavsky
2009-07-27 22:28:29
doh! writing "repeating" was a total typo. Fixed now as "non-repeating".
abelenky
2009-07-27 22:30:02
+1
A:
I believe it rounds it to a very small number, and is most likely a constant. If you use PHP, this is how PI renders:
echo pi(); // 3.1415926535898 echo M_PI; // 3.1415926535898
Just like you only need 3.14159 in High School, computers only need so much to get it fairly accurate.
Chacha102
2009-07-27 22:08:05
+1
A:
Computers just use rounded values of pi, unless of course there is a special case such as scientific computing. For example, in python pi is represented as:
>>> import math
>>> math.pi
3.1415926535897931
You can test this out for yourself in IDLE, pythons interactive interpreter.
Javed Ahamed
2009-07-27 22:08:50
Interesting: C# reports the value as 3.14159265358979323846.Note the digits: 97931 vs. 97932That looks like something more than a rounding error to me.
abelenky
2009-07-27 22:20:23
Only about 17 of the digits are meaningful. The last digit is little more than noise coming back from the platform's binary-to-decimal conversion algorithm.
S.Lott
2009-07-27 22:32:10
+11
A:
Mathematically, computers are both finite and non-continuous and therefore can neither know PI completely, nor correctly render a circle.
However, in the digital realm neither of these exist anyway, so it is sufficient to approximate PI and then use that to approximately render the circle, resulting in exactly the same pixels that would have been calculated from an exact PI anyway.
Either way, the resulting pixels aren't really a circle either, because they are a finite collection of digital points and a circle is a curve made up of an infinite number of points, most with irrational values.
(It has been pointed out to me that PI is not normally used to plot a circle, which is true, however, the methods used to plot a circle are related to the formulas used to express and/or calculate the value of PI, which still have the same issues).
RBarryYoung
2009-07-27 22:16:17
I'd be curious to know why the random downvote is for? If there's something wrong with my answer, please tell me. But AFAIK, it is precisely correct, both mathematically and wrt computation theory.
RBarryYoung
2009-07-27 22:27:48
Indeed, this is a perfectly good high-level explanation. What makes it so good is that it doesn't just apply to circles, but is effectively universal.
Pavel Minaev
2009-07-27 23:12:40
Actually, since humans are both non-discrete and "real" they are technically considered continuous.
RBarryYoung
2009-10-09 00:11:38
+1
A:
Programming languages use a rounded constant for pi and similar "infinite" numbers.
In order to get higher precision you use iterative algorithms that are looped for as long as is required.
Console
2009-07-27 22:26:30
+5
A:
You don't need PI at all to draw a circle. There are many ways to draw a circle. The naive way is with sine and cosine.
The algorithm I saw most often on 8-bit machines was Bresenham's circle. You don't even need floating point math for that.
Nosredna
2009-07-27 22:35:20
RBarryYoung
2009-07-27 23:20:04
Sine and cosine are related to e, pi, and imaginary numbers, but I disagree that it's _typical_ that they are calculated from any of those, or from tables representing those. I'd say it's typical for Sine and Cosine to be calculated by a series expansion that converges quickly. But the C language doesn't say how they should be calculated. You can find Taylor expansions that calculate the circular functions without any reference to pi or e. Although, as I said, all these entities are inter-related.
Nosredna
2009-07-28 01:28:20
My apologies, you are correct Nosredna. In my dotage, I seem to have forgotten how Taylor series, et al, are constructed.
RBarryYoung
2009-07-28 14:25:49
+1
A:
Pi is not infinite it is irrational, what mean that you can not express it as quotient. It has infinite number of digits. http://en.wikipedia.org/wiki/Proof_that_π_is_irrational
About computing find some informations here. http://en.wikipedia.org/wiki/Computing_π
Nice page is also this http://3.141592653589793238462643383279502884197169399375105820974944592.com/
ralu
2009-07-27 22:46:20
+2
A:
Somewhere I saw a proof that to draw a circle around the universe to millimetre accuracy, you'd need less than 100 digits of pi, in other words, far fewer digits than have been calculated by people with too much time on their hands (or too much computing power...). Now, if only I could find that proof... (edit) found it
jford
2009-07-28 01:27:02