# Finding a function that computes a point on the unit-circle

I can't find the definition of a function $f(x); x \in [-1;1]$ where $(x|f(x))$ is a point on the unit-circle.

Can you please give me a hint?

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Background:

I want to render a filled circle (programatically) onto a screen using an iteration from $x-rad \rightarrow x+rad$ where $x$ is the x-position of the circle on the screen, and $rad$ is the radius.

Solution:

$f(x) = cos(sin^{-1}(x))$

seems to work pretty well. For the programmers under us, here is Python-code to demonstrate my approach:

import math

for x in xrange(radius * 2):

ytop = math.cos(math.asin( x / radius ))
ybot = -ytop

x = x + xpos
drawLine( x, ypos+ytop, x, ypos+ybot)

Result:

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What does $(x|f(x))$ mean? Did you intend that to be $(x,f(x))$ a co-ordinate on the unit circle? –  user21436 Jan 23 '12 at 20:56
Draw the unit circle on a piece of paper, pick a point on it, consider its projections on the axes, and apply Pythagoras' rule. –  Emre Jan 23 '12 at 20:57
Do you know of any functions that map to the unit circle? You knew enough to tag this "trigonometry", so it should come as no surprise if trig ratios come into play. –  The Chaz 2.0 Jan 23 '12 at 20:59
@Emre Oh man, simpler than I thought. Headache, you know? ;-) $f(x) = cos(sin(x))$ ? –  Niklas R Jan 23 '12 at 21:02
Sorry, mistyped. I meant $f(x) = cos(sin^{-1}(x))$ which actually seems to work well. –  Niklas R Jan 23 '12 at 21:36

For $-1\le x\le 1$, $$\cos(\sin^{-1}x)=\sqrt{1-x^2}.$$
A circle with radius $r$, centered at the origin can be described by $$x^2+y^2=r^2.$$ Solving for $y$ in terms of $x$ gives $$y=\pm\sqrt{r^2-x^2}.$$
Definately a nicer than using $cos$ and $sin$. I sure wouldn't have come to this equation. Thank you! –  Niklas R Jan 23 '12 at 22:28