Tuesday, January 08, 2008

Equation for a circle

Postulate 1: The distance d between two points (x1,x2) and (y1,y2)

d = √(x2 - x1)2 + (y2 - y1)2

This postulate assumes all the postulates from Euclid but that works fine for the standard Cartesian coordinates.

Definition 1: A circle

A circle is the set of points equidistant from a given point. This given point is called the center. The distance from the center to any point in the set of points is called the radius.

Theorem 1: Equation of a circle

For any given circle with center at (centerX, centerY) and radius r, the equation is:

(x - centerX)2 + (y - centerY)2 = r2

Proof:

(1) By Definition 1 above and Postulate 1 above, we have:

r = √(x - centerX)2 + (y - centerY)2

(2) Squaring both sides gives us:

r2 = (x - centerX)2 + (y - centerY)2

QED

Monday, January 07, 2008

Geometric Interpretation of Roots of Unity

The nth roots of unity can be thought of as n points evenly dividing up a circle.

Wolfram's MathWorld web site has a wonderful graphic that demonstrates this:

Here's the reason that it's true:

Theorem: The N-th Roots of Unity correspond to n points evenly dividing up a circle

Proof:

(1) The number of radians in a circle is [See here for review of radians, if needed]

(2) The equation for roots of unity is (see Corollary 1.1, here):

where 0 ≤ k ≤ n-1.

(3) So each root of unity is:

cos[ (2kπ)/n] + isin[(2kπ)/n]

where 0 ≤ k ≤ n-1

(4) Which is the same as:

cos[(k/n)2π] + isin[(k/n)2π]

where 0 ≤ k ≤ n -1

(5) Now, complex values can be graphed on the Cartesian coordinate system on x + iy [this is called the complex plane].

(6) Each of the roots above maps to a point on the circumference of a unit circle since:

(a) The equation for a unit circle at (0,0) is [See Theorem 1, here]:

x2 + y2 = 1

(b) Since we are mapping cos[(k/n)2π] + isin[(k/n)2π] to x + iy, this gives us:

x = cos[(k/n)2π]

y = sin[(k/n)2π]

(c) From a previous result (see Corollary 2, here), we have:

cos2[(k/n)2π] + sin2[(k/n)2π] = 1

(d) So, applying step #6b above gives us:

x2 + y2 = 1

(7) So, all we have left to prove is that each of these n points is equidistant from the adjacent points on the circle.

(8) Clearly, we have points at based on the following n values:

(0/n)2π, (1/n)2π, (2/n)2π, ..., [(n-1)/n]2π

where

y = cos[(k/n)2π]

x = sin[(k/n)2π]

(9) Now, if we use a unit circle, it is clear that each of these points is equidistant from each other since:

Consider a circle which has the radius r = 1.

It is clear that plotting lines at (0/n)2π, (1/n)2π. ... ([n-1]/n)2π divides up the total circle (2π radians) into n equal portions.

Using a unit circle, it is clear that x=[cos(k/n)2π] and y = [sin(k/n)2π] are the places of intersection when the circle is evenly divided since:

sin θ = y/r = y

cos θ = x/r = x

In other words, each y=cos[(k/n)2π] and x=sin[(k/n)2π] is a point at the place where the (k/n)th part of the circle sweeps out against the circumference of the circle.

Since the length of the circumference is πD=(2r)πr = 2(1)π, [see Corollary 1, here], this means that the length of each subtended arc is (k/n)*2π.

This results in the patterns above depending upon the value of n.

QED

For example, if we graph the third roots of unity. Then, we have:

lines sweeping out at:

0, (1/3)2π, and (2/3)2π.

which results in the following graph:

References