$x= rcos\theta$

$y=rsin\theta$

where $0\le \theta \le 2\pi$........................(1)

If uniformly spaced points lie on the periphery of the circle, then $\delta \theta=$consatnt (increament in angle)

$\therefore$ (1) can be expressed as:-

$x_{i+1}=rcos(\theta _i+\delta \theta)$

$y_{i+1}=rsin(\theta _i+\delta \theta)$

$\therefore x_{i+1}=rcos\theta _icos \delta \theta-rsin\theta _isin \delta \theta$

$\therefore y_{i+1}=rsin\theta _icos \delta \theta+rcos\theta _isin \delta \theta$

Simplifying:

$x_{i+1}=x_icos\delta \theta-y_isin \delta \theta$

$y_{i+1}=x_isin\delta \theta-y_icos \delta \theta$

A not origin circle is obtained by considering a single origin centered unit circle(generation); then scaling and translation.

E.g. Generate a circle of radius 2 with centre located at (2,2).

Solution

1. Generate an origin centered circle (unit radius)
2. Scale by factor of 2
3. Translate by (2,2) units

Assuming 8 number of points on the circle equally spaced.

(Note: Normally, a much larger number of points is required for display, depends on the radius of circle.)

$\therefore \delta \theta =\frac{2\pi ^c}{8}=\frac{\pi ^c}{4}=45^o$

Starting with point 1;

$x_1=rcos\theta _1=1cos 0=1$

$y_1=rsin\theta _1=1sin 0=0$

$\therefore$ Using parametric equations;

we get the x and y values of the other points also.

$\therefore x_2=x_1cos\delta \theta-y_1sin\delta \theta$

$\therefore y_2=x_1sin\delta \theta+y_1cos\delta \theta$

Here; $sin\delta \theta=cos\delta \theta (\because \theta = 45)$

$=\frac{1}{\sqrt 2}$

$\therefore x_2=\frac{1}{\sqrt 2}$

$y_2=\frac{1}{\sqrt 2}$

Hence; the results for the unit circle can be translated as:-

Q. Generate a circle of radius 4 width center located at (2, 2)

Q. Generate a circle (0 to $90^o$ ) of radius 4 width center located at (2, 2) in incremental co-ordinate.