Scan Conversion of a Circle Computer Graphics

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A circle is an eight-way symmetric shape. All quadrants of a circle are the same. There are two octants in each quadrant of a circle. If we know the value of any point, then we can easily calculate the values of the remaining seven points by using the eight-way symmetry method. A circle is a shape consist of a line called the circumference. All the straight lines drawn from a particular point within the circle to the circumference are always equal. A circle is a round shape that has no corner or sides.

“A circle can be defined as a combination of points that all points are at the same distance (radius) from the center point.” We can express a circle by the following equation-

                                (P – Pc)2 + (Q – Qc)2 = r2­­­

The above equation can be written as-

                                (P)2 + (Q)2 = r2­ ­                                                  {r = radius}

If we want to draw a circle, then we will consider it by its origin point. Let us assume a point P1(R, S) then we can represent the other seven points as follows-

P2(R, -S)

P3(-R, -S)

P4(-R, S)

P5(S, R)

P6(S, -R)

P7(-S, -R)

P8(-S, R)

Scan Conversion of a Circle

We can also represent eight points of the circle on the computer screen by using of put pixel function ().

Putpixel (R, S, Color)                                    

Putpixel (R, -S, Color)

Putpixel (-R, -S, Color)

Putpixel (-R, S, Color)

Putpixel (R, S, Color)

Putpixel (R, -S, Color)

Putpixel (-R, -S, Color)

Putpixel (-R, S, Color)

Example: Let, we have taken a point (4, 6) of a circle. We will calculate the remaining seven points by using of reflection method as follows-

The seven points are – (4, -6), (-4, -6), (-4, 6), (4, 6), (4, -6), (-4, -6), (-4, 6)

Scan Conversion of a Circle

There are two following standard methods to define a circle mathematically.

  • A circle with a second-order polynomial equation.
  • A circle with trigonometric/ polar coordinates.

A circle with the second-order polynomial equation

                                  y2 =r2 – x2

Here, x = The coordinates of x

         y = The coordinates of y

r = The radius of the circle

In this method, we will find the x coordinate (90° to 45°) by moving x from 0 to r/√2, and we will find each y coordinate by calculating r2-x2 for each step.

It is an ineffective method because for each point x coordinate and radius r must be squaredand subtracted from each other. 

Scan Conversion of a Circle

A circle with trigonometric/polar coordinate

x = r cos θ 

y = r sin θ

Here, x = The coordinate of x

        y = The coordinate of y

          r = The radius of the circle

          θ = Current angle

In this method, the value of θ moves from 0 to π/4. We will calculate each value of x and y.

Scan Conversion of a Circle

Algorithms used in Circle Drawing

There are the following two algorithms used to draw a circle.

  • Bresenham’s Circle drawing Algorithm  
  • Mid-point Circle Drawing Algorithm
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