ELLIPSE-GENERATING ALGORITHMS

Ellipse-generating algorithms are methods used to draw ellipses in computer graphics. These algorithms calculate the points on the ellipse’s perimeter and plot them on the screen.

There are various ellipse-generating algorithms available, but two common ones are:

1. Midpoint Ellipse Algorithm:
   • This algorithm is an extension of the midpoint circle algorithm and uses a similar approach.
   • It iteratively calculates points in one quadrant of the ellipse and mirrors them to the other quadrants.
   • The algorithm uses a decision parameter to determine the next point to plot.
2. Bresenham’s Ellipse Algorithm:
   • This algorithm is an adaptation of Bresenham’s line algorithm for drawing ellipses.
   • It uses the concept of incremental calculations to determine the points on the ellipse’s perimeter.
   • The algorithm avoids costly floating-point calculations and provides accurate results.

Midpoint Ellipse Algorithm

The Midpoint Ellipse Algorithm is a method used to draw ellipses in computer graphics. It is an extension of the Midpoint Circle Algorithm and follows a similar approach to efficiently generate points on the ellipse’s perimeter.

Here’s an explanation of how the Midpoint Ellipse Algorithm works:

1. Input:
   • Center coordinates (h, k) where (h, k) represents the center point of the ellipse
   • Major axis length (a)
   • Minor axis length (b)
2. Initialization:
   • Set the initial decision parameter (p) based on the given major and minor axis lengths:
     • p = b^2 – a^2*b + (a^2)/4
3. Generating Points:
   • Initialize the coordinates of the current point on the ellipse:
     • x = 0
     • y = b
   • Calculate the decision parameters for the first region (Region 1):
     • dx = 2 * b^2 * x
     • dy = 2 * a^2 * y
4. Plotting Points:
   • Iterate through Region 1:
     • Plot the points (x, y) and their symmetrical counterparts in the other three regions:
       • (x, -y), (-x, y), (-x, -y)
   • Update the coordinates and decision parameters:
     • If p < 0:
       • Increment x and update dx and p:
         • x = x + 1
         • dx = dx + 2 * b^2
         • p = p + dx + b^2
     • If p ≥ 0:
       • Increment both x and y and update dx, dy, and p:
         • x = x + 1
         • y = y – 1
         • dx = dx + 2 * b^2
         • dy = dy – 2 * a^2
         • p = p + dx – dy + b^2
   • Repeat the plotting and updating steps until x ≤ y

Bresenham’s Ellipse Algorithm

Bresenham’s Ellipse Algorithm is another method used to draw ellipses in computer graphics. It is an adaptation of Bresenham’s line algorithm and provides an efficient way to generate points on the ellipse’s perimeter.

Here’s an explanation of how Bresenham’s Ellipse Algorithm works:

1. Input:
   • Center coordinates (h, k) where (h, k) represents the center point of the ellipse
   • Major axis length (a)
   • Minor axis length (b)
2. Initialization:
   • Set the initial values:
     • x = 0
     • y = b
     • a2 = a * a
     • b2 = b * b
     • d = b2 – (a2 * b) + (0.25 * a2)
3. Generating Points:
   • Calculate the points on the ellipse’s perimeter in the first region (Region 1):
     • Plot the points (x, y) and their symmetrical counterparts in the other three regions:
       • (x, -y), (-x, y), (-x, -y)
   • Update the coordinates and decision parameter:
     • If d < 0:
       • Increment x and updated:
         • x = x + 1
         • d = d + (2 * b2 * x) + (b2)
     • If d ≥ 0:
       • Increment both x and y and updated:
         • x = x + 1
         • y = y – 1
         • d = d + (2 * b2 * x) – (2 * a2 * y) + (b2)
   • Repeat the plotting and updating steps until x ≤ y

Bresenham’s Ellipse Algorithm

Bresenham’s Ellipse Algorithm is another method used to draw ellipses in computer graphics. It is an adaptation of Bresenham’s line algorithm and provides an efficient way to generate points on the ellipse’s perimeter.

Here’s an explanation of how Bresenham’s Ellipse Algorithm works:

1. Input:
   • Center coordinates (h, k) where (h, k) represents the center point of the ellipse
   • Major axis length (a)
   • Minor axis length (b)
2. Initialization:
   • Set the initial values:
     • x = 0
     • y = b
     • a2 = a * a
     • b2 = b * b
     • d = b2 – (a2 * b) + (0.25 * a2)
3. Generating Points:
   • Calculate the points on the ellipse’s perimeter in the first region (Region 1):
     • Plot the points (x, y) and their symmetrical counterparts in the other three regions:
       • (x, -y), (-x, y), (-x, -y)
   • Update the coordinates and decision parameter:
     • If d < 0:
       • Increment x and updated:
         • x = x + 1
         • d = d + (2 * b2 * x) + (b2)
     • If d ≥ 0:
       • Increment both x and y and updated:
         • x = x + 1
         • y = y – 1
         • d = d + (2 * b2 * x) – (2 * a2 * y) + (b2)
   • Repeat the plotting and updating steps until x ≤ y