Core Algorithms for Computer Graphics Rendering

Circle Generating Algorithms

Circle generating algorithms are fundamental in computer graphics for rendering circles efficiently on raster displays. These algorithms calculate the points on the circumference of a circle and plot them. The most commonly used circle generating algorithms are the Midpoint Circle Algorithm and Bresenham's Circle Algorithm.

Midpoint Circle Algorithm Steps

The following steps outline the basic process for generating points in one octant, which are then mirrored to complete the circle:

1. Step 1: Initialization. Set starting coordinates $x = 0$ and $y = r$ (radius). Calculate the initial decision parameter $p = 1 - r$.
2. Step 2: Iteration. Repeat the following steps while $x \le y$:
   • Plot the current point $(x, y)$ and its octant symmetric points.
   • Decision Parameter Update:
     • If $p < 0$: Set $p = p + 2x + 3$.
     • Else ($p \ge 0$): Set $p = p + 2(x - y) + 5$. Decrease $y$ by 1 ($y = y - 1$).
   • Increment $x$ by 1 ($x = x + 1$).
3. Step 3: Termination. End the process when $x > y$.

Line Clipping in Computer Graphics

Line clipping in computer graphics is the process of removing portions of lines that are outside a specified rectangular region, known as the clipping window. This is essential in rendering scenes efficiently, ensuring that only the visible parts of objects are processed and displayed.

The Cohen-Sutherland Algorithm

The Cohen-Sutherland algorithm is a divide-and-conquer method used for line clipping. It uses a region code (OutCode) for each endpoint of the line. These region codes indicate the location of the endpoints relative to the clipping window. The process involves:

Algorithm Steps

1. Step 1: Calculate Region Codes. Determine the 4-bit region codes for both endpoints of the line segment.
2. Step 2: Visibility Check (OR Operation). Perform a logical OR operation on both endpoint region codes.
   • If the result is 0000, the line is entirely visible (trivially accepted).
3. Step 3: Trivial Rejection Check (AND Operation). If the line is not trivially accepted, perform a logical AND operation on both endpoint region codes.
   • If the result is not 0000, the line is entirely invisible (trivially rejected) because both endpoints lie on the same side outside the window.
4. Step 4: Clipping Required. If the line is neither trivially accepted nor rejected, it is considered a clipped case.
   • Find the intersection point(s) with the boundaries of the clipping window using the line slope $m = (y_2 - y_1) / (x_2 - x_1)$.
   • The region code bits indicate which boundary to clip against:
     • i) If bit 1 is "1": The line intersects with the left boundary of the rectangle window.
     • ii) If bit 2 is "1": The line intersects with the right boundary.
     • iii) If bit 3 is "1": The line intersects with the bottom boundary.
     • iv) If bit 4 is "1": The line intersects with the top boundary.

Polygon Clipping Techniques

Polygon clipping in computer graphics is the process of removing portions of a polygon that lie outside a specified rectangular region, known as the clipping window. This process ensures that only the visible parts of the polygon are processed and displayed.

The most widely used algorithms for polygon clipping are the Sutherland-Hodgman algorithm and the Weiler-Atherton algorithm.

Sutherland-Hodgman Algorithm

The Sutherland-Hodgman algorithm is used to clip a polygon against a rectangular clipping window. The algorithm processes each edge of the input polygon against each edge of the clipping window in turn (typically left, right, bottom, then top), producing a sequence of output vertices that define the clipped polygon.

Process Overview

The algorithm relies on processing the input polygon vertices sequentially against the clipping boundaries:

• Input polygon vertices are provided.
• The algorithm generates a new set of vertices that define the clipped polygon.