Core Algorithms in Computer Graphics: Clipping, Drawing & Displays

Posted on Dec 4, 2025 in Computers

Cohen-Sutherland Line Clipping Algorithm

What is the Cohen-Sutherland Algorithm?

The Cohen–Sutherland algorithm is a line clipping method used to clip a line segment against a rectangular clipping window. It divides the area around the window into regions and assigns a 4-bit region code (Outcode) to each endpoint of the line.

Algorithm Steps

1. Assign 4-bit codes to each endpoint. Each bit represents a position relative to the window:

   • Bit 1: Top
   • Bit 2: Bottom
   • Bit 3: Right
   • Bit 4: Left

   For example:

   • 0000 → Inside the window
   • 1000 → Above the window
   • 0100 → Below the window
   • 0010 → Right of the window
   • 0001 → Left of the window

2. Evaluate the line based on the codes:

   • If both endpoints have a code of 0000, the line is completely inside and is accepted.
   • If the logical AND of the two endpoint codes is not 0000, the line is completely outside and is rejected.
   • Otherwise, the line is partially inside. The algorithm calculates intersection points with the window boundary and repeats the process for the clipped segment.

Example

Let the clipping window be: (xmin=10, xmax=50, ymin=10, ymax=40).

Consider a line from P1(5, 20) to P2(60, 30).

• The code for P1 (left of the window) is 0001.
• The code for P2 (right of the window) is 0010.
• Since the line is partially inside, the algorithm finds the intersection points with the left and right boundaries. The clipped line inside the window would be from (10, 20) to (50, 30).

Limitations

1. It only works for rectangular clipping windows.
2. Repeated intersection calculations can make it slower for complex scenes.
3. It is not suitable for curved or concave shapes.
4. It requires recomputation if the window changes.

Cyrus-Beck Line Clipping Algorithm

What is the Cyrus-Beck Algorithm?

The Cyrus–Beck algorithm is a line clipping method used to clip a line segment against a convex polygon window. It is an improvement over the Cohen–Sutherland algorithm, which only works for rectangles, as Cyrus–Beck can handle any convex-shaped window.

Core Concept

The main idea is to find the part of the line that lies inside the polygon and remove the portions that are outside. It does this by checking where the line enters and where it leaves the boundary of the polygon.

How the Algorithm Works

1. Input: A line (with two endpoints) and a convex polygon (the clipping window).
2. Check each edge of the polygon: The algorithm looks at each side (edge) of the polygon and decides whether the line is moving into the polygon or out of it.
3. Find entry and exit points:
   • The point where the line first crosses from outside to inside is the entering point.
   • The point where it crosses from inside to outside is the leaving point.
4. Keep the visible part: The portion of the line between the entering and leaving points is inside the polygon and is displayed. The rest is clipped away.

Conceptual Example

Imagine a line passing through a pentagon-shaped window:

• The line starts outside, goes through the pentagon, and exits from another side.
• The Cyrus–Beck algorithm will keep only the part inside the pentagon and cut off the rest.

Advantages

• Works for any convex polygon, not just rectangles.
• More accurate because it considers the direction of the line and edges.

Limitations

• Does not work for concave polygons (shapes that bend inward).
• Slightly more complex to understand and implement.

Random Scan vs. Raster Scan Displays

Bresenham’s Circle Drawing Algorithm

Bresenham’s Circle Drawing Algorithm is an efficient method for drawing a circle on a computer screen using only integer calculations, avoiding floating-point or trigonometric functions. It is based on the midpoint circle principle, which determines the next pixel position by evaluating a decision parameter.

Core Principle

The main idea is to draw only one octant (1/8th of the circle) and then use eight-way symmetry to plot the remaining points, since a circle is symmetrical in all directions.

How It Works

The algorithm starts at the top of the circle (0, r) and moves outward along the x-axis. At each step, it determines whether the next pixel in the y-direction should remain the same or decrease by 1 based on a decision parameter.

At each step, the algorithm evaluates a decision parameter p to choose between two possible pixels:

• If p < 0, the next pixel is directly to the right.
• If p ≥ 0, the next pixel moves diagonally (right and down), and y decreases by 1.

This process continues until x ≥ y, which means all points of the octant have been plotted. By mirroring these points using eight-way symmetry, the complete circle is drawn.

Key Advantages

Bresenham’s circle drawing algorithm is highly efficient because it:

• Avoids floating-point arithmetic, using only integer addition, subtraction, and bit-shifting (multiplication by 2).
• Uses symmetry to reduce computation significantly.
• Determines pixel positions using a simple decision parameter.
• Produces a smooth and accurate circle on a raster display.

Understanding Antialiasing in Graphics

What is Antialiasing?

Antialiasing is a technique used in computer graphics to reduce the jagged edges or “stair-step” effect (called aliasing) that appears when diagonal or curved lines are displayed on low-resolution screens made up of square pixels. When smooth lines or curves are drawn on a pixel grid, some pixels are either completely on or off, which causes the edges to look rough. Antialiasing helps make these edges appear smoother and more realistic.

Why is Antialiasing Needed?

• To make images, text, and lines appear smoother and more natural.
• To improve the visual quality of graphics, especially in games, 3D models, and digital art.
• To reduce flickering or distortion when an image moves or scales.

Common Antialiasing Techniques

Supersampling (SSAA)

• The image is rendered at a higher resolution and then scaled down to the desired size.
• This process averages the colors of multiple pixels, producing smoother edges.
• It provides high quality but is computationally expensive.

Multisampling (MSAA)

• A more efficient version of supersampling.
• Only the edges of polygons are sampled multiple times instead of the whole image.
• It provides good quality with less processing cost.

Post-Processing Antialiasing (FXAA / SMAA)

• This technique works after the image has been rendered.
• It detects edges in the final image and smooths them using blur and color blending.
• It is very fast but can slightly reduce image sharpness.

Area Sampling

• This method determines how much of each pixel is covered by the line or shape and adjusts the color intensity accordingly.
• It produces smooth edges without heavy computation.

Conceptual Framework of Interactive Graphics

The conceptual framework of interactive graphics describes how computer graphics systems allow users to create, modify, and view graphical images interactively using hardware and software components. It defines the relationship between the input, processing, and output components that work together to produce visual results on a display.

Core Components of the Framework

1. Input Devices

   These devices allow the user to communicate with the system. Examples include a keyboard, mouse, light pen, joystick, touch screen, and graphic tablet.

2. Central Processing Unit (CPU)

   The CPU acts as the brain of the system. It interprets user commands, processes data, and controls the display process.

3. Graphics System / Display Processor

   This component handles all graphic operations like drawing lines, circles, and shapes. It converts geometric data into pixels (a process known as scan conversion).

4. Frame Buffer (Video Memory)

   A part of memory that stores the image as a set of pixel values before displaying it on the screen.

5. Output Devices

   These devices display the final image generated by the system. Examples include monitors (CRT, LCD), projectors, and plotters.

6. Software / Graphics Packages

   These provide libraries and tools for drawing, transformations, animation, and modeling. Examples include OpenGL, AutoCAD, and Blender.

Advantages of Interactive Graphics

• User Control: Users can directly manipulate objects (e.g., rotate, scale, or move them).
• Immediate Feedback: Any change is reflected instantly on the screen.
• Enhanced Understanding: Makes complex data or designs easier to visualize.
• Improved Accuracy: Allows for precise drawing and modeling.
• Ease of Editing: Modifications can be made quickly without redrawing the entire image.
• Wide Application: It is widely used in CAD, animation, simulations, games, and education.

Bresenham’s Line Drawing Algorithm

Purpose of the Algorithm

Bresenham’s Line Drawing Algorithm is used to draw a straight line on a raster display (a pixel-based screen). It is highly efficient because it uses only integer calculations, avoiding slow floating-point operations. The algorithm decides which pixels to turn on so that the line appears as straight as possible.

How It Works

Suppose you want to draw a line from point (x0, y0) to (x1, y1). Let dx = x1 - x0 and dy = y1 - y0. The slope of the line is m = dy / dx. For simplicity, we assume the slope is gentle, i.e., 0 < m < 1. The algorithm starts by plotting the first point (x0, y0).

Step 1: Decide the Next Pixel

At each step along the x-direction, there are two candidate pixels for the next point:

1. The pixel directly to the right: (xk+1, yk)
2. The pixel to the right and one step up: (xk+1, yk+1)

The algorithm chooses the pixel that is closer to the actual line.

Step 2: Use a Decision Parameter

A decision parameter pk is defined to decide which pixel to choose.

• The initial value is: p0 = 2 * dy - dx
• At each step, check the value of pk:
  • If pk < 0, choose the pixel (xk+1, yk) and update the parameter: pk+1 = pk + 2 * dy
  • If pk >= 0, choose the pixel (xk+1, yk+1) and update the parameter: pk+1 = pk + 2 * dy - 2 * dx

Step 3: Repeat the Process

This process continues for every x-coordinate until the endpoint x1 is reached. For slopes greater than 1 or negative slopes, the same logic applies, but the roles of x and y may be swapped.

Advantages

• It only uses integer addition and subtraction, with no multiplication or division in the main loop.
• It is fast and highly optimized for computer graphics hardware.
• It produces a visually straight line on a pixel grid.