Line Representations in Computer Graphics: Algorithms and Techniques

Posted on May 21, 2024 in Computers

Lines and Line Representations

Lines are fundamental geometric primitives in computer graphics, often represented by mathematical equations or digital approximations. They play a crucial role in rendering images, creating shapes, and defining boundaries in graphics applications.

Mathematical Representation

Slope-Intercept Form

y = mx + b, where m is the slope and b is the y-intercept.

Point-Slope Form

y – y1 = m(x – x1), where (x1, y1) is a point on the line and m is the slope.

Vector Form

r = p + td, where r is a point on the line, p is a point on the line, d is the direction vector of the line, and t is a parameter.

Parametric Form

x = x1 + t⋅Δx, y = y1 + t⋅Δy, where (x1, y1) is a point on the line, and (Δx, Δy) is the direction vector of the line.

Rasterization

Digital lines are represented as sequences of pixels, approximating the mathematical line equation.

Bresenham’s Line Algorithm

An efficient algorithm for rasterizing lines, avoiding floating-point arithmetic and providing accurate results.

Clipping Algorithms

Ensure that only the portions of lines that are within the viewport or clipping region are displayed, improving efficiency and reducing unnecessary computation.

Intersection of Lines

Represent the lines as linear equations using the endpoints:

For Line 1: y = mx + c where m = (y2 – y1) / (x2 – x1) and c = y1 – m ⋅ x1

For Line 2: y = mx + c, where m = (y4 – y3) / (x4 – x3) and c = y3 – m ⋅ x3

Normalized Device Coordinates (NDC)

Normalized Device Coordinates (NDC) is a coordinate system used in computer graphics to represent the position of objects on the screen or in a rendering pipeline. NDC is a standardized coordinate space that makes it easier to perform rendering operations across different devices and resolutions.

NDC is a 3D coordinate system typically represented as a cube where x, y, z range from -1 to 1. The origin (0, 0, 0) is at the center of the cube.

  • x increases from left to right.
  • y increases from bottom to top.
  • z increases from near to far (depth).

Device Independence

NDC allows rendering operations to be performed independently of the device’s resolution or aspect ratio.

Simplified Clipping

Since NDC is a standardized space, it simplifies clipping operations, ensuring that objects outside the view frustum are efficiently discarded.

Depth Buffering

NDC facilitates depth testing and depth buffering, where objects can be correctly occluded based on their Z-coordinate in the NDC space.

DDA Algorithm

The Digital Differential Analyzer (DDA) algorithm is a method used for drawing lines on a computer screen or raster graphics display. It’s one of the simplest algorithms for line drawing and is easy to understand and implement.

Algorithm Steps:

Input: Start point (x0, y0) End point (x1, y1)

Output: Plot the line between the start and end points.

Algorithm:

  1. Calculate the differences between the x and y coordinates: dx = x1 – x0 and dy = y1 – y0
  2. Determine the number of steps required for the line: Let steps = max(|dx|, |dy|)
  3. Calculate the increment in x and y for each step: x(increment) = dx / steps and y(increment) = dy / steps
  4. Starting from the initial point (x0, y0), for each step k from 0 to steps: Calculate the coordinates of the current point: xk = x0 + k ⋅ x(increment) and yk = y0 + k ⋅ y(increment)

Bresenham’s Line Drawing Algorithm

Bresenham’s line drawing algorithm is a method used for drawing lines on a computer screen or raster graphics display. It’s more efficient than the Digital Differential Analyzer (DDA) algorithm, especially for lines with a steep slope, as it avoids floating-point arithmetic.

Algorithm Steps

  1. Calculate differences between the x and y coordinates: dx = x1 – x0 and dy = y1 – y0
  2. Determine the increments: x(increment) = 1 if dx > 0 and y(increment) = 1 if dy > 0
  3. Calculate Decision Parameters: d = 2 ⋅ |dy| – |dx|
  4. Plot the start point (x0, y0)
  5. For each step from 1 to |dx|:
    1. If d ≥ 0, increment y and update d: d = d + 2 ⋅ |dy| – 2 ⋅ |dx| and y = y + y(increment)
    2. Else, only update d: d = d + 2 ⋅ |dy|
    3. Increment x by x(increment).

Bresenham’s Line and Midpoint Circle Drawing Algorithms

Bresenham’s line drawing algorithm and the midpoint circle drawing algorithm are two classic methods used for drawing lines and circles on a computer screen or raster graphics display. Both algorithms are efficient and widely used in computer graphics due to their simplicity and effectiveness.

Algorithm Steps:

  1. Given two endpoints (x0, y0) and (x1, y1).
  2. Calculate differences: dx = x1 – x0 and dy = y1 – y0
  3. Determine the increments: x(increment) = ±1 and y(increment) = ±1 based on the signs of dx and dy.
  4. Calculate the decision parameter: d = 2 ⋅ |dy| – |dx|.
  5. Plot the start point (x0, y0)
  6. For each step from 1 to |dx|:
    • If d ≥ 0, increment y and update d.
    • Else, only update d.
    • Increment x.
  7. Plot the next point using the updated coordinates.

Midpoint Circle Drawing Algorithm:

Algorithm Steps:

  1. Given the center (xc, yc) and radius r.
  2. Initialize variables: x = 0, y = r, and the decision parameter d = 1 – r.
  3. Plot the eight symmetric points (x, y), (-x, y), (x, -y), (-x, -y), (y, x), (-y, x), (y, -x), and (-y, -x).
  4. For each step:
    • If d < 0, update d and increment x.
    • Else, update d and both x and y.