Weiler-Atherton Polygon Clipping in Computer Graphics
Introduction
Weiler-Atherton polygon clipping is a fundamental algorithm used in computer graphics to clip one polygon against another. It is commonly employed in rendering engines to determine which parts of polygons should be rendered based on the view frustum or other clipping boundaries.
Algorithm Overview
The Weiler-Atherton algorithm operates by intersecting two polygons and generating a new set of polygons representing the clipped regions. Here's a step-by-step guide:
- Identify Polygon Vertices: Begin by identifying all vertices of the two polygons.
- Generate Intersection Points: Find the intersection points between the edges of the polygons.
- Classify Intersection Points: Classify the intersection points as entering or exiting points for each polygon based on the direction of the intersecting edges.
- Construct Clipped Polygons: Use the classified intersection points to construct new polygons representing the clipped regions of the original polygons.
Step-by-Step Process
Identify Polygon Vertices
Consider two polygons, P and Q, with vertices P1, P2, ..., Pn and Q1, Q2, ..., Qm respectively.
Generate Intersection Points
For each edge of polygon P, check for intersections with the edges of polygon Q, and vice versa. Compute the intersection points using line intersection algorithms such as the parametric equation of a line or the line-line intersection formula.
Classify Intersection Points
Determine whether each intersection point is an entering or exiting point for each polygon. This classification is based on the order of vertices and the direction of traversal.
Construct Clipped Polygons
Using the classified intersection points, construct new polygons representing the clipped regions of the original polygons. Traverse the edges of each polygon, starting from an entering point and ending at an exiting point, to form the clipped polygons.
Example
Consider two polygons:
- Polygon P: [(0,0), (2,0), (2,2), (0,2)]
- Polygon Q: [(1,1), (3,1), (3,3), (1,3)]
Intersecting the edges of P and Q yields intersection points at (1,0), (2,1), (1,2), and (0,1).
Classifying these intersection points results in the following:
- Entering points for P: (1,0), (0,1)
- Exiting points for P: (2,1), (1,2)
- Entering points for Q: (1,0), (0,1)
- Exiting points for Q: (2,1), (1,2)
Using these points, construct the clipped polygons representing the regions of intersection.
Conclusion
The Weiler-Atherton polygon clipping algorithm is a powerful tool for determining the intersection between two polygons in computer graphics. Its step-by-step process allows for precise clipping and rendering of complex shapes, making it a valuable asset in various applications.