(Geometry: Graham's algorithm for finding a convex hull)...

(Geometry: Graham's algorithm for finding a convex hull) Section 22.10.2 introduced Graham's algorithm for finding a convex hull for a set of points. Assume that the Java's coordinate system is used for the points. Implement the algorithm using the following method: /*t Return the points that form a convex hu17 */ public static Arraylist<MyPoint> getConvexHul1(double[][] s) MyPoint is a static inner class defined as follows: private static class MyPoint implements Comparable<MyPoint> \{ double $x, y$; MyPoint rightMostLowestPoint; MyPoint(double $x$, double $y$ ) \{ \} this. $x=x$; this. $y=y$; public void setRightMostLowestPoint(MyPoint p) \{ \} rightMostLowestPoint $=p$; Goverride public int compareTo(MyPoint o) \{ // Implement it to compare this point with point o // angularly along the $x$-axis with rightMostlowestPoint $/ /$ as the center, as shown in Figure 22.10b. By implementing // the Comparable interface, you can use the Array.sort // method to sort the potnts to stmplify coding. \} \} Write a test program that prompts the user to enter the set size and the points and displays the points that form a convex hull. Here is a sample run:FIGURE CANT COPY

(Geometry: Graham's algorithm for finding a convex hull) Section 22.10.2 introduced Graham's algorithm for finding a convex hull for a set of points. Assume that the Java's coordinate system is used for the points. Implement the algorithm using the following method:
/*t Return the points that form a convex hu17 */ public static Arraylist<MyPoint> getConvexHul1(double[][] s)
MyPoint is a static inner class defined as follows:
private static class MyPoint implements Comparable<MyPoint> \{ double $x, y$;
MyPoint rightMostLowestPoint;
MyPoint(double $x$, double $y$ ) \{ \} this. $x=x$; this. $y=y$;
public void setRightMostLowestPoint(MyPoint p) \{ \} rightMostLowestPoint $=p$;
Goverride
public int compareTo(MyPoint o) \{
// Implement it to compare this point with point o
// angularly along the $x$-axis with rightMostlowestPoint $/ /$ as the center, as shown in Figure 22.10b. By implementing // the Comparable interface, you can use the Array.sort // method to sort the potnts to stmplify coding.
\}
\}
Write a test program that prompts the user to enter the set size and the points and displays the points that form a convex hull. Here is a sample run:FIGURE CANT COPY

Key Concepts

Orientation Test

The orientation test involves determining the relative position of three points to decide whether they form a left turn, a right turn, or are collinear. This test is essential in convex hull algorithms like Graham's, as it helps in deciding whether a point should be included in or excluded from the convex hull by checking if moving from one point to the next maintains a convex (usually left-turn) progression.

Polar Angle Sorting

Polar angle sorting is the process of ordering points based on the angle they make with a reference point, typically measured relative to the horizontal axis. This sorting is crucial in Graham's algorithm, as it arranges the points in the order they appear around the reference, allowing the algorithm to easily check and maintain the convexity of the hull as points are processed.

Java Comparable Interface

The Java Comparable interface is used to define a natural ordering for objects. In the context of implementing Graham's algorithm, it allows points to be sorted by their polar angles relative to a reference point. By overriding the compareTo method, developers ensure that the Array.sort or Collections.sort method can order the points properly, which is a critical step in the algorithm.

Convex Hull

A convex hull is the smallest convex polygon that encloses a set of points in the plane. It can be visualized as the shape formed by a rubber band stretched around the outermost points. In computational geometry, finding the convex hull is a fundamental problem with applications in pattern recognition, image processing, and collision detection.

Graham's Algorithm

Graham's algorithm is an efficient method for computing the convex hull of a finite set of points by sorting the points angularly and then iteratively processing them to remove those that do not contribute to the hull. The algorithm identifies a base or reference point (often the point with the lowest y-value) and sorts the other points by polar angle relative to this point, ensuring an organized traversal to detect and discard concave turns.

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