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Programming Discrete Math Concepts for Beginners

Cartesian Point Class

Explore how to implement the Cartesian Point class and use vector cross products to calculate the position of a point relative to a line segment. Understand the findPosition() method and how to work with coordinates and vectors in Java to solve geometric problems programmatically.

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Introduction to the Cartesian system

Let’s dive into the discrete mathematical concept of vector space. In computer science, vectors represent a couple of things, such as lists of only rows or of rows and columns. Sometimes, we can place a vector point with two coordinates, x and y, and find the direction of the point concerning a line segment.

This algorithm is used in producing maps and directions, finding the area of polygons, and much more using vector cross products.

“How do we get the direction of a point on a map? Should we turn right or left? We can figure that out using the x- and y-coordinates: x for right, y for left. We can also move forward if that point lies on the same line.”

In geometry, three concepts are vital: point, line, and plane. To understand these concepts, let’s first see the illustration below.

Using the cross product for the position of points

Looking at the illustration, we can say that point C is on the left side of the line segment AB (moving from point A to B). Is there any formula to find that in vector space?

Yes, there is. We can use the vector cross product.

In our next Java program, let’s calculate the direction of C from line segment AB.

OnLeftSide
OnRightSide
OnSameLine
class PointClass{
    double x,y;
}
//GetDirection
class Main{
    public static void main(String[] args) {
        PointClass a = new PointClass();
        a.x = -4;
        a.y = 3;
        PointClass b = new PointClass();
        b.x = 2;
        b.y = -3;
        PointClass c = new PointClass();
        c.x = -5;
        c.y = -3;
        PointClass ba = new PointClass();
        ba.x = b.x - a.x;
        ba.y = b.y - a.y;
        PointClass ca = new PointClass();
        ca.x = c.x - a.x;
        ca.y = c.y - a.y;
        double crossProductsOfBAndC =  (ba.y * ca.x) - (ba.x * ca.y);
        if(crossProductsOfBAndC > 0){
            System.out.println("The point C is on the left of line AB.");
        } else if(crossProductsOfBAndC < 0){
            System.out.println("The point C is on the right of line AB.");
        } else {
            System.out.println("The point C is on the same line of AB.");
        }
    }
}
Java code for figuring out the location of a point

Take a look at the output of the OnLeftSide.java file:

The point C is on the left of line AB.

Next, let’s change all the coordinates of C to negative values in the OnRightSide.java file:

The point C is on the right of line AB.

Next, let’s change the value of C to (0,-1) in the OnSameLine.java file:

The point C is on the same line of AB.

  • Lines 7–15: This code makes use of Java’s built-in PointClass class to initialize three points A, B, and C, in the Cartesian plane, represented by a, b, and c in the code.

  • Lines 16–21: Then, we take two vectors between these points, AB→\overrightarrow{\rm AB} and AC→\overrightarrow{\rm AC} ...