Step by Step solution about How to resolve the algorithm Find the intersection of two lines step by step in the C++ programming language

This C++ program calculates the intersection point of two lines using the line-line intersection algorithm. Here's how it works:

1. Header Includes:

   • The program includes necessary header files for input/output, mathematical functions, and assertions.

2. Determinant Calculation:

   • The Det function is defined to calculate the determinant of a 2x2 matrix [a b; c d], which is used later in the line-line intersection calculations.

3. Line-Line Intersection Function:

   • The LineLineIntersect function takes the start and end points of two lines as input and calculates their intersection point, if it exists.
   • It calculates the determinant of each line and uses Cramer's rule to solve for the intersection point.
   • If the determinant is zero, the lines are parallel and don't intersect, so it returns false.
   • If the intersection point is valid and finite, it returns true and sets the output parameters ixOut and iyOut to the coordinates of the intersection point.

4. Main Function:

   • In the main function:
     • It defines the start and end points of two lines (Line 1 and Line 2).
     • It calls the LineLineIntersect function to calculate the intersection point of the two lines and stores the result in result, ix, and iy.
     • It prints the result.
     • It uses assertions to check the correctness of the intersection point.

5. Program Execution:

   • For the given input lines (Line 1 and Line 2), the program calculates the intersection point and prints the result.
   • The assertions verify that the intersection point is correct, with a tolerance of eps.

In summary, this program demonstrates how to calculate the intersection point of two lines using the line-line intersection algorithm. It handles the case of parallel lines (no intersection) and performs checks to ensure the validity of the intersection point.

#include <iostream>
#include <cmath>
#include <cassert>
using namespace std;

/** Calculate determinant of matrix:
	[a b]
	[c d]
*/
inline double Det(double a, double b, double c, double d)
{
	return a*d - b*c;
}

/// Calculate intersection of two lines.
///\return true if found, false if not found or error
bool LineLineIntersect(double x1, double y1, // Line 1 start
	double x2, double y2, // Line 1 end
	double x3, double y3, // Line 2 start
	double x4, double y4, // Line 2 end
	double &ixOut, double &iyOut) // Output 
{
	double detL1 = Det(x1, y1, x2, y2);
	double detL2 = Det(x3, y3, x4, y4);
	double x1mx2 = x1 - x2;
	double x3mx4 = x3 - x4;
	double y1my2 = y1 - y2;
	double y3my4 = y3 - y4;

	double denom = Det(x1mx2, y1my2, x3mx4, y3my4);
	if(denom == 0.0) // Lines don't seem to cross
	{
		ixOut = NAN;
		iyOut = NAN;
		return false;
	}

	double xnom = Det(detL1, x1mx2, detL2, x3mx4);
	double ynom = Det(detL1, y1my2, detL2, y3my4);
	ixOut = xnom / denom;	
	iyOut = ynom / denom;
	if(!isfinite(ixOut) || !isfinite(iyOut)) // Probably a numerical issue
		return false;

	return true; //All OK
}

int main()
{
	// **Simple crossing diagonal lines**

	// Line 1
	double x1=4.0, y1=0.0;
	double x2=6.0, y2=10.0;
	
	// Line 2
	double x3=0.0, y3=3.0;
	double x4=10.0, y4=7.0;

	double ix = -1.0, iy = -1.0;
	bool result = LineLineIntersect(x1, y1, x2, y2, x3, y3, x4, y4, ix, iy);
	cout << "result " <<  result << "," << ix << "," << iy << endl;

	double eps = 1e-6;
	assert(result == true);
	assert(fabs(ix - 5.0) < eps);
	assert(fabs(iy - 5.0) < eps);
        return 0;
}