Algorithm to find all line segment intersections given n lines - algorithm

Question

I'm looking for a algorithm to find all the intersection points given n line segments.
Below is the pseudo code from http://jeffe.cs.illinois.edu/teaching/373/notes/x06-sweepline.pdf
The input S[1 .. n] is an array of line
segments.
label[i] is the label of the ith leftmost endpoint.
sort the endpoints of S from left to right
create an empty label sequence
for i ← 1 to 2n
line ← label[i]
if isLeftEndPoint[i]
Insert(line)
if Intersect(S[line], S[Successor(line)])
return TRUE
if Intersect(S[line], S[Predecessor(line)])
return TRUE
else
if Intersect(S[Successor(line)], S[Predecessor(line)])
return TRUE
Delete(label[i])
return FALSE
Apply the algorithm to the line set below, only one intersection point is checked. What should I do to know the existence of the other 2 intersection points?
line[1] enters
line[2] enters, intersection between line[1] and line[2] is checked.
line[3] enters, intersection between line[2] and line[3] is checked.
line[4] enters, intersection between line[4] and line[1] is checked. Intersection A is found.
line[4] leaves, nothing is checked.
line[1] leaves, nothing is checked.
line[2] leaves, nothing is checked.
line[3] leaves, nothing is checked.

Answer 1

Standard line equation
Ax+By=C
The slope(m) of a line defined by the standard line of equation is
m = -(A/B)
Point-slope line equation
y-y1=m(x-x1)
Substituting m = (-A/B) in the point-slope line equation
y2-y1 = (A/-B)*(x2-x1)
(y2-y1)/(x2-x1) = A/-B
thus:
A = y2-y1
B = x1-x2
C = Ax+By
x = (C-By)/A
y = (C-Ax)/B
Given two lines with equation
A1x1+B1y1=C1 and A2x2+B2y2=C2.
Then the point of intersection between the lines is specified
by the points that make A1x+B1y-C1 = A2x+B2y-C2
A1x+B1y=C1
A2x+B2y=C2
A1B2x+B1B2y=B2C1 (multiply the first equation by B2)
A1B2x+B1B2y-B2C1=0
A2B1x+B1B2y=B1C2 (multiply the second equation by B1)
A2B1x+B1B2y-B1C2=0
Equating the two equations
A1B2x+B1B2y-B2C1=A2B1x+B1B2y-B1C2
A1B2x+B1B2y-B2C1-A2B1x-B1B2y+B1C2=0
A1B2x-B2C1-A2B1x+B1C2=0
A1B2x-A2B1x=B2C1-B1C2
x(A1B2-A2B1)=B2C1-B1C2
x = (B2C1-B1C2)/A1B2-A2B1
A1x+B1y=C1
A2x+B2y=C2
A1A2x+A2B1y=A2C1 (multiply the first equation by A2)
A1A2x+A2B1y-A2C1=0
A1A2x+A1B2y=A1C2 (multiply the second equation by A1)
A1A2x+A1B2y-A1C2=0
Equating the two equations
A1A2x+A2B1y-A2C1=A1A2x+A1B2y-A1C2
A1A2x+A2B1y-A2C1-A1A2x-A1B2y+A1C2=0
A1C2-A2C2=A1B2y-A2B1y
A1B2y-A2B1y=A1C2-A2C2
y(A1B2-A2B1)=A1C2-A2C1
y(A1B2-A2B1)=A1C2-A2C1
y = (A1C2-A2C1)/(A1B1-A2B1)
the denominator in y and in x are the same so
denominator = A1B1-A2B1
thus:
x = (B2C1-B1C2)/denominator
y = (A1C2-A2C1)/denominator
These are the x and y coordinates of the intersection of two lines with points (x1, y1), (x2, y2) and (x3, y3), (x4, y4)
Now for a line segment it's the same but we need to check that the x or y coordinate is in both segments. That means between the x coordinate of both segments with lesser value and the x coordinate of both segments with greater value
This is a C++ program that returns true if the segments intersect and returns false if they don't. If the segments intersect it stores the point of intersection in a variable i.
struct Point
{
float x, y;
};
//p1 and p2 are the points of the first segment
//p3 and p4 are the points of the second segment
bool intersection(Point p1, Point p2, Point p3, Point p4, Point &i)
{
float max1; //x-coordinate with greater value in segment 1
float min1; //x-coordinate with lesse value in segment 1
float max2; //x-coordinate with greater value in segment 2
float min2; //x-coordinate with lesser value in segment 2
float A1 = p2.y - p1.y;
float B1 = p1.x - p2.x;
float C1 = A1 * p1.x + B1 * p1.y;
float A2 = p4.y - p3.y;
float B2 = p3.x - p4.x;
float C2 = A2 * p3.x + B2 * p3.y;
float denom = A1 * B2 - A2 * B1;
if (denom == 0.0) //When denom == 0, is because the lines are parallel
return false; //Parallel lines do not intersect
i.x = (C1 * B2 - C2 * B1) / denom;
i.y = (A1 * C2 - A2 * C1) / denom;
if (p1.x > p2.x)
{
max1 = p1.x;
min1 = p2.x;
}
else
{
max1 = p2.x;
min1 = p1.x;
}
if (p3.x > p4.x)
{
max2 = p3.x;
min2 = p4.x;
}
else
{
max2 = p4.x;
min2 = p3.x;
}
//check if x coordinate is in both segments
if (i.x >= min1 && i.x <= max1 &&
i.x >= min2 && i.x <= max2)
return true;
return false; //Do no intersect, intersection of the lines is not between the segments
}
Now you just need to compare on a loop all the segments and store the intersection point on array.