Intersection

Intersection means the point where two lines meet or cross.

In other words an intersection is a single point where two lines meet or cross each other. The above figure says that point k is an intersection of line segments AB and PQ. In another way it may be expressed as the line segment PQ intersects line segment AB at point K.

 

 

 

Circle – Line intersection:

 

 

An infinite line determined by 2 points that is (x1 , x2) and (x2 , y2). It may intersect a circle of radius and also center (0,0 ) in 2 imaginary points(figure a). A degenerative single point that is corresponding to the line being tangent to the circle (figure b) or two real points (figure c).

In geometry a tangent line known as a line meeting a circle in exactly one point. The secant line is defined as a line meeting a circle in exactly two points.

 

Intersection of a line with a circle:

 

Given the line,

Ax + By + C = 0 and

A circle of arbitrary center and radius,
 `x^2` + `y^2` - 2fx - 2gy + d = 0
where,
               d = f2 + g2 - r2
with,
               (f,g): Circle's Center
                   r: Circle's Radius
 
The general equation for the intersection between circle and line:
The circle points are (x1,x2) and (y1,y2)
 
x = x1 + t(y1 –x1)
y = x2 + t(y2 –x2)
 
Then we replace x and y in the circle equation (x-h)2+(y-k)2-r2 = 0
By the expression x = x1 + t(y1-x1) and y = x2 + t(y2-x2) and then solve for t. In general substitute these equations into parametric equations for the intersecting points.
If The discriminant is positive the two intersecting points are occur or if the discriminant is zero, just one intersecting points occur or if the discriminant is negative there is no intersecting points will occur.
So, the formula for intersecting line and circle.
 
 
t = (2x12 + 2x22 - 2x1y1 - 2x2y2 - 2x1h + 2y1h - 2x2k + 2y2k - 
sqrt(-2x12 - 2x22 + 2x1y1 + 2x2y2 + 2x1h - 2y1h + 2y2k - 2y2k)2 - 4(x12 + x22 - 2x1y1 + y12 - 2x2y2 + y22)(x12 + x22 - 2x1h + h2 - 2x2k + k2 - r2)) / (2(x12 + x22 - 2x1y1 + y12 - 2x2y2 + y22))}.