Graphics Reference
In-Depth Information
10.5 Intersection of a Circle with a Straight Line
The equation of a circle has already been given in the previous chapter, so we
will now consider how to compute its intersection with a straight line.
We begin by testing the equation of a circle with the normal form of the
line equation:
x 2 + y 2 = r 2 and y = mx + c
By substituting the line equation in the circle's equation we discover the two
intersection points:
r 2 (1 + m 2 )
x 1 , 2 =
mc
±
c 2
1+ m 2
m r 2 (1 + m 2 )
y 1 , 2 = c
±
c 2
(10.57)
1+ m 2
Let's test this result with the scenario shown in Figure 10.27. Using the normal
form of the line equation, we have
y = x +1 where m =1and c =1
Substituting these values in (10.57) yields
x 1 , 2 =
1 , 0and y 1 , 2 =0 , 1
The actual points of intersection are (
1, 0) and (0,1).
Y
y = x + 1
x y + 1 = 0
0.707 x + 0.707y
0.707 = 0
1
X
1
1
x 2 + y 2 = r 2
Fig. 10.27. The intersection of a circle with a line defined in its normal form, general
form, and the Hessian normal form.
 
Search WWH ::




Custom Search