Graphics Reference
In-Depth Information
Testing the equation of the circle with the general equation of the line
ax
+
by
+
c
= 0 yields intersections given by
b
r
2
(
a
2
+
b
2
)
x
1
,
2
=
−
ac
±
−
c
2
a
2
+
b
2
a
r
2
(
a
2
+
b
2
)
y
1
,
2
=
−
bc
±
−
c
2
(10.58)
a
2
+
b
2
From Fig. 10.27, the general form of the line equation is
x
−
y
+1=0 where
a
=1
,b
=
−
1and
c
=1
Substituting these values in (10.58) yields
x
1
,
2
=
−
1
,
0and
y
1
,
2
=0
,
1
which gives the same intersection points found above.
Finally, using the Hessian normal form of the line
ax
+
by
−
d
= 0 yields
intersections given by
b
r
2
x
1
,
2
=
ad
±
−
d
2
a
r
2
y
1
,
2
=
bd
±
−
d
2
(10.59)
From Fig. 10.27, the Hessian normal form of the line equation is
−
0
.
707
x
+0
.
707
y
−
0
.
707 = 0
where
a
=
−
0
.
707
,b
=0
.
707 and
d
=0
.
707. Substituting these values in
(10.59) yields
x
1
,
2
=
−
1
,
0and
y
1
,
2
=0
,
1
which gives the same intersection points found above. One can readily see the
computational benefits of using the Hessian normal form over the other forms
of equations.
10.6 3D Geometry
3D straight lines are best described using vector notation, and it is a good
idea to develop strong skills in vector techniques if you wish to solve problems
in 3D geometry.
Let's begin this short survey of 3D analytic geometry by describing the
equation of a straight line.
