Graphics Reference
In-Depth Information
To develop a generic equation for line equations in R
2
, we write the original curve equation as
y
−
fx
=
0
or
F xy
=
0
(8.5)
where the implicit function F is dependent upon two variables, but still describes a planar curve.
If we make small changes to F using dx and dy, and remain on the curve, F xy must still
equal zero, and dF
=
0. Therefore,
F
x
dx
F
y
dy
=
+
=
dF
0
(8.6)
But Eq. (8.6) can be written as a vector scalar product:
F
x
i
y
j
F
dF
=
+
·
dx
i
+
dy
j
=
0
Figure 8.3 illustrates why dx
i
+
dy
j
is the tangent vector to the curve F.
Y
y
=
f
(
x
)
t
dy
j
dx
i
X
Figure 8.3.
If the product
F
x
i
y
j
F
+
·
dx
i
+
dy
j
=
0
then
F
x
i
F
y
j
+
must be the normal vector to the curve F:
F
x
i
F
y
j
n
=
+
So the curve F is defined as the scalar product of two perpendicular vectors: the tangent and
normal vectors.
