Graphics Reference
In-Depth Information
In the limit as t
→
0, we have
f
t
=
dx
dt
i
dy
dt
j
dz
dt
k
+
+
lim
t
→
0
Therefore,
d
f
dt
=
f
t
=
f
t
+
t
−
f
t
lim
t
lim
t
t
→
0
→
0
From Fig. 8.1 we can see that as Q approaches P,
−
QP becomes tangential to the point P.
As an illustration, let's define a helix coiling upwards about the vertical y-axis:
f
t
=
cos t
i
+
t
j
+
sint
k
and
d
f
dt
=
d
dt
cos t
i
d
dt
t
j
d
dt
sint
k
+
+
d
f
dt
=
−
sint
i
+
j
+
cos t
k
which is the slope of the curve's tangent at the point
f
t. For example, when t
=
2, we have
d
f
dt
=
j
+
k
Without proof, the differentials of vector operations are
d
dt
p
d
p
dt
+
d
q
dt
+
q
=
d
dt
p
d
p
dt
·
d
q
dt
·
q
=
q
+
p
·
d
dt
p
d
p
dt
×
d
q
dt
×
q
=
q
+
p
×
d
dt
p
d
p
dt
·
d
q
dt
×
d
r
dt
·
q
×
r
=
q
×
r
+
p
·
r
+
p
·
q
×
d
q
dt
×
r
q
d
dt
p
d
p
dt
×
d
r
dt
×
q
×
r
=
q
×
r
+
p
×
+
p
×
×
