Graphics Reference
In-Depth Information
Figure 9.27.
Geodesics on a surface of
revolution.
z
meridian
y
h(t)=(x(t),y(t))
a
b
x
and
()
¥
()
=
()
¢
()
-
()
(
-
()
)
F
t
t
,
q
F
t
,
q
ytyt xt
,
cos
q
,
xt
sin
q
.
q
The vector F
t
¥F
q
is a surface normal. Although it may not have unit length, using
it in formulas, as we shall do, rather than a normalized version will simplify expres-
sions but not change the validity of when certain dot products are zero.
Consider the meridian
()
=
()
a
t
F
t
,
q
and its derivatives
¢
()
=¢
()
¢
()
(
¢
()
)
a
t
xt
,
yt
cos
q
,
yt
sin
q
≤
()
=≤
()
≤
()
(
≤
()
)
a
t
xt
,
yt
cos
q
,
yt
sin
q
It is easy to check that
(
(
)
¥¢
)
=
a
≤∑
FF
¥
a
0,
t
q
and so a(t) is a geodesic by the theorems mentioned.
Next, consider the circle of latitude
()
=
()
bq
F t,
q
and its derivatives
¢
()
=-
()
(
()
)
bq
0
0
,
yt
sin
q
,
yt
cos
q
≤
()
=-
()
(
-
()
)
bq
,
yt
cos
q
,
yt
sin
q
.
This time
(
(
)
¥¢
)
=¢
3
b
≤∑
FF
¥
b
yy
.
t
q
