Graphics Reference
In-Depth Information
Definition.
The function l(u) is called the
distribution parameter
of the ruled surface.
Note that the distribution parameter is constant along each ruling.
If we use Lemma 9.15.3 again and also equation (9.88) we have that
(
)
∑¢ + ¢
(
)
=¢∑¢+
2
Ep p
=∑=¢ +¢
q v
a
q v
a
qq v
a
¢∑¢
a
uu
(
)
∑=¢ ∑
Fp p
=∑=¢ +¢
q v
aa
q
a
uv
Gp p
=∑=
1.
v
v
From this, the vector identity in Proposition 1.10.4(4), and the definition of l we get
2
2
2
(
)
EG
-
F
=
v
aa
¢∑
¢+
q
¢∑ ¢-
q
q
¢∑
a
2
2
(
)
(
)
-¢ ∑
(
)
=¢ ∑
v
aa
¢ +¢ ∑
q
q
¢
aa
∑
q
a
2
2
(
)
∑¢ ¥
(
)
=¢ ∑
v
aa
¢ +¢ ¥
q
a
q
a
(
)
2
(
)
=+
l
v
a
¢∑
a
¢
Furthermore, equations (9.86), (9.87), and the fact that (a¢ ¥ a)•a¢ = 0 implies
2
2
(
(
(
)
¥
)
∑¢
)
(
(
)
∑¢
)
2
(
)
qv
pp
¢+
aaa
¢
q
¢¥
aa
la a
l
¢∑
¢
2
2
(
)
M
=∑¢
n
a
=
=
=
.
(
)
2
2
2
(
)
2
2
l
+
v
a
¢∑
a
¢
+
v
¥
u
v
We collect all these facts together in the theorem we were after.
9.15.4. Theorem.
The Gauss curvature of the ruled surface defined by equation
(9.85) with a¢ π 0 is given by
2
2
M
EG
l
(
)
=-
KKuv
=
,
=-
,
2
2
-
F
(
)
2
2
l
+
v
where l is defined by equation (9.89).
It follows from the theorem that the Gauss curvature of a ruled surface is £ 0.
Definition.
A ruled surface is called a
developable surface
if the tangent plane is
constant along each ruling.
Developable surfaces have also been defined as the envelope of a one-parameter
family of planes. What this means is that the points p(u,v) of the surface satisfy an
equation of the form
(
)
∑
()
=
()
puv
,
Nt
qt
,
(9.90)
for a nonzero vector-values function N(t) and a real function q(t).
If
S
is a surface in
R
3
, then the following statements about
S
are
9.15.5. Theorem.
equivalent:
