Graphics Reference
In-Depth Information
Torsion t(s):
B¢ (s) =-t(s)N(s)
¢
()
≤
()
¢≤
()
g
g
g
s
Ê
ˆ
1
1
()
=
(
¢
()
¥≤
()
)
∑¢≤
()
=
Á
Á
˜
˜
t
s
g
s
g
s
g
s
det
s
s
2
()
2
()
k
s
k
s
Ë
¯
¢
()
g
t
Ê
ˆ
(
¢
()
¥≤
()
)
∑¢≤
()
¢
()
¥≤
()
g
t
g
t
g
t
1
()
=
Á
Á
≤
()
¢≤
()
˜
˜
t
t
=
det
g
g
t
t
2
2
¢
()
¥≤
()
g
t
g
t
g
t
g
t
Ë
¯
The Serret-Frenet Formulas:
T¢=
kN
N¢=-kT
+ tB
B¢=
-tN
Curve g in
R
n
:
u
¢
=
k
u
1
1
2
u
¢
=-
k
u
+
k
u
2
1
1
2
3
u
¢
=-
k
u
+
k
u
3
2
2
3
4
.
.
.
.
.
.
u
¢
=-
k
u
+
k
u
n
-
1
n
-
2
n
-
2
n
-
1
n
u
¢
=-
k
u
n
n
--
11
n
Involute of curve p(s):
p*(s) = p(s) + (c - s)T(s), where c is a constant
22
kt
k
+
Curvature of involute p*(s):
k
* =
2
(
)
2
cs
-
1
1
(
)
s
()
=
()
+
()
+
Ú
()
Evolute of curve p(s):
qs
ps
Ns
cot
t
+
cBs
,
()
()
k
s
k
s
0
for some constant c.
1
c
s
()
=
()
+
()
+
()
Evolute of planar curve p(s):
qs
ps
Ns
Bs
,
()
()
k
s
k
for some constant c
1
()
=
()
+
()
Plane evolute of planar curve p(s):
qs
ps
Ns
()
k
s
Parallel or offset curves:
p
d
(t) = p(t) + dn(t)
p
d
¢(t) = p¢(t) (1 + k(t)d)
k
k
()
t
td
()
=
k
d
t
()
1
+
