Graphics Reference
In-Depth Information
9.18
Summary of Curve Formulas
Ú
a
b
Length of a curve g : [a,b] Æ
R
n
:
g¢
Arc-length parameterization g(s):
|g¢(s)| = 1
(this equation defines it)
g¢(s) • g≤ (s) = 0
d
ds
1
d
dt
In general:
=
¢
()
g
t
T(s) =g¢(s)
Curve g in
R
2
:
Principal normal N(s):
|N(s)| = 1
(T(s),N(s)) determines the standard
orientation of
R
2
N(s) = (-T
2
(s),T
1
(s))
Signed curvature k
S
(s):
T¢(s) = k
S
(s)N(s)
Curvature k(s):
k(s) = |k
S
(s)| = |T¢(s)|
()
¢
()
Ts
Ê
Ë
ˆ
¯
()
=
k
S
s
det
Ts
¢
()
¢¢
()
-¢
()
¢¢
()
¢
()
≤
()
g
g
t
gg
t
t
g g
t
t
1
Ê
Ë
ˆ
¯
1
2
2
1
()
=
k
S
t
=
det
(
)
32
3
t
¢
()
2
2
g
t
¢
()
+¢
()
g
t
g
t
1
2
1
If n
()
=
¢
()
-¢
()
¢
()
(
)
¢
()
=
()
¢
()
g
t
,
g
t
,
then n t
k
t
g
t
.
2
1
S
g
t
d
ds
()
=
()
Total curvature k
T
of g:
k
s
k
s
T
S
Curve g in
R
3
:
Curvature vector:
K(s) =g≤(s)
Curvature:
k(s) = |K(s)|
¢
()
¥≤
()
¢
()
g
t
g
t
()
=
k t
3
g
t
Inflection point:
Point on curve where curvature is zero.
Principal normal N(s):
T¢(s) = k(s)N(s)
Binormal B(s):
B(s) = T(s) ¥ N(s)
