Graphics Reference
In-Depth Information
3.3 Parametric Representation of Curves
In the following sections, we provide a brief introduction to the elementary differential geome-
try of curves terms used in the manuscript. For details, please refer to some standard textbooks
on differential geometry such as Pressley (2001).
3
Definition 3.3.1
A parametrized curve
A parametrized curve in
R
is a map
γ
:[
a
,
b
]
→
3
, for some a, b with
R
−∞ ≤
a
<
b
≤+∞
. The symbol
[
a
,
b
]
denotes the open interval
[
a
,
b
]
={
t
∈ R
,
a
<
t
<
b
}
.
3
Definition 3.3.2
Reparametrization
A parametrized curve
˜
γ
:[
a
,
b
]
→ R
is a repara-
3
metrization of a parametrized curve
γ
:[
a
,
b
]
→ R
if there is a smooth bijective map
:
b
]
b
]
−
1
[
a
,
→
[
a
,
b
]
(the reparametrization map) such that the inverse map
:[
a
,
b
]
→
[
a
,
also smooth and
b
]
(
t
)
(
t
)) for all
t
γ
˜
=
γ
(
∈
[
a
,
.
(
3.2)
is also called a diffeomorphism from [ a,b] to itself.
Note that, since
has a smooth inverse, ˜
γ
is a reparametrization of
γ
:
−
1
(
t
))
−
1
(
t
)))
γ
˜
(
=
γ
(
(
=
γ
(
t
) for all
t
∈
[
a
,
b
]
.
(3.3)
Two curves that are reparametrizations of each other have the same image, so they should
have the same geometric properties.
by
x
2
y
2
Example 3.3.3
One
parametrization
of
a
circle
defined
+
=
1is
γ
=
(cos(
t
)
,
sin(
t
)), another parametrization of a circle is
γ
=
˜
(sin
t
,
cos
t
). To see that
γ
˜
is a
reparametrization of
γ
, we have to find a reparametrization map such that
(cos(
(
t
))
,
sin(
(
t
)))
=
(sin(
t
)
,
cos(
t
))
.
One solution is
(
t
)
=
π/
2
−
t
.
In general the analysis of a curve is simplified when it is known to be unit-speed.
Definition 3.3.4
Arc-Length
The arc-length of a curve
γ
starting at the point
γ
(
t
0
)
is the
functions(t) given by
t
t
0
s
(
t
)
=
γ
˙
(
u
)
d
u
.
(
3.4)
˙
Definition 3.3.5
A point
γ
(
t
)
of a parametrized curve
γ
is called a regular point if
γ
(
t
)
=
0
;
otherwise
γ
(
t
)
is a singular point of
γ
. A curve is
regular
if all of its points are regular.
