space curve

space curve

Kinematic definition.

A

parameterized space curve

is a

parameterized curve

taking
values in 3-dimensional

Euclidean space

. It may be interpreted as the
trajectory of a particle moving through space. Analytically, a smooth
space curve is represented by a sufficiently

differentiable mapping

γ

:

I

→

ℝ

3

,

of an

interval

I

⊂

ℝ

into
3-dimensional Euclidean space

ℝ

3

. Equivalently, a
parameterized space curve can be considered a 3-vector of

functions

:

γ

⁢

(

t

)

=

(

x

⁢

(

t

)

y

⁢

(

t

)

z

⁢

(

t

)

)

,

t

∈

I

.

Regularity hypotheses.

To preclude the possibility of kinks and corners, it is
necessary to add the

hypothesis

that the

mapping

be

regular

(

http://planetmath.org/Curve

), that is
to say that the

derivative

γ

′

⁢

(

t

)

never

vanishes

. Also, we say
that

γ

⁢

(

t

)

is a

point of inflection

if the first and

second
derivatives

γ

′

⁢

(

t

)

,

γ

′′

⁢

(

t

)

are

linearly dependent

. Space curves
with points of inflection are beyond the scope of this entry.
Henceforth we make the

assumption

that

γ

⁢

(

t

)

is both and
lacks points of inflection.

Geometric definition.

A

space curve

, per se, needs to be conceived of as a subset of

ℝ

3

rather than a mapping. Formally, we could define a space
curve to be the image of some parameterization

γ

:

I

→

ℝ

3

. A
more useful

concept

, however, is the notion of an

oriented space
curve

, a space curve with a specified

direction

of motion.
Formally, an oriented space curve is an

equivalence class

of
parameterized space curves; with

γ

1

:

I

1

→

ℝ

3

and

γ

2

:

I

2

→

ℝ

3

being judged

equivalent

if there exists a
smooth, monotonically increasing reparameterization function

σ

:

I

1

→

I

2

such that

γ

1

⁢

(

t

)

=

γ

2

⁢

(

σ

⁢

(

t

)

)

,

t

∈

I

1

.

Arclength parameterization.

We say that

γ

:

I

→

ℝ

3

is an arclength parameterization of an
oriented space curve if

∥

γ

′

⁢

(

t

)

∥

=

1

,

t

∈

I

.

With this hypothesis the
length of the space curve between points

γ

⁢

(

t

2

)

and

γ

⁢

(

t

1

)

is
just

|

t

2

-

t

1

|

. In other words, the parameter in such a
parameterization measures the relative

distance

along the
curve.

Starting with an arbitrary parameterization

γ

:

I

→

ℝ

3

,
one can obtain an arclength parameterization by fixing a

t

0

∈

I

,
setting

σ

⁢

(

t

)

=

∫

t

0

t

∥

γ

′

⁢

(

x

)

∥

⁢

𝑑

x

,

and using the

inverse function

σ

-

1

to reparameterize the curve. In other
words,

γ

^

⁢

(

t

)

=

γ

⁢

(

σ

-

1

⁢

(

t

)

)

is an arclength
parameterization. Thus, every space curve possesses an arclength
parameterization, unique up to a choice of

additive

constant

in the
arclength parameter.

Title

space curve

Canonical name

SpaceCurve

Date of creation

2013-03-22 12:15:03

Last modified on

2013-03-22 12:15:03

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

15

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 53A04

Synonym

oriented space curve

Synonym

parameterized space curve

Related topic

Torsion

Related topic

CurvatureOfACurve

Related topic

MovingFrame

Related topic

SerretFrenetFormulas

Related topic

Helix

Defines

point of inflection

Defines

arclength parameterization

Defines

reparameterization

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