A parametrized differentiable curve is simply a specific subset of
R^{3} with which certain aspects of differntial
calculus can be applied. The curve is described by a set of differentiable
functions with only one variable. The curves generated can be thought
of as 1dimensional, and by definition are a map from the real number
line:
Definition.
The term differentiable in this case means that every function describing
the curve is differentiable (or smooth). For example, take the curve
described by f(t) = { x(t),y(t),z(t) }. The curve f is differeniable
if each of the functions x(t), y(t), and z(t) are each themselves differentiable.
The trace of a curve is the image set in
R^{3} which is generated by the Curve in a given interval. One must not
confuse the meanings of Curve and Trace. A Curve is map (in other words,
a fuction, or equation), while a trace is just a picture. The trace
can be thought of as how the curve "looks" in R^{3}. (See example
3)
The vector {x'(t),y'(t),z'(t)} = f'(t) is called the tangent vector
of f(t) at the point t. Where x'(t) denotes the derivative of x(t) with
respect to t.
Example 1: The graph above, next to the table
of contents, is called a "helix". It is parameterized by f(t)
= {Sin[t],Cos[t],t}. In this case the trace was generated over the interval
u = 0 … 4Pi (this is obvious due to the two complete twists). It's
velocity vector at a point t is f'(t) = {Cos[t],Sin[t],1}. It is interesting
to note that the magnitude f'(t) = Sqrt[2] for all t. From a physics
point of view we would say that this parameterization has "constant
velocity".
Example 2: Consider the Curve
f(t) = {t^{3} 2t^{2}, t^{2}}. It is a parameterized differentiable curve
with its trace shown in Fig 1.1. Notice that f'(0) = {0,0}. In other
words, the velocity vector is zero at t = 0. Notice that velocity vectors
shown get smaller as t approaches 0, agreeing with the calculated result.
Example 3: Let a(t) = {t,t^{2}} and b(t) = {2t,
4t^{2}}. Both of these curves trace a parabola. This illustrates the difference
between a curve and thetrace of a curve. We can see from
fig. 12 that the velocity vector a'(t) is half the length of b'(t).
Despite the identical trace, these are two different curves.
The Orientation of a curve concerns
which direction the velocity vector is pointing at a given t. We say
that two curves, f and g, differ in orientation if they have the same
trace and at a given point on the curve, f'(t1) = g'(t2). In other
words, the vector points in the opposite direction. In order to change
the orientation of a curve f(t) = {x(t), y(t)}is take g(t) = f(t) =
{x(t), y(t)}.
In the study of Differential Geometry it is important to understand
the concept of a tangent line. A tangent line of f(t) at a given
point t_{1} is a line that passes through f(t_{1}) and contains the vector
f'(t_{1}). It can be parameterized by g(n) = f(t_{1}) + n f'(t_{1}). It is important
that any curve we study have a tangent line at every point, we must
exclude Curves which have a velocity vector of 0 at any point. We call
any point that satisfies f'(t) = 0 a singular point and we will
ristrict our study to curves without singular points.
Definition.
In other words, a regular curve is one where the velocity vector never
goes to zero. Thankfully, regular curves have tangent lines at every
point along the curve. Notice that Example 2 in the
above section has its velocity vector equal to 0 at t = 0, therefor
it is not a regular curve.
The Arc Length of a curve can be described as the "length"
of a piece of string if it were layed upon the curve. The Arc Length
of a regular differentiable curve a(t) from point t_{0}
to t is defined to be:
where a'(t) is the length of the vector a'(t).
Proof.

Take a(t) to be a regular differentiaible curve.
A small section of this curve from t to t+Dt is shown. For sufficiently
small Dt, the arc length can given simply by s =
Sqrt[Dx^{2} +
Dy^{2}].
If we divide the curve a(t) up into n peices, then add them togeather,
we can approximate the arc length as:
Multiplying through by Dt/Dt and taking the limit yeilds our finial
equation:

Note that since a'(t) > 0, the arc length s is a differentiable function
of t and ds/dt = a'(t).
As can be seen from Example 2 above, a single trace
can be parameterized many different ways. A paticularly useful parameterization
of a Curve is called parameterization by arc length. In other
words, instead of an arbitrary variable t, the variable represents arclength
of the Curve. For example, if f(s) were a curve parameterized by arclength,
the trace of f(s), 0 < s < 1, would have an arclength of 1.
Curves parametrized by arc length: It can happen that the parameter
t is already the arc length measured from some point. In this case,
ds/dt = 1 = a'(t); that is, the velocity vector has constant length
equal to 1. Definition: A curve a from (a,b) > R is said to be parametrized
by arc length s if a'(s) = 1 for all s in (a,b).
Propositions: (1) a(t) is parametrized by arc length if and only if
t is the arc length of a measured from some point. (proof) (2) Let a:
I > R be a curve parametrized by arc length, then a"(s) is normal
to a'(s). (proof)Since a is parametrized by arc length, a'(s)
= 1 for all s in I. By differentiating
a'(s)·a'(s)
= 1 we obtain a"(s)·a'(s)
= 0. Thus, a"(s) is normal to a'(s).
Reparametrization by arc length: Given a regular parametrized curve
a: I > R (not necessarily parametrized by arc length), it is possible
to obtain a curve b: J ® R parametrized by arc length which has the
same trace as a. In fact, let s = (*junk*), t, t_{0} in I. Recall ds/dt
= a'(t) does not equal 0, the function s = s(t) has a differentiable inverse t =
t(s), s in s(I) = J, where, by an abuse of notation, t also denotes the
inverse function s^{1} of s. Now set b = a(t) : J > R . Clearly, b(J) =
a(I) and b'(s) = a'(t)·(dt/ds) = a'(t) =1. This shows that b has
the same trace as a and is parametrized by arc length. It is usual to
say that b is a reparametrization of a(I) by arc length.