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Parameterized Curves -- Regular Curves -- Arc Length -- |
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Parametrized Curves:
A parametrized differentiable curve is simply a specific subset of R3 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 1-dimensional, 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 R3 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 R3. (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) = {t3- 2t2, t2}. 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,t2} and b(t) = {2t, 4t2}. Both of these curves trace a parabola. This illustrates the difference between a curve and thetrace of a curve. We can see from fig. 1-2 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)}.
Regular Curves:
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 t1 is a line that passes through f(t1) and contains the vector f'(t1). It can be parameterized by g(n) = f(t1) + n f'(t1). 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.
Arc Length:
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 t0 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[Dx2 + Dy2]. 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, t0 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.
