Curves Definitions and Examples

7/31/2019 Curves Definitions and Examples

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Chapter 1

Curves

1.1 Denitions and Examples

Denition 1.1 A curve is a continuous mapping α : I → R n , where I is an interval inthe real line. The interval could be closed [ a, b], open (a, b), half-open [a, b), an open ray(a, ∞ ), or a closed ray ( −∞ , b]. The function α is called a parameterization for the curve.The curve is called smooth if α has derivatives of all orders. It is called differentiable of order k if all derivatives of order ≤ k exist and are continuous. The curve is called regularif dα

dt = 0 for all t (a, b).

We will be mostly interested in curves in R 3 . Then we can represent the curve in termsof its coordinate functions:

α (t) = ( x1 (t), x 2 (t), x 3 (t)) and sodα

dt

= x1 (t), x 2 (t), x 3 (t) .

Note that the notation in our text (and hence throughout the remainder of these notes)is:

α(t) = α 1 (t), α 2 (t), α 3 (t) and α (t 0 ) =dαdt

=dα 1

dt t = t 0

,dα 2

dt t = t 0

,dα 3

dt t = t 0

.

The image of α in R n is called the trace of the curve, and this is the geometric object.We often identify a curve with its trace, but realize that there may be many curves for agiven trace.

Example 1.1 We know that two points determine a unique line, and in the plane we knowhow to nd a unique representation of this line using the slope-intercept equation for a line.The slope is a unique number that is associated to a line in the plane. We have no suchnumber in three space. What we do have joining two points P and Q is the following. Let p denote the vector terminating at P and q denote the vector terminating at Q. Then thevector joining P to Q is the vector q − p. The line joining P and Q must run along thisvector, i.e. , it has the same direction as the vector q − p. We can use a parameter t totell us how far along the direction of q − p we are and put this together to get that theparametrized equation for a line in R 3 is

α (t) = p + t(q − p).

Note that when t = 0 you are at the point P and when t = 1 you are at the point Q.

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2 CHAPTER 1. CURVES

α as we have dened it is called the velocity vector of α . Note that for our line,α (t) = p + t(q − p), the velocity vector is our direction vector α (t) = q − p. We may seethis written as α(t) = p + tv .

By going through the usual denitions of the derivative, we can show that

α (t 0 ) = limt → t 0

α(t) − α(t 0 )t − t 0

,

which looks like our slope denition of the derivative.

Denition 1.2 Given a smooth curve α : I → R 3 , the tangent vector to α at α(t) isgiven by

dαdt

(t) = α (t).

The speed of α at α (t) is the norm, |α (t)| . The unit tangent vector to α is

T (t) =α (t)

|α (t)|.

If α (t) = α 1 (t), α 2 (t), α 3 (t) is a curve in R 3 , then its acceleration vector is given by

α (t) =d2 α 1

dt 2 (t),d2 α 2

dt 2 (t),d2 α 3

dt 2 (t) .

Proposition 1.1 The curve α is a line if and only if α = 0 .

Proof: If α (t) = p+ tv is a line, then α (t) = v and, hence, α (t) = 0 since v is a constant.

If α (t) = 0 for all t, thend2 α i

dt 2 = 0 , i = 1 , 2, 3.

This means that each component of the vector α is constant

dα i

dt= vi ,

and hence

α i (t) = pi + tv i

and

α(t) = ( p1 + tv 1 , p2 + tv 2 , p3 + tv 3 ) = p + tv .

Thus, the equation can be parametrized as a line.

What else can calculus tell us? We know that distance is the integral of speed, so thiswould imply that

L(α) = b

a|α (t)| dt

is the arclength of the curve α : [a, b] → R 3 from α(a) to α(b).





4 CHAPTER 1. CURVES

be the line segment above. To measure how much this is, we can measure the angle that αmakes with the vector in the direction of q − p. We will do this by taking the dot productof α with this direction vector. Let v = ( q − p)/ |q − p| be the unit vector in the direction

of q − p. The total deviation of α from the line segment can be added up by integration.Precisely, we need to compute

b

aα (t) · v dt,

to see why L(α) is larger than L( ).Note that

(α(t) · v ) = α (t) · v + α(t) · v = α (t) · v

because v is a constant. Thus, by the Fundamental Theorem of Calculus

b

aα (t) · v dt =

b

a(α(t) · v ) dt = α(b) · v − α(a) · v

= q · v − p · v = ( q − p) · v

=(q − p) · (q − p)

|q − p|= |q − p|

= L( )

Now, by Schwarz’s Inequality

b

aα (t) · v dt ≤

b

a|α (t)|| v | dt

= b

a|α (t)| dt

= L(α)

Thus, by these two, we have that L( ) ≤ L(α). Now, α (t) · v = |α (t)|| v | only whencos(θ) = 1 or θ = 0. Thus, α (t) is parallel to q − p for all values of t. In this case, α (t) isthe line from P to Q. Therefore the strict inequality holds, unless the curve is the line.

Other Curves

There are a lot of curves that we have and have not studied.

The circle of radius r α (t) = ( r cos(t), r sin(t))The cycloid α(t) = ( a(t − sin(t)) , a (1 − cos(t))

The astroid α(t) = ( a cos3 (t), a sin3 (t))

The Witch of Agnesi α (t) = (2 a tan( t), 2a cos2 (t))

A circular helix α(t) = ( a cos(t), a sin(t), bt)

Tractrix or pursuit curve α(t) = ( a sin(t), a ln(tan( t/ 2)) + a cos(t))

Additionally, we have the polar curves — the cardiod, the limacon of Pascal, the lem-niscate, the roses, the Archimedean spiral. We will look at many of these, and more.

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Curves Definitions and Examples