Arc Length and Curvature

Differential geometry(MODULE III)

arc length and curvature

PREPARED :RASHMI.M

CONTENT EDITOR :NANDAKUMAR.M

PRESENTER :RASHMI.M

Objectives

From this unit a learner is expected to achieve the following

Learns about arclength
Familiarizes with unit tangent vector
Familiarizes with curvature of a curve
Understands principal unit normal vector for a curve in the plane

Arc length Along a Curve

One of the special features of smooth space curves is that they have a measurable length. This enable us to locate points along these curves by giving their directed distance along the curve from some base point, the way we locate points on coordinate axes by giving their directed distance from the origin (Fig.1). Time is the natural parameter for describing a moving body’s velocity and acceleration, but is the natural parameter for studying a curve’s shape. Both parameters appear in analyses of space flight.

To measure distance along a smooth curve in space, we add a z-term to the formula we use for curves in the plane.

Definition

The length of a smooth curve that is traced exactly once as t increases from is

…(1)

We usually take then

…(2)

Just as for plane curves, we can calculate the length of a curve in space from any convenient parametrization that meets the stated conditions. The square root in either or both of Eqs.(1) and (2) is , the length of the velocity vector . Hence we have the Length Formula (Short Form)

…(3)

Example 1 Find the length of one turn of the helix

Solution

The helix makes one full turn as t runs from 0 to 2(Fig.2).

Using the Length Formula (short form), the length of this portion of this curve is

This is times the length of the circle in the xy-plane over which the helix stands.

If we choose a base point on a smooth curve C parametrized by t, each value of t determines a point on C and a “directed distance”

…(4)

measured along C from the base point (Fig.3). If is the distance from to . If is the negative of the distance. Each value of s determines a point on C and this parametrizes C with respect to s. We call s an arc length parameter for the curve. The parameter’s value increases in the direction of increasing t.

Arc length parameter with base point is given by

…(5)

Example 2If the arc length parameter along the helix

from is Using Eq. (4)

Thus, , and so on (Fig.4).

Example 3 (Distance along a Line) Show that if is a unit vector, then the directed distance along the line

from the point where is t itself.

Solution

So (noting that, being unit vector, )

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Speed on a Smooth Curve

Since the derivatives beneath the radical in Eq.(5) are continuous (the curve is smooth), the Fundamental Theorem of Calculus tells us that s is a differentiable function of t with derivative

.…(6)

As we expect, the speed with which the particle moves along its path is the magnitude of v.

Notice that while the base point plays a role in defining in Eq. (5), it plays no role in Eq.(6). The rate at which a moving particle covers distance along its path has nothing to do with how far away the base point is.

Notice also that since, by definition, is never zero for a smooth curve. We see once again that s is an increasing function of t.

The Unit Tangent Vector T

Since for the curves we are considering, s is one-to-one and has an inverse that gives t as a differentiable function of s. The derivative of the inverse is

…(7)

This makes r a differentiable function of s whose derivative can be calculated with the Chain Rule to be

…(8)

Equation (8) says that is a unit vector in the direction of v. We call the unit tangent vector of the curve traced by r and denote it by T (Fig.5).

Definition The unit tangent vector of a differentiable curve is

…(9)

The unit tangent vector T is a differentiable function of t whenever v is a differentiable function of t. As we will see in the next chapter T is one of three unit vectors in a traveling reference frame that is used to describe the motion of space vehicles and other bodies moving in three dimensions.

Example 4 Find the unit tangent vector of the helix

Solution

Example 5 Find the unit tangent vector to the curve

at the point t =1.

Solution

The position vector of a point on the curve is given by

Then

Hence

Therefore,

Therefore at the unit tangent vector is

Example 6 Find the unit tangent vector at a point t to the curve

Solution

Hence

Example 7(The involute of a circle) (Fig. 6)

Find the unit tangent vector of the curve

Solution

Definition The involute of a circle is the path traced by the endpoint P of a string unwinding from a circle. In the above example (Fig.6) it is the unit circle in the xy-plane.

Example 8 For the counterclockwise motion

around the unit circle,

is already a unit vector, so . (Fig. 7)

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CURVATURE

A frame of mutually orthogonal unit vectors that always travels with a body moving along a curve in space (Fig.8). The frame has three vectors. The first is T, the unit tangent vector. The second is N, the unit vector that gives the direction of. The third is. These vectors and their derivatives, when available, give useful information about a vehicle’s orientation in space and about how the vehicle’s path turns and twists.

For example, tells how much a vehicle’s path turns to the left or right as it moves along; it is called the curvature of the vehicle’s path. The numbertells how much a vehicle’s path rotates or twists out of its plane of motion as the vehicle moves along; it is called the torsion of the vehicle’s path. Look at Fig.8again. If P is a train climbing up a curved track, the rate at which the headlight turns from side to side per unit distance is the curvature of the track. The rate at which the engine tends to twist out of the plane formed by T and N is the torsion.

Every moving body travels with a TNB frame that characterizes the geometry of its path of motion.

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The Curvature of a Plane Curve

As a particle moves along a smooth curve in the plane, turns as the curve bends. Since T is a unit vector, its length remains constant and only its direction changes as the particle moves along the curve. The rate at which T turns per unit of length along the curve is called the curvature (Fig.9). The traditional symbol for the curvature function is the Greek letter (“Kappa”).

Definition

If T is the unit tangent vector of smooth curve, the curvature function of the curve is

If is large, T turns sharply as the particle passes through P, and the curvature at P is large. If is close to zero, T turns more slowly and the curvature at P is smaller. Testing the definition, we see in the following Examples 9 and 10 that the curvature is constant for straight lines and circles.

Example9 (The curvature of a straight line is zero)

On a straight line, the unit tangent vector T always points in the same direction, so its components are constants. Therefore. (Fig. 10).

Example10 (The curvature of a circle (Fig.11) of radius a is 1/a)

The parametrization for a circle having radius a is

and substitute to parametrize in terms of arc length s. (Note that if the radius of the circle is , and is the angle between two rays emanating from the centre, then the length of the arc of the circle included between the rays is given by )

Then

and

Hence, for any value of s,

The Principal Unit Normal Vector for Plane Curves

Since T has constant length, the vector is orthogonal to T. This conclusion is using the result “If u is a differentiable vector function of t of constant length, then ”

Therefore, if we divide by the length , we obtain a unit vector orthogonal to T (Fig.11) and is given in the following definition.

Definition

At a point where the principal unit normal vector for a curve in the plane is

The vector points in the direction in which T turns as the curve bends. Therefore, if we face in the direction of increasing arc length, the vector points toward the right if T turns clockwise and toward the left if T turns counterclockwise. In otherwords, the principal normal vector N will point toward the concave side of the curve (Fig.11).

Because the arc length parameter for a smooth curve is defined with positive, and the Chain Rule gives

… (10)

This formula enables us to find N without having to find and s first.

Example 11Find T and N for the circular motion

SolutionWe first find T:

From this we find

and

Summary

In this unit we discussed the idea of length of smooth

Curves and speed on a smooth curve also discussed the

Concept of curvature which is very useful in the

Coming sessions

Glossary

involute:The involute of a circle is the path traced by the endpoint P of a string unwinding from a circle.

Curvature:The rate at which unit tangent vector T turns per unit of length along the curve is called the curvature

curvature function If T is the unit tangent vector of smooth curve, the curvature function of the curve is

FAQ

1.Is there any relation between rate of a moving particle and the distance to the base point

Answer:The rate at which a moving particle covers distance along its path has nothing to do with how far away the base point is.

2.Does unit tangent vector has any practical application

Yes ofcourse,here we have mentioned that in a traveling reference frameto describe the motion of space vehicles and other bodies moving in three dimensions the idea of unit tangent vectors is used.

Quiz

Consider any smooth curve ,for a smooth curve is

(a)Is always a positive number

(b)Is always a negative number

(c)Is always zero

(d)Is never zero

2.The curvature of a circle of diameter 5 is

(a) 5

(b) 2.5

(d) 1\5

Answers

1.(d) is never zero

2.(c) 1\2.5

Assissnments

1. Find the point on the curve

at a distance units along the curve from the origin in the direction of increasing arc length.

2. Find the length of the curve

from.

3. a) Show that the curve , is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Find equations for the cylinder and plane

b) Write an integral for the length of the ellipse (Evaluation of the integral is not required, as it is nonelementary).