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TERM PAPER

MTH 151

CALCULUS-I

DATE OF ALLOTMENT:22 September2010DATE OF SUBMISSION:19 October2010

Topic :Radius of curvature, Arclength and Circle of curvature

Mr.Ratesh Kumar NameChayan ToshniwalDepartment of SectiA4005

Mathematics RolRA4005A02

Registration11006878

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No.Corse code 1258D

CONTENT

1 INTRODUCTION

2 DERIVATION AND DEFINATION

2.1 ARC LENGTH

2.2 TECHNICAL DEFINATION OF

CURVATURE

2.3 PARAMETRIC FORM OF

CURVATURE

2.4 POLAR FORM OF

CURVATURE

2.5 DEFINATION OF RADIUS OF

CURVATURE

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2.6 DEFINATION OF CIRCLE OF

CURVATURE

3 APPLICATION OF THIS

PHENOMENON IN DAILY LIFE

3.1 RADIUS OF CURVATURE ON

EARTH

3.2 RADIUS OF CURVATURE TO

DETECT MAXIMUM

DEVIATION BY TRAIN DUE TO

CURVED TRACK

INTRODUCTION

In mathematics curvature refers to any of a number of loosely related concept in different areas

of geometry. Curvature may be defined as the amount by which any geometric object deviate its

path from being flat, or being a straight line. In other words, it can be defined as The shape of

a curve depends very largely upon the rate at which the direction of the tangent changes as

the point of contact describes the curve. This rate of change of direction is called curvature

and is denoted by K. The curvature at any point is inversely proportional to the radius of anosculating circle. So, from the above definition it can be easily said that, straight line have no

curvature orzero curvature

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DERIVATION

Arc Length And Derivation

In this section we are going to concentrate at computing the arc length of a function.Because its easy enough to derive the formula that we will use in this section.

We want to determine the length of the continuous function

y = f(x)on interval [a,b].Initially we will need to find the length of the curve. We will do this by

dividing the interval up into n equal subintervals each of width x and each point

on a curve is denoted by P. We can then approximate the curve by a series

of straight lines connecting the points. The following figure will make it

clear.

Fig. 1

Now denote the length of each of this line segment by

and the length of curve will then be approximately,

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and we can get the exact length by taking n as larger as possible. In other sense, the exact length

will be,

First ,on each segment lets define . We can then compute

directly the length of the line segment as follow

By the Mean Value Theorem we know that on the interval there is a point so that,

Therefore ,the length can now be written as,

The exact length of the curve is then,

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However ,using the definition of the definite integral, this is nothing more than

A slightly more convenient notation is following.

The above is the relation for arc length

But instead of using this to find arc length, we need to make a small change in notation . Instesdof having two formula for the arc length of a function we are going to reduce it , in part, to a

single formula.From this point on we are going to use the following formula for the length of the curve:

Where,

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Technical definition of curvature

Consider any smooth curve. Curvature measure the rate at which the tangent line turns per unitdistance moved along the curve. Or, more simply, it measures the rate of turns per unit distance

moved along the curve. Or, more simply, it measure the rate of change of direction of curve.

Let P and P be two points on a curve, separated by an arc length .Then the curvature,of the

arc from P to P is expressed by the fraction

where, = '- is the angle turned through by the tangent line moving from P to P. The

curvature K at point P is defined as

To find ds/d

To compute d/dx first observe that tan =dy/dx ,so =arctan (dy/dx).Consequently,

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The relation is given by

So the formula for curvature is given by,

The sign of K will be positive if d2y/dx2 is positive and negative if it is negative.

Polar Form

Polar form of curvature is given by

Here, the prime now refers to differentiation with respect to .

Parametric FormPolar form of curvature is represented as;

Defination For Radius of curvatureThe radius of curvature for apoint P on a curve is defined as

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Here K is radius of curvature

Defination For Circle Of CurvatureLet R be the radius of curvature at a point on a curve at a point P on a curve .The circle ofcurvature or Osculating Circle of the curve at point P is the circle of radius R on the concave side

of the curve and tangent to it at P (fig.3).

To construct the circle of curvature, on the concave side of the curve construct the

Normal at p and on it lay off PC=R. The point C is the centre of the required circle.

The circle of curvature of a curve at a point P is that particular circle which has the sameCurvature as the curve itself at a point P. Of the indefinitely large number of circle that can be

drawn tangent to the curve at point P, this is the only one whose curvature is the same as that of

the curve at the point of contact.It can be shown that this circle fit the curve more closely in the

neighborhood of P than any other circle ,just as the tangent line fits it more closely than anyother line.

Another definition of the circle of curvature at point P is as follow;

Suppose we pass a circle through P and two arbitrarily selected near by P and P of the curve.

The limiting position of this circle as P and P both approach to P along the curve can be shown

to be identical with that of the circle of curvature as explained above

APPLICATION OF THIS PHENOMENON IN

DAILY LIFE

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Radius Of Curvature On EarthThis section gives the formulas needed to calculate the radius of curvature of the spheroid usedto approximate the surface of the Earth. The notation used in this section may or may not agree

with that used in the rest of the article.

Specifying the spheroid

Numerous spheroids have been used in the past to approximate the Earths surface ; each of them

is defined by two numbers .usually one is a,the distance from the centre to the spheroid to the

equator; defined by two numbers. Usually one is a, the distance from the centre of the spheroid

to the equator; the second may be b, the slightly smaller distance from the spheroid centre to the

pole. Or it may be the dimensionless number r expressing the difference between the two

spheroid dimensions

r =

(where, r is reciprocal of flattening)

For the WGS84 spheroid ,now commonly used ,a is set to be 6378137 meters exactly and r is set

to be 298.257223563 exactly (which makes b about 6356752.3142 meters).

Another dimensionless number is , the eccentricity squared of the spheroid

(Irrelevant aside; if we look at a cross-sectional ellipse containing the spheroids pole-to-pole

axis, the eccentricity e is the distance from the centre of the ellipse to a focus, divided by a, the

longer half-axis of the ellipse. In other words, if the eccentricity of an ellipse is 0.5,each focus is

halfway from the centre of the ellipse to its end.)

At any given point on the spheroid, a vertical plane is a plane containing the vertical live through

that point; we are going to pretend that the vertical line is perpendicular to the surface of the

spheroid at that point .

If we cross-sectional the spheroid with a vertical north-south plane ,the radius of curvature of the

resulting ellipse at latitude is

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The radius of curvature of the ellipse cross-sectioned by a

vertical east-west plane through a point at latitude

N =

(Irrelevant aside: N is also the distance from the point to the spheroids axis, measured along the

straight line that is vertical at the point.)

Another relation given by Euler gives the radius of curvature of the ellipse cross-sectioned by avertical plane in some direction other than north-south or east-west

R =

Where is the azimuth of the line at the point making angle of 90 with north, 90 with east .At

the pole M=N, but at any other point M is the minimum radius of curvatureof all the possible

vertical cross-sectiona through that point, while N is the maximum.

Radius Of Curvature To Detect Maximum Deviation

By Train Due To Curved Railway Track

The radius Of Curvature in railways detect how speedly the track is changing direction. It is the

radius of a circle that matches the particular section of track involved

This information is important for many reasons :It is used to calculate the maximum speed that atrain can have when entering the curve. Part of this is knowing rapidly the radius changes

usuallya curved section of track is gradually tightened up. So that, the left-right acceleration of the train

does not change suddenly.

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It is used to calculate the maximum deviation from centerline that a train will have going

through the curve ,due to the fact that each car has distance between wheels and the car will be a

chord on the circle of track that are adjacent to other curved tracks.

It also helps the engineer to design a railway track

We can draw acircle that closely fits nearby points on a local section of a curve ,as follow

Application

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When engineers design train tracks, they

need to ensure the curvature of the track will be safe and provide

a comfortable ride for the given speed of the trains.

The radius of curvature of the curve is defined as the radius of the

approximatingcircle. This radius changes as we shift along the curve. How do

we find this variable radius curvature?

The formula for the radius of curvature at any point x for the curve y = f(x)

is given by:

Radius of curvature =