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CURVATURE OF A CURVE

TERM PAPERMTH 151

CALCULUS- II

TOPIC- RADIUS OF CURVATURE, LENGTH OF ARC ANDCIRCLE OF CURVATURE

DOA:

DOR:

DOS:

MR.RATNESH KUMAR MR. ASHISH MITTAL

DEPARTMENT OF MATHEMATICS

ROLL. NO- RA4005A20

CLASS- A4005

REG. NO- 11001449COURSE CODE- 1258D


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CURVATURE OF A CURVECurvature of any general curve

For any general plane curve C, the curvature at a given point P has a magnitude

equal to the reciprocal of the radius of an osculating circle (a circle that closely

touches the curve at the given point P, its center shaping the curve's evolute), and

is a vector pointing in the direction of that circle's center. The smaller the radius r

of the osculating circle, the larger the magnitude of the curvature (1/r) will be; so

that where a curve is "nearly straight," the curvature will be close to zero, and

where the curve undergoes a tight turn, the curvature will be large in magnitude.
http://www.newworldencyclopedia.org/entry/Image:Osculating_circle.svghttp://www.newworldencyclopedia.org/entry/Image:Osculating_circle.svghttp://www.newworldencyclopedia.org/entry/Image:Osculating_circle.svghttp://www.newworldencyclopedia.org/entry/Image:Osculating_circle.svg


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CURVATURE OF A CURVEThe magnitude of curvature at points on physical curves can be measured in

diopters (also spelled dioptre); a diopter has the dimension length-1

A straight line has curvature 0 everywhere; a circle of radius r has curvature 1/r

everywhere.

MATHEMATICAL REPRESENTATION OF CURVATURE

For any function y= f(x) representing any general curve

=

) ))

Meanwhile the radius of the curve

p =

= =

))

= )

))

.


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CURVATURE OF A CURVEGENERAL MEANING OF CURVATURE

If we suppose that the particle moves in any arbitrary plane with a unit

velocity, then the locus of the particle will trace out a curve. Moreover, taking

the time as the parameter, this provides a natural parametrization for C.At any

motion and the curvature measures how fast this vector rotates. If a curve keeps

close to the same direction, the unit tangent vector changes very little and the

curvature is small; where the curve undergoes a tight turn, the curvature is

large.

If we consider any two points on the circle P & N and the tangents at the points

P & Q makes an angle Q and Q with the X-axis, then the angle b/w the two

tangents is Q-Q= dq and let the distance b/w two points be ds.

Then the rate of bending of the curve or the curvature can be

expressed as

LENGTH OF THE ARC OF A CURVE


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CURVATURE OF A CURVEFor any curve ,the length of the locus of the tells us the length of arc of the

curve by using line segments to create a polygonal path. Now summing over

the length of such line segment over the entire curve will give us theapproximation of the length of the curve.

LENGTH OF ANY GENERAL CURVE

LetC be acurveinEuclideanspace X =Rn, such thatC is theimageof a

continuous function f: [a, b] X of theinterval[a, b] into X.

)) ) ) ) We calculate the distance from (to, f(to)) to, (to+i ,f(to+i )) denote it by d which is

the length of the segment of the curve.

LENGTH OF THE SMALL SEGMENT =

) ) ))

So length of each segment = ) ) = dxNow integrating over the whole length of the curve ,we get the length of curve as

If a function is defined by parametric coordinates (r,)

Then length of the arc is )

i
http://d/wiki/Curvehttp://d/wiki/Curvehttp://d/wiki/Curvehttp://d/wiki/Euclidean_spacehttp://d/wiki/Euclidean_spacehttp://d/wiki/Euclidean_spacehttp://d/wiki/Image_(mathematics)http://d/wiki/Image_(mathematics)http://d/wiki/Image_(mathematics)http://d/wiki/Continuous_functionhttp://d/wiki/Continuous_functionhttp://d/wiki/Interval_(mathematics)http://d/wiki/Interval_(mathematics)http://d/wiki/Interval_(mathematics)http://d/wiki/Interval_(mathematics)http://d/wiki/Continuous_functionhttp://d/wiki/Image_(mathematics)http://d/wiki/Euclidean_spacehttp://d/wiki/Curve


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Approximation by multiple linear segments

A curve can be approximated by connecting the finite number of points on the

curve by using line segments to create a polygonal path.Since it is easy to find the

length of each segment of the curve, then the approximated length of curves can be

predicted out by summing the total length of the line segments

If the curve is not already a polygonal path, better approximations to the curve can

be obtained by following the shape of the curve increasingly more closely. The

approach is of use when there are larger number of segments of smaller lengths.

As the length of the successive line segments gets arbitrarily small, then the

summated length of the line segments over the curve will approach to the length of

the smooth curve.

For some curves there is a smallest number L that is an upper bound on the length

of any polygonal approximation. If such a number exists, then the curve is said to

berectifiable and the curve is defined to havearc length L.

Let C be acurveinEuclidean(or, more generally, ametric) space X =Rn, so C is

theimageof acontinuous functionf: [a, b] X of theinterval[a, b] into X.
http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/File:Arclength.svghttp://en.wikipedia.org/wiki/Interval_(mathematics)http://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Curve


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CURVATURE OF A CURVEFrom apartitiona = t0 < t1 < ... < tn1 < tn = b of the interval [a, b] we obtain a

finite collection of points f(t0), f(t1), ..., f(tn1), f(tn) on the curve C. Denote the

distancefrom f(ti) to f(ti+1) by d(f(ti), f(ti+1)), which is the length of theline segmentconnecting the two points.

.

The arc length L is eitherfiniteorinfinite. If L < then we say that C is

rectifiable, and isnon-rectifiable otherwise. This definition of arc length does notrequire that C is defined by adifferentiablefunction.

To find arc length through integration

Consider a realfunctionf(x) such that f(x):AB where A{R}.

Consider an infinitesimal part of the curve be ds. According to Pythagoras'

theorem ds2

= dx2

+ dy2, from which:

ds2 = dx2 + dy2
http://en.wikipedia.org/wiki/Partition_of_an_intervalhttp://en.wikipedia.org/wiki/Partition_of_an_intervalhttp://en.wikipedia.org/wiki/Partition_of_an_intervalhttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Finitehttp://en.wikipedia.org/wiki/Finitehttp://en.wikipedia.org/wiki/Finitehttp://en.wikipedia.org/wiki/Infinitehttp://en.wikipedia.org/wiki/Infinitehttp://en.wikipedia.org/wiki/Infinitehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/File:Booglengte.PNGhttp://en.wikipedia.org/wiki/File:Booglengte.PNGhttp://en.wikipedia.org/wiki/File:Booglengte.PNGhttp://en.wikipedia.org/wiki/File:Booglengte.PNGhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Infinitehttp://en.wikipedia.org/wiki/Finitehttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Partition_of_an_interval


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If a curve is defined parametrically by x = X(t) and y = Y(t), then its arc between t = a and t = b is

S =

)

)

S along f between two points a and b in that curve, construct a series of right

triangles whose concatenated hypotenuses "cover" the arc of curve chosen as

shown in the figure. For convenience, the bases of all those triangles can be set

equal to x, so that for each one an associated y exists. The length of any givenhypotenuse is given by thePythagorean Theorem:

The summation of the lengths of the n hypotenuses approximates S:

Multiplying the radicand by produces:

Then, our previous result becomes:
http://en.wikipedia.org/wiki/Pythagorean_Theoremhttp://en.wikipedia.org/wiki/Pythagorean_Theoremhttp://en.wikipedia.org/wiki/Pythagorean_Theoremhttp://en.wikipedia.org/wiki/Pythagorean_Theorem


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CURVATURE OF A CURVEAs the length x of these segments decreases, the approximation improves. The

limit of the approximation, as x goes to zero, is equal to S:

APPLICATION OF THESE CONCEPTSIN REAL LIFE

A real-world example of the radius of curvature is when we are

driving along a curved road. We are forced to hold the steering

wheel in a certain position. If at Point A on the road we were to

keep the steering wheel in a fixed position, the car wouldtravel

in a circle. That circle is the curvature of the function at Point A,

and the radius of that circle is the radius of curvature of that

function.
http://www.ehow.com/travel/http://www.ehow.com/travel/


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The radius of curvature in railways signifies how fast the track is

changing direction. It is the radius of a circle that matches the

particular section of track involved.

This is important in calculating the maximum speed that a train can

have while entering the curve. Knowing how rapidly the radius changes,

a 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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CURVATURE OF A CURVEIt is used to calculate the maximum deviation from centre that a train

will have going through the curve. This is due to the fact that each carhas a set distance between wheels and the car will be a chord on the

circle of track. This has application in positioning platforms in relation

to tracks, and in positioning curved tracks that are adjacent to other

curved tracks

.


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