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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 =
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