orthogonal trajectories in cartesian coordinates
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ORTHOGONAL TRAJECTORIESIN CARTESIAN COORDINATES
Submitted by:
RAFAE ASHFAQ
08-EE-40
ORTHOGONAL TRAJECTORIESIN CARTESIAN COORDINATES
Submitted by:
SHEHZEB ASHRAF
08-TE-24
CONTENTS
• This presentation includes the following– INTRODUCTION TO ORTHOGONAL
TRAJECTORIES.– FINDING ORTHOGONAL TRAJECTORIES– APPLICATIONS– EXAMPLES
INTRODUCTION
ORTHOGONAL TRAJECTORIESIn dictionary
ORTHOGONAL: Derived from greek word “Ortogonios” meaning right angled. TRAJECTORY: Derived from the latin word “Trajectoria” meaning cut across.
So, orthogonal trajectories means cutting something across the right angle.
CARTESIAN COORDINATES locate a point in terms of its perpendicular distance from (two or three) mutually perpendicular axes
ORTHOGONAL TRAJECTORIESIN GEOMETRY
• In geometry, orthogonal means perpendicular and trajectories mean curves or surfaces cutting another family of curves or surfaces at a constant angle. As we are relating it with the word “orthogonal” so “ORTHOGONAL TRAJECTORIES” means curves or surfaces intersecting a family of curves or surfaces at right angle where the angle is defined as the angle between the tangents of the two curves.
ORTHOGONAL FAMILIES
Similarly two families of curves will be orthogonal if each curve in one family intersect the other family at right angle. In other words, two families
F(x, y, c) = 0 and G(x, y,K) = 0
are orthogonal families if each member
of one family is an orthogonal trajectory of
the other family.
FINDING
ORTHOGONAL TRAJECTORIES
METHODS TO FIND ORTHOGONAL TRAJECTORIES
There are two ways in which we can find the differential equations to obtain orthogonal trajectories.
Before going into the detail of these methods we should be familiar with some of the useful concepts which are related to these concepts.
DERIVATIVE AS THE SLOPE OF THE CURVE
• Let AB be the arc of the graph of f defined by the equation y=f(x).
• Let P(x,f(x)) and Q(x+δx,f(x+δx))
be two neighbouring points on
the arc AB. The line PQ is
secant of the curve and it
makes <α with the x-axis.
Drawing the ordinates of PM,
QN and the perpendicular
PR to NQ, we have
O
y
M NS x
A R
B
P(x,f(x))
Q(x+δx),f(x+δx))
and
RQ = NQ –NR =NQ –MP= f(x+δx) – f(x)
PR = MN= ON –OM = x+δx-x = δx
Thus, tan m <α =tan m <RPQ
Revolving the secant line
PQ about P towards P,
some of its successive
positions PQ1, PQ2 , PQ3 , . . .
are shown in the figure.
δx
f(x)δx)f(x
PR
RQ
O
y
MS x
Q
P
Q1
Q2
Q3
R3
T
Points Qi (i=1,2,3,…) are getting closer and closer to the point P and PRi i.e.,δ xi (i=1,2,3,…) are approaching to zero, that is,
Thus, the slope of the tangent line to the graph of f at (x,f(x)) is f’(x) or dy/dx.
XTPmtan(x)f'or
XTPmtanδx
f(x)δx)f(xlim
0δxasXTPmtanδx
f(x)δx)f(xor
0δxwhenXTPmtanαmtan
0δx
SLOPE OF A CURVE PERPENDICULAR TO ANOTHER CURVE
• Consider two curves C1 and C2 which intersect each other at an angle α
at an arbitrary point P.
In other words the
angle between their
tangents at P will be
equal to angle α.φ
C1
C2
Ψ
α
P
x
y
O
• To relate it to orthogonal trajectories, we consider that C2 is perpendicular to C1. So,
angle α =90° or π/2 rad
The angle between x-axis and the tangent of C1 at P is Φ and the angle between x-axis and the tangent of C2 at P is Ψ. So,
angle α = Ψ – Φ
According to the basic trigonometric identity
φΨ
φΨ
φ)(ψ
πα
tantan
tantan
tan
tantan
1
2
As α = π/2, so tan α will be infinite, but the numerator of the above equation is finite. So the above equation will be infinite if and only if
1+ tanΨ tanΦ = 0
Showing that )(1tanψ
1Φtan
• Now, if dy/dx is the slope of the curve C1 at P,
dy/dx = tanΦ
and if dyo/dx is the slope of its orthogonal trajectory at P,
dyo/dx = tanΨ
Thus from eq 1, we see that these two slopes are related by the expression
dxdy
1dx
dy
0
FINDING THE ORTHOGONAL TRAJECTORIES
An important application of first order DE’s is to find the orthogonal trajectories of a family of curves F1, given by the equation:
f(x, y,C) = 0
This can be done by two ways1. By using differential 2. By differentiating with respect to x.
BY USING THE DIFFERENTIAL
Suppose we have to find the orthogonal trajectories of a family of curves F1, given by the equation:
f(x, y,C) = 0 ….. (1)To find the orthogonal trajectories, we may derive the differential equations, whose solutions are described by these trajectories. For this, we proceed as follows
The differential of eq. 1 will be dy
y
fdx
x
fdf
• Solving for dy/dx, we have
If F2 is the family of curves perpendicular to F1,which
is given by
f(x0, y0,C0) = 0
then as we have proved earlier, its slope will be related to the slope of F1 as below,
)(2
δyδfδxδf
dx
dy
• Putting the value of dy/dx from eq. 2 in the above equation
or
dxdy
1dx
dy
0
0
δyδfδxδf
1dx
dy
)(3x
y
f
f
δxδfδyδf
dx
dy
• By integrating eq.3, we can get the required equation of the family F2 which is orthogonal to the family F1.
BY DIFFERENTIATING WITH RESPECT TO x
Suppose we have to find the orthogonal trajectories of a family of curves, given by the equation:
f(x, y,c) = 0 ….. (1)To find the orthogonal trajectories, we would have to find the differential equations, and by integrating them we shall find the equations described by these trajectories. For this, we proceed as follows,
Differentiating both sides with respect to x, we shall get the constant canceled, terms
• Containing y will give dy/dx. Solving for dy/dx we shall get a function in c and y variables with no constant. If constant is not eliminated yet, then by solving the equation.1 for c and then putting the value of c in dy/dx will give us the derivative or slope of the given family.
• Now, as we have proved, the family of curves passing perpendicularly through the given family of curves will have slope or derivative related to the given family will be
• By integrating this equation, we can get the required family of curves.
dxdy
1
dX
dY
EXAMPLE• Find the orthogonal trajectory of the
following:
x2 + y2 = c2
SOLUTION
We have the family
x2 + y2 = c2
Differentiating both sides with respect to x, we have
2x + 2y dy/dx = 0
The orthogonal family will have the slope
Re arranging and integrating both sides
y
x
dx
dy
x
y
dx
dy
yx
1dx
dy
lnylnclnx
dyy
1dx
x
1
which is the required
family of lines. These
two families can be
sketched as shown in
the figure.
cxy
APPLICATIONS
APPLICATIONS• Orthogonal trajectories has vast applications in
Physics, Engineering and many other fields. Some of them are– Heat conduction– Fluid dynamics/hydrodynamics – Electromagnetic theory– Electrostatics– Gravitation– Elasticity– Meteorology (Weather mapping)– Geometric optics
HEAT CONDUCTION
• Orthogonal trajectories arise in thermodynamics as well. In heat conduction, isotherms are orthogonal to the heat flow.
Fluid dynamics/hydrodynamics
• Orthogonal trajectories arise in fluid dynamics as well. In two dimensional fluid dynamics, orthogonal trajectories express the relationship between the curves followed by the fluid particles called streamlines, and the associated equipotential lines along which a function called the fluid potential is constant.
GRAVITATION
• Orthogonal trajectories also occur in gravitational study. If the original family represent the lines of force in a gravitational field, its orthogonal trajectories represent the equipotentials which are the curves along which gravitational potential is constant.
METEOROLOGY
• As another example, consider a weather map. The curves represent the isobars, which are curves connecting all cities that report the same barometric pressure at the weather bureau.
WORLD GLOBE
• Meridians and parallels on the world globe are orthogonal trajectories of each other.
ELASTICITY
• Orthogonal trajectories arise in elasticity as well. When stress is applied in a body, strain is produced. It is seen that the lines of stress are orthogonal to the lines of constant strain.
Geometric Optics• In geometric optics wave fronts are orthogonal
trajectories of the rays.Light rays
Wave fronts
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