Orthogonal Trajectories

Introduction:

ORTHOGONAL TRAJECTORIES In 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.

ORTHOGONAL TRAJECTORIES IN 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.

Examples of Orthogonal Trajectories:

Cartesian Coordinates:

Example: Find the equation of the orthogonal trajectories of the

family of curves

c1x2 – y2 = 1

Step1:

2c1x – 2y dy/dx = 0 c1x2 – y2 = 1

2y dy/dx = 2c1x c1x2 = y2 +1

Dy/dx = c1 x/y c1 = y2+1/x2

Dy/dx = (y2+1/x2)(x/y)

f(x,y) = 1+y2/xy

Step2:

Diff. Eq of the Orthogonal Trajectories is

Dy/dx = - 1/f(x,y) >>> dy/dx = - xy/1+y2

Step3:

Solving the diff. Eq by method of Separation Variables

[1+y2/y] dy = -xdx

[1/y+y] dy = -xdx

End result;

2ln |y| + y2 + x2 = c

Example 2: 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

y

x

dx

dy

The orthogonal family will have the slope

x

y

dx

dy

y

x1

dx

dy

Re arranging and integrating both sides

lnylnclnx

dyy

1dx

x

1

Term Report on:

Orthogonal Trajectories

Submitted to:

Madam Naila

Submitted by:

Ghulam Murtaza Roll No # 101

Muhammad Waleed Hussain Roll No # 102

Muhammad Junaid Jamil Roll No # 103

Muhammad Ammar Mohsin Roll No # 104

Muhammad Usama Iqbal Roll No # 105

Hafiz Muhammad Zia Roll No # 106

Class: BSc Chemical Engineering 2nd year

Section: B

Session: 2012-16

Date: Wednesday 15th January; 2014

NFC Institute of Engineering & Technology, Multan