Transcript 1.7 - Parametric Equations

Parametric Equations
Cartesian Equations
Equations defined in terms of x and y.
These may or may not be functions. Some
examples include:
x2 + y2 = 4
y = x2 + 3x + 2
Parametric Equations
Equations where x and y are functions of
a third variable, such as t. That is,
x = f(t) and y = g(t).
The graph of parametric equations are
called parametric curves and are defined
by (x, y) = (f(t), g(t)).
Example
The path of a particle in two-dimensional space
can be modeled by the parametric equations:
x = 2 + cos t and y = 3 + sin t. Sketch a graph of
the path of the particle for 0  t  2.
t
0.00
0.52
1.05
1.57
2.09
2.62
3.14
3.67
4.19
4.71
5.24
5.76
6.28
x = 2 + cos(t)
3.00
2.87
2.50
2.00
1.50
1.13
1.00
1.13
1.50
2.00
2.50
2.87
3.00
y = 3 + sin (t)
3.00
3.50
3.87
4.00
3.87
3.50
3.00
2.50
2.13
2.00
2.13
2.50
3.00
Plot of x = 2 + cos t and y = 3 + sin t
4.50
How is t
represented
on this
graph?
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
Plot of x = 2 + cos t and y = 3 + sin t
t=
t=0
Parametric Equations and Technology
Graphing calculators and other mathematical
software can plot parametric equations much
more efficiently then we can. Put your graphing
calculator and plot the following equations. In
what direction is t increasing?
(a) x = t2, y = t3
(b) x  ln t, y  t ; t  1
(c) x = sec θ, y = tan θ; -/2 < θ < /2
Converting Equations
Parametric equations can easily be converted to
Cartesian equations by solving one of the equations
for t and substituting the result into the other
equation.
x  ln t, y  t for t  1
y  t t  y
x  ln t  ln y
2
2
x  ln y  y  e  y   e  y  e ; x  0
2
2
x
x
x
You Try It
You try it for x = sec θ, y = tan θ
where -/2 < θ < /2
Hint: sec2 θ – tan2 θ = 1
Cycloids
A cycloid is the graph of the path of a
fixed point P on a circle of radius r that
rolls along a straight line.
x = r( – sin )
y = r(1 – cos )
An ant is walking along...only to be crushed by a
rolling wheel.
Question: What is the path traced out by its
bloody splat?
Why would we ask such a question?
Mathematicians are sick !!!
Problem Posed Again
(in a less gruesome manner)
A wheel with a radius of r feet is marked at
its base with a piece of tape. Then we allow the
wheel to roll across a flat surface.
a) What is the path traced out by the tape
as the wheel rolls?
b) Can the location of the tape be determined at
any particular time?
Questions:
•What is your prediction for the
shape of the curve?
•Is the curve bounded?
•Does the curve repeat a pattern?
Picture of the Problem
Finding an Equation
•f(x) = y may not be good enough to express the
curve.
•Instead, try to express the location of a point, (x,y),
in terms of a third parameter to get a pair of
parametric equations.
•Use the properties of the wheel to our advantage.
The wheel is a circle, and points on a circle can be
measured using angles.
WARNING: Trigonometry ahead!
Diagram of the Problem
2r
r
C

P
Q
r
O
X
T
We would like to
find the lengths
of OX and PX,
since these are
the horizontal and
vertical distances
of P from the
origin.
The Parametric Equations
r
C
r

r cos
P
|OX| = |OT| - |XT|
= |OT| - |PQ|
x() = r - r sin
Q
r sin
|PX| = |CT| - |CQ|
y() = r - r cos
r
O
T
X
r
Graph of the Function
If the radius r=1,
then the parametric equations become:
x()=-sin, y()=1-cos
Real-World Example:
Gears