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4 Solved Questions on Parametric Equations - Project | MATH 1511, Study Guides, Projects, Research of Calculus

Material Type: Project; Professor: Wachsmuth; Class: Honors Calculus II; Subject: Mathematics; University: Seton Hall University; Term: Spring 2005;

Typology: Study Guides, Projects, Research

Pre 2010

Uploaded on 08/08/2009

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MATH 1511 – Final Project
Instructions: You must type all answers to the following questions into a Maple worksheet (which should also
contain your name). You must send me that Maple worksheet as an attachment to an email message to me
(wachsmut@shu.edu) on or before May 6, 2005. You are required to complete this assignment on your own, but
you may use Maple, and/or any type of calculator, and/or any Calc book you like.
It is perfectly okay to use Maple to help you with any calculation, but if you rather do the calculations manually, that
would be fine as well as long as you type up sufficient explanations in your Maple worksheet.
To answer the various “conjectures”, type as much as you feel is necessary to convey your ideas. Support your
conjectures with Maple plots or computations, if appropriate.
I know it is exceedingly easy to “copy” assignments like this – any plagiarism or cheating I detect it will result in 0
points for every project involved and any further action that seems appropriate. Note that all projects differ slightly.
If you have any questions, please send me electronic mail as soon as possible.
Question 1: Below are four parametric curves, and four named parametric equations. Which
curve belongs to which equation?
Lissajous curve:
)cos(4)( ttx
)2sin(2)( tty
Evolute of ellipse:
)(cos)(
3
ttx
)(sin2)(
3
tty
Involute of circle:
)sin()cos()( ttttx
)cos()sin()( tttty
Serpentine curve:
)cot()( ttx
)cos()sin(4)( ttty
Conjecture: Describe how the Lissajous curves
,
)sin(2)( ntty
will look for
different values of n, where n is a positive integer.
Question 2: In Greek, the word cycloid means “wheel”, the word hypocycloid means “under the
wheel”, and the word epicycloid means “upon the wheel”. You can obtain an epicycloids, for
example, by tracing a point on a small circle as it rolls around the outside circumference of a
larger circle.
A Hypocycloid H(A, B) has the following parametric equation:
pf3

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MATH 1511 – Final Project

Instructions: You must type all answers to the following questions into a Maple worksheet (which should also contain your name). You must send me that Maple worksheet as an attachment to an email message to me (wachsmut@shu.edu) on or before May 6, 2005. You are required to complete this assignment on your own, but you may use Maple, and/or any type of calculator, and/or any Calc book you like. It is perfectly okay to use Maple to help you with any calculation, but if you rather do the calculations manually, that would be fine as well as long as you type up sufficient explanations in your Maple worksheet. To answer the various “conjectures”, type as much as you feel is necessary to convey your ideas. Support your conjectures with Maple plots or computations, if appropriate. I know it is exceedingly easy to “copy” assignments like this – any plagiarism or cheating I detect it will result in 0 points for every project involved and any further action that seems appropriate. Note that all projects differ slightly. If you have any questions, please send me electronic mail as soon as possible.

Question 1 : Below are four parametric curves, and four named parametric equations. Which

curve belongs to which equation?

 Lissajous curve:

x ( t ) 4 cos( t ) y ( t ) 2 sin( 2 t )

 Evolute of ellipse :

x ( t )cos^3 ( t ) y ( t ) 2 sin^3 ( t )

 Involute of circle :

x ( t )cos( t ) t sin( t ) y ( t )sin( t ) t cos( t )

 Serpentine curve:

x ( t )cot( t ) y ( t ) 4 sin( t )cos( t )

Conjecture : Describe how the Lissajous curves x (^^ t )^4 cos( t ), y^ (^ t )^2 sin( nt )will look for

different values of n , where n is a positive integer.

Question 2 : In Greek, the word cycloid means “wheel”, the word hypocycloid means “under the

wheel”, and the word epicycloid means “upon the wheel”. You can obtain an epicycloids, for

example, by tracing a point on a small circle as it rolls around the outside circumference of a

larger circle.

A Hypocycloid H(A, B) has the following parametric equation:

( ) ( )cos() cos( t ) B

A B

x t A B t B

( ) ( )sin() sin( t ) B

A B

y t A B t B

An Epicycloid E(A, B) has the following parametric equation:

( ) ( )cos() cos( t ) B

A B

x t A B t B

( ) ( )sin() sin( t ) B

A B

y t A B t B

Use Maple to create the following four graphs:

 H(10,3) (i.e. a hypocycloid where A = 10 and B = 3)

 H(29, 9) (i.e. a hypocycloid where A = 29 and B = 9)

 E(10, 3) (i.e. an epicycloids where A = 10 and B = 3)

 E(29, 9) (i.e. an epicycloids where A = 29 and B = 9)

Note that the parameter t must be in a reasonably large interval, like [^0 ,^100 ], to give good

pictures.

Conjecture A : Consider a hypocycloid H(A, B) and set B = 3. Describe what the impact of the

parameter A is, as A is 3, 5, 7, 11, 13, etc.

Conjecture B: Consider a hypocycloid H(A, B) and set A = 13. Describe what the impact of the

parameter B is, as B is 1, 2, 3, …, 13. What will happen if B becomes larger than A?

Question 3 : The following functions are given in polar coordinates where r denotes the radius

and t denotes the angle.

 Limacons : r^ ^ a  b cos( t )or r^  a  b sin( t )

 Rose curves : r^  a^ cos( nt ) or r^  a sin( nt )

Use Maple to draw three Limacons using the cos function, where

 a = 2 and b = 3

 a = 3 and b = 3

 a = 3 and b = 2

Use Maple to draw three Rose curves using the cos function, where

 a = 2 and n = 3

 a = 2 and n = 5

 a = 2 and n = 4