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Parametric Equations. Example: x t, y t 2, 2 t 2

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Consider an object moving in two-dimensional space (the plane) whose position at time t has coordinates (x(t), y(t)). The equations x = x(t), y = y(t) are called parametric equations -- t in this case is the parameter. A plot showing these coordinates as time varies will trace the path of the object.

Example: x = t, y = t 2 , -2 £ t £ 2

ParametricPlot@8t, t^2<, 8t, -2, 2<, AxesLabel ® 8"x", "y"<D

-2 -1 1 2 x

1 2 3 4 y

Manipulate@ParametricPlot@8t, t ^ 2<,8t,-2, s<, AxesLabel®8"x", "y"<, PlotRange®88-2, 2<,80, 4<<, AxesOrigin®80, 0<D,8s,-1.999, 2, .1<D

(2)

Example: x = 1 - t, y = 1 - 2 t + t 2 , -1 £ t £ 3

ParametricPlot@81 - t, 1 - 2 t + t^2<, 8t, -1, 3<, AxesLabel ® 8"x", "y"<D

-2 -1 1 2 x

1 2 3 4 y

Manipulate@ParametricPlot@81-t, 1-2 t+t ^ 2<,8t,-1, s<, AxesLabel®8"x", "y"<, PlotRange®88-2, 2<,80, 4<<, AxesOrigin®80, 0<D,8s,-.999, 3, .1<D

Example: x = 1

4 t 3 - t, y = t - 1, -¥ £ t £ ¥

ParametricPlot@8H1  4L t^3 - t, t - 1<, 8t, -3, 3<, AxesLabel ® 8"x", "y"<D

-3 -2 -1 1 2 3 x

-4 -3 -2 -1 1 2 y

(3)

Manipulate@ParametricPlot@8H14Lt ^ 3-t, t-1<,8t,-3, s<, AxesLabel®8"x", "y"<, PlotRange®88-4, 4<,8-4, 4<<, AxesOrigin®80, 0<D,8s,-2.999, 3, .1<D

Example: x = 2 cos Θ, y = 3 sin Θ, 0 £ t £ 2 Π

ParametricPlot@82 Cos@tD, 3 Sin@tD<, 8t, 0, 2 Π<,

AxesLabel®8"x", "y"<, PlotRange ® 88-3, 3<, 8-3, 3<<D

-3 -2 -1 1 2 3 x

-3 -2 -1 1 2 3 y

Manipulate@ParametricPlot@82 Cos@tD, 3 Sin@tD<,8t, 0, s<, AxesLabel®8"x", "y"<, PlotRange®88-3, 3<,8-3, 3<<, AxesOrigin®80, 0<D,8s, .001, 2Π, .1<D

Example (Cycloid): x = Θ - sin Θ, y = 1 - cos Θ, 0 £ t £ 4 Π

ParametricPlot@8t - Sin@tD, 1 - Cos@tD<, 8t, 0, 4 Π<,

AxesLabel®8"x", "y"<, PlotRange ® 880, 14<, 80, 3<<, AspectRatio ® 3  14D

0 2 4 6 8 10 12 14x

0.0 0.5 1.0 1.5 2.0 2.5 3.0 y

Manipulate@ParametricPlot@8t-Sin@tD, 1-Cos@tD<,

8t, 0, s<, AxesLabel®8"x", "y"<, PlotRange®880, 14<,80, 3<<, AspectRatio®314, AxesOrigin®80, 0<D,8s, .001, 4Π, .1<D

Example (Projectile): x = H v 0 cos Θ L t, y = h + H v 0 sin Θ L t - 1

2 g t 2 , 0 £ t

(4)

Example (Projectile): x = H v 0 cos Θ L t, y = h + H v 0 sin Θ L t - 1

2 g t 2 , 0 £ t

g= 32;

h= 10;

v0= 20;

Θ = А 3;

ParametricPlot@8v0Cos@ΘD t, h + v0Sin@ΘD t - H1  2L g t^2<, 8t, 0, 4<,

AxesLabel®8"x", "y"<, PlotRange ® 880, 20<, 80, 20<<, AspectRatio ® 1, PlotStyle ® ThickD

0 5 10 15 20x

0 5 10 15 20 y

Manipulate@ParametricPlot@8v0Cos@ΘDt, h+v0Sin@ΘDt-H12Lg t ^ 2<, 8t, 0, s<, AxesLabel®8"x", "y"<, PlotRange®880, 20<,80, 20<<, AspectRatio®1, PlotStyle®ThickD,8s, .001, 4, .01<D

Polar Coordinates

Some Polar Plots

Ÿ r = 2

Clear@r,ΘD

(5)

PolarPlot@2,8Θ, 0, 2Π<, PlotRange®88-3, 3<,8-3, 3<<, PlotStyle®ThickD

-3 -2 -1 1 2 3

-3 -2 -1 1 2 3

r=2;

ParametricPlot@8r Cos@ΘD, r Sin@ΘD<,8Θ, 0, 2Π<, PlotRange®88-3, 3<,8-3, 3<<, PlotStyle®ThickD Clear@

rD

-3 -2 -1 1 2 3

-3 -2 -1 1 2 3

(6)

Ÿ Θ = Π  4

Θ = А4;

ParametricPlot@8r Cos@ΘD, r Sin@ΘD<,8r,-5, 5<, PlotRange®88-3, 3<,8-3, 3<<, PlotStyle®ThickD Clear@

ΘD

-3 -2 -1 1 2 3

-3 -2 -1 1 2 3

Ÿ r sin Θ = 3

Clear@r,ΘD

(7)

PolarPlot@3Sin@ΘD,8Θ,- Π,Π<, PlotRange®88-5, 5<,8-5, 5<<, PlotStyle®ThickD

-4 -2 2 4

-4 -2 2 4

Ÿ r = 3 sin Θ

PolarPlot@3 Sin@ΘD,8Θ, 0,Π<, PlotRange®88-5, 5<,8-5, 5<<, PlotStyle®ThickD

-4 -2 2 4

-4 -2 2 4

(8)

A More Complicated Graph: r = 1 + 2 cos Θ

Ÿ Table of Values

TableForm@Table@8Θ, 1+2 Cos@ΘD<,8Θ, 0, 2Π,А12<D, TableHeadings®8None,8Θ, r<<D

Θ r

0 3

Π

12 1+1+ 3

2 Π

6 1+ 3

Π

4 1+ 2

Π

3 2

5Π

12 1+-1+ 3

2 Π

2 1

7Π

12 1--1+ 3

2 2Π

3 0

3Π

4 1- 2

5Π

6 1- 3

11Π

12 1-1+ 3

2

Π -1

13Π

12 1-1+ 3

2 7Π

6 1- 3

5Π

4 1- 2

4Π

3 0

17Π

12 1--1+ 3

2 3Π

2 1

19Π

12 1+-1+ 3

2 5Π

3 2

7Π

4 1+ 2

11Π

6 1+ 3

23Π

12 1+1+ 3

2

2Π 3

(9)

Ÿ Polar Graph Paper

g1=PolarPlot@Table@r,8r, 0, 3, 12<D,8Θ, 0, 2Π<,

PlotRange®88-3.5, 3.5<,8-3.5, 3.5<<, PlotStyle®8Black<, Axes®NoneD; g2=ParametricPlot@Table@8r Cos@ΘD, r Sin@ΘD<,8Θ, 0, 2Π,А12<D,

8r,-3, 3<, PlotRange®88-3.5, 3.5<,8-3.5, 3.5<<, PlotStyle®BlackD;

g3=Show@g1, g2, Epilog®8Text@Style@"1", 16D,81,-.2<D, Text@Style@"2", 16D,82,-.2<D, Text@Style@"3", 16D,83,-.2<D, Text@Style@"0", 16D,83.2, 0<D,

Text@Style@"А2", 16D,80, 3.2<D, Text@Style@"Π", 16D,8-3.2, 0<D, Text@Style@"3А2", 16D,80,-3.2<D<, ImageSize®500D

1 2 3 0

Π 2

Π

3Π 2

(10)

Ÿ Plot Points

g4=ListPolarPlot@

Table@8Θ, 1+2 Cos@ΘD<,8Θ, 0, 2Π,А12<D, PlotStyle®8PointSize@LargeD<D;

Show@g1, g2, g4, Epilog®8Text@Style@"1", 16D,81,-.2<D, Text@Style@"2", 16D,82,-.2<D, Text@Style@"3", 16D,83,-.2<D, Text@Style@"0", 16D,83.2, 0<D,

Text@Style@"А2", 16D,80, 3.2<D, Text@Style@"Π", 16D,8-3.2, 0<D, Text@Style@"3А2", 16D,80,-3.2<D<, ImageSize®500D

1 2 3 0

Π 2

Π

3Π 2

(11)

Ÿ Connect the Dots

g5=PolarPlot@1+2 Cos@ΘD,8Θ, 0, 2Π<, PlotStyle®8Thick<D; Show@g1, g2, g4, g5,

Epilog®8Text@Style@"1", 16D,81,-.2<D, Text@Style@"2", 16D,82,-.2<D, Text@Style@"3", 16D,83,-.2<D, Text@Style@"0", 16D,83.2, 0<D,

Text@Style@"А2", 16D,80, 3.2<D, Text@Style@"Π", 16D,8-3.2, 0<D, Text@Style@"3А2", 16D,80,-3.2<D<, ImageSize®500D

1 2 3 0

Π 2

Π

3Π 2

(12)

Ÿ Without the Graph Paper

g5

0.5 1.0 1.5 2.0 2.5 3.0

-1.5 -1.0 -0.5 0.5 1.0 1.5

Ÿ Animation!

Manipulate@PolarPlot@1+2 Cos@ΘD,8Θ, 0, s<,

PlotRange®88-3, 3<,8-3, 3<<, PlotStyle®8Thick<D,8s, .01, 2Π<D

(13)

Another Complicated Graph: r = 4 cos Θ

Ÿ Table of Values

TableForm@Table@8Θ, 4 Cos@ΘD<,8Θ, 0,Π,А12<D, TableHeadings®8None,8Θ, r<<D

Θ r

0 4

Π

12 2 J1+ 3N

Π

6 2 3

Π

4 2 2

Π

3 2

5Π

12 2 J-1+ 3N

Π

2 0

7Π

12 - 2 J-1+ 3N

2Π

3 -2

3Π

4 -2 2

5Π

6 -2 3

11Π

12 - 2 J1+ 3N

Π -4

(14)

Ÿ Plot the Points

g1=PolarPlot@Table@r,8r, 0, 4, 12<D,8Θ, 0, 2Π<,

PlotRange®88-4.5, 4.5<,8-4.5, 4.5<<, PlotStyle®8Black<, Axes®NoneD; g2=ParametricPlot@Table@8r Cos@ΘD, r Sin@ΘD<,8Θ, 0, 2Π,А12<D,8r, 0, 4<,

PlotRange®88-4.5, 4.5<,8-4.5, 4.5<<, PlotStyle®BlackD; g4=

ListPolarPlot@Table@8Θ, 4 Cos@ΘD<,8Θ, 0, 2Π,А12<D, PlotStyle®8PointSize@LargeD<D; Show@g1, g2, g4, Epilog®8Text@Style@"1", 16D,81,-.2<D, Text@Style@"2", 16D,82,-.2<D,

Text@Style@"3", 16D,83,-.2<D, Text@Style@"4", 16D,84,-.2<D, Text@Style@"0", 16D,84.2, 0<D, Text@Style@"А2", 16D,80, 4.2<D,

Text@Style@"Π", 16D,8-4.2, 0<D, Text@Style@"3А2", 16D,80,-4.2<D<, ImageSize®500D

1 2 3 4 0

Π 2

Π

3Π 2

(15)

Ÿ Connect the Dots

g5=PolarPlot@4 Cos@ΘD,8Θ, 0, 2Π<, PlotStyle®8Thick<D; Show@g1, g2, g4, g5,

Epilog®8Text@Style@"1", 16D,81,-.2<D, Text@Style@"2", 16D,82,-.2<D, Text@Style@"3", 16D,83,-.2<D, Text@Style@"4", 16D,84,-.2<D,

Text@Style@"0", 16D,84.2, 0<D, Text@Style@"А2", 16D,80, 4.2<D,

Text@Style@"Π", 16D,8-4.2, 0<D, Text@Style@"3А2", 16D,80,-4.2<D<, ImageSize®500D

1 2 3 4 0

Π 2

Π

3Π 2

Ÿ Animation

Manipulate@PolarPlot@4 Cos@ΘD,8Θ, 0, s<,

PlotRange®88-4, 4<,8-4, 4<<, PlotStyle®8Thick<D,8s, .01, 2Π<D

(16)

Interesection Points of r = 1 + 2 cos Θ and r = 4 cos Θ

Ÿ Connect the Dots

g5=PolarPlot@81+2 Cos@ΘD, 4 Cos@ΘD<,

8Θ, 0, 2Π<, PlotStyle®88Thick, Red<,8Thick, Blue<<D;

Show@g1, g2, g5, Epilog®8Text@Style@"1", 16D,81,-.2<D, Text@Style@"2", 16D,82,-.2<D, Text@Style@"3", 16D,83,-.2<D, Text@Style@"4", 16D,84,-.2<D,

Text@Style@"0", 16D,84.2, 0<D, Text@Style@"А2", 16D,80, 4.2<D,

Text@Style@"Π", 16D,8-4.2, 0<D, Text@Style@"3А2", 16D,80,-4.2<D<, ImageSize®500D

1 2 3 4 0

Π 2

Π

3Π 2

Solve@1+2 Cos@ΘD Š4 Cos@ΘD,ΘD

Solve::ifun : Inverse functions are being used by Solve,

so some solutions may not be found; use Reduce for complete solution information. ‡

::Θ ® -Π

3>,:Θ ® Π 3>>

Manipulate@PolarPlot@81+2 Cos@ΘD, 4 Cos@ΘD<,8Θ, 0, s<,

PlotRange®88-4, 4<,8-4, 4<<, PlotStyle®88Thick, Red<,8Thick, Blue<<D,8s, .01, 2Π<D

(17)

Polar Plots as Parametric Plots: r = 4 cos Θ

ParametricPlot@84 Cos@tD^ 2, 4 Cos@tDSin@tD<,8t, 0, 2Π<, PlotStyle®ThickD

1 2 3 4

-2 -1 1 2

(18)

Vectors and Vector-Valued Functions

The Resultant

m1=2;

m2=3;

Θ1= А3;

Θ2= А4;

F1=m18Cos@Θ1D, Sin@Θ1D<

F2=m28Cos@Θ2D, Sin@Θ2D<

Show@Graphics@8Arrow@880, 0<, F1<D, Arrow@880, 0<, F2<D, Red, Arrow@880, 0<, F1+F2<D, Black, Text@Style@"F1", Bold, Italic, FontFamily®"Arial", 16D, F12,81,-1<D, Text@Style@"F2", Bold, Italic, FontFamily®"Arial", 16D, F22,8-2, 0<D, Red,

Text@Style@"F1+F2", Bold, Italic, FontFamily®"Arial", 16D, 2HF1+F2L 3,81.5, 0<D <D, PlotRange®880, 4<,80, 4<<, Axes®True, AxesLabel®8"x", "y"<D

:1, 3>

: 3 2

, 3

2

>

F

1

F

2

F

1

+F

2

0 1 2 3 4 x

0 1 2 3 4 y

Norm@F1+F2D N 4.95894

HF1+F2L Norm@F1+F2D N 80.629433, 0.777055<

(19)

ArcCos@HHF1+F2L Norm@F1+F2DL@@1DDD N 0.889973

ArcCos@HHF1+F2L Norm@F1+F2DL@@1DDD180 Π N 50.9917

Example: r H t L = H 1 - t L i + H 2 + 2 t L j

Manipulate@Show@Graphics@8PointSize@LargeD, Point@81-t, 2+2 t<D<D, Axes®True, PlotRange®88-10, 10<,8-10, 10<<D,8t,-3, 3<D

Manipulate@

Show@Graphics@8PointSize@LargeD, Point@81-t, 2+2 t<D, Arrow@880, 0<,81-t, 2+2 t<<D<D, Axes®True, PlotRange®88-10, 10<,8-10, 10<<D,8t,-3, 3<D

ParametricPlot@81-t, 2+2 t<,8t,-10, 10<,

PlotRange®88-3, 3<,8-3, 3<<, PlotStyle®8Blue, Thick<, AxesLabel®8"x", "y"<D

-3 -2 -1 1 2 3 x

-3 -2 -1 1 2 3 y

Example: r H t L = cos t i + sin t j

Manipulate@Show@Graphics@8PointSize@LargeD, Point@8Cos@tD, Sin@tD<D<D, Axes®True, PlotRange®88-1, 1<,8-1, 1<<D,8t, 0, 2Π<D

Manipulate@Show@Graphics@

8PointSize@LargeD, Point@8Cos@tD, Sin@tD<D, Arrow@880, 0<,8Cos@tD, Sin@tD<<D<D, Axes®True, PlotRange®88-1, 1<,8-1, 1<<D,8t, 0, 2Π<D

(20)

ParametricPlot@8Cos@tD, Sin@tD<,8t, 0, 2Π<,

PlotRange®88-1, 1<,8-1, 1<<, PlotStyle®8Blue, Thick<, AxesLabel®8"x", "y"<D

-1.0 -0.5 0.5 1.0x

-1.0 -0.5 0.5 1.0 y

References

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