TI-84 Plus CE: Tracing Polar Equations

TI-84 Plus CE:  Tracing Polar Equations

Traveling Around the Graphs

The programing BLINKING animates a dot around one of four polar equations:

1.  Ellipse

r = b / √(1 - ε^2 * cos^2 θ), where ε = √(a^2 - b^2)/a, a > b

2.  Cardioid

r = a + (1 + cos θ)

3.  3-Leaf Rose

r = a * cos(3 * θ)

4.  Log Spiral

r = a * e^(b * θ)

Notes:

The point alternates through blue (TI-84 Plus CE color 10), green (14), orange (15), and black (12).   The graph itself is light gray (21).  

r1 is the polar equation r1(θ).

Plotting points (not the polar graph) must be in cartesian coordinates (x,y).   

TI-84 Plus CE Program:  BLINKING

"2023-01-15 EWS"

Radian: Polar: FnOff

ZStandard

{10,14,15,12}→L6

Menu("PLOT","ELLIPSE",10,"CARDIOID",11,"3-LEAF ROSE",12,"LOG SPIRAL",13)

Lbl 10

Disp "A≥B"

Prompt A, B

√(A^2-B^2)/A→E

"B/√(1-E^2*cos(θ)^2)"→r1

Goto 20

Lbl 11

Prompt A

"A*(1+cos(θ))"→r1

Goto 20

Lbl 12

Prompt A

"A*cos(3*θ)"→r1

Goto 20

Lbl 13

Disp "A>0, B>0"

Prompt A,B

"A*e^(B*θ)"→r1

Goto 20

Lbl 20

Input "MAX? ",M

Input "NO. STEPS? ",S

M/S→N

FnOn 1

GraphColor(1,21)

0→P: r1(P)→Q

P>Rx(Q,P)→F: P>Ry(Q,P)→G

Pt-On(F,G,1,12)

For(K,1,S)

Wait 0.25

Pt-On(F,G,1,21)

remainder(K,4)+1→C

K*N→P: r1(P)→Q

P>Rx(Q,P)→F: P>Ry(Q,P)→G

Pt-On(F,G,1,L6(C))

End 

Disp "END"

Examples

In all examples, a = 3, b = 2, maximum = 4 * π, steps = 24

Ellipse

Cardioid

 

3-Leaf Rose

Log Spiral

Source

Harris, John W. and Horst Stocker.  Handbook of Mathematics and Computational Science  Springer:  New York.  2006.  ISBN 0-387-94746-9

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

March Calculus Madness Sweet Sixteen - Day 15: r = α * (1 - cos Θ)

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Welcome to March Calculus Madness!

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r = α * (1 - cos Θ)

r = α * (1 - cos Θ)

r^2 = α^2 * (1 - 2 * cos Θ + cos^2 Θ)

dr/dΘ = α * sin Θ

(dr/dΘ)^2 = α^2 * sin^2 Θ

Area from 0 ≤ Θ ≤ 2*π

1/2 * ∫ α^2 * (1 - cos Θ)^2  dΘ from Θ = 0 to Θ = 2*π

= α^2/2 * ∫ 1 - 2 * cos Θ + cos^2 Θ dΘ from Θ = 0 to Θ = 2*π

= α^2/2 * ∫ 1 - 2 * cos Θ + 1/2 * cos(2*Θ) + 1/2 dΘ from Θ = 0 to Θ = 2*π

= α^2/2 * (3/2 * Θ - 2 * sin Θ + 1/4 * sin(2*Θ) from Θ = 0 to Θ = 2*π)

= 3/2 * π * α^2

Arc Length from 0 ≤ Θ ≤ 2*π

r^2 + (dr/dΘ)^2

= α^2 * (1 - 2 * cos Θ + cos^2 Θ) + α^2 * sin^2 Θ

= α^2 - 2  * α^2 * cos Θ + α^2 * (cos^2 Θ + sin^2 Θ)

= α^2 - 2  * α^2 * cos Θ + α^2 

= 2 * α^2 - 2 * α^2 * cos Θ

= 2 * α^2 *(1  - cos Θ)

Arc Length:

∫ 2 * α^2 *(1  - cos Θ) dΘ from Θ = 0 to Θ = 2*π

= α * √2 * ∫ √(1  cos Θ) dΘ from Θ = 0 to Θ = 2*π

= α * √2 * 4 * √2

= 8 * α

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.