/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 11.08.0 ] */

/* [wxMaxima: title   start ]
Parametric and Polar Equations 
with a Figure Skater
   [wxMaxima: title   end   ] */

/* [wxMaxima: comment start ]
Chapter 13 Technology Application Project
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Introduction
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
OBJECTIVE: Represent curves and analyze motion in parametric 
and polar form.

Parametric equations are very powerful, and the purpose of 
this module is to help you get used to the idea of expressing 
curves using parametric equations and analyzing motion in the 
plane using these parametric equations. Polar plots can also 
be expressed parametrically, and that can be translated easily
into Cartesian coordinates.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Technology Guidelines
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
NOTE:  If you have just finished a document, restart Maxima 
before executing a new document.  This can be done by choosing
"Restart" from the Maxima menu.

TO OPEN OR CLOSE CELLS
Click on the arrow at the top of the cell bracket. 

TO STOP AN EXECUTION
Click on STOP button from the toolbar. 

ORDER OF EXECUTION
Execute commands in the order given. Do not skip any Maxima
Input lines within a given document. 

Alternatively, you can execute the entire worksheet by 
selecting the "Evaluate All Cells" command from the "Cell" drop
down menu or simply press Ctrl-r. 

SAVING WORKSHEETS
You can save anytime to any directory you choose, and it is 
wise to save often.

EXPERIENCING MAJOR PROBLEMS
Save if appropriate, and then shut down Maxima and start it up 
again. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Part I:  Parametric Equations of a Curve in Two-
Dimentional Space
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
DEFINING THE FUNCTION

First, we define the x and y coordinates parametrically. 
Suppose that time, t, is the independent variable. Once x and 
y are defined, we can write the position vector r(t). 
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
reset()$
kill(all)$
load(draw)$
ratprint:false$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
x(t) := cos(t)^3$
y(t) := 1-exp(sin(t))$
r(t) := [x(t), y(t)]$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Now we plot the resulting curve in blue. In what direction are
you moving on the curve as t increases?
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
print("")$
print("The position vector is: ")$
print(r(t))$
wxdraw2d(
    nticks=200,
    color=blue,
    parametric(x(t), y(t), t, 0, 2*%pi),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
TAKING THE VELOCITY AND ACCELERATION INTO ACCOUNT

If you consider the parametric equation as a vector equation 
for the motion of a particle, the velocity vector is found by 
differentiating each component of the position vector. 
Similarly, the acceleration vector is found by differentiating
the components of the position vector twice. We do this with 
Maxima and plot the velocity in red and the acceleration in 
green.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
xd1: diff(x(t),t,1)$
yd1: diff(y(t),t,1)$
vel: parametric(xd1,yd1,t,0,2*%pi)$
print("")$
print("The velocity function is: ")$
print([xd1, yd1])$
wxdraw2d(
    nticks=200,
    color=red,
    vel,
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
xd2: diff(x(t),t,2)$
yd2: diff(y(t),t,2)$
acc: parametric(xd2,yd2,t,0,2*%pi)$
print("")$
print("The acceleration function is: ")$
print( [xd2, yd2])$
wxdraw2d(
    nticks=200,
    color=green,
    acc,
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Note that the components of the velocity and acceleration 
functions are more complicated than the components of the 
postion function.

Let's look at the speed function and see what it tells us. 
The following plot shows the speed in black, the x-coordinate 
in orange, and the y-coordinate in magenta. Contrasting that to 
your parametric plot, identify the places where the speed 
function is 0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
speed(t) := sqrt((diff(r(t),t,1).diff(r(t),t)))$
print("The speed is: ")$
print(speed(t))$
wxdraw2d(
    nticks=200,
    color=black,
    explicit(speed(t), t, 0, 2*%pi),
    color=orange,
    explicit(x(t), t, 0, 2*%pi),
    color=magenta,
    explicit(y(t), t, 0, 2*%pi));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
print("")$
print("The position vector is: ")$
print(r(t))$
wxdraw2d(
    nticks=200,
    color=blue,
    parametric(x(t), y(t), t, 0, 2*%pi),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Identify the places on your path where the speed is 0 and the 
speed is a maximum.
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
COMPUTING THE DISTANCE TRAVELED ON A CURVED PATH

Suppose that you are walking along the path given above. The 
distance traveled can be found by integrating the speed 
function over a particular interval.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
distance: quad_qags(speed(t), t, 0, 2*%pi)[1]$
wxdraw2d(
    nticks=200,
    color=blue,
    parametric(x(t), y(t), t, 0, 2*%pi),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=black,
    explicit(speed(t), t, 0, 2*%pi),
    xlabel="t",
    ylabel="speed");
print("")$
print("The distance traveled around the closed path is ")$
print(distance, " units")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Think of this answer as either the distance around the curve 
or as the area under the speed function over the interval t 
from 0 to 2*%pi.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
You Try It: Part I
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
DEFINING A FUNCTION

Select your own functions for x(t) and for y(t) in the cell 
below, and then execute the command below. You need not select
a closed path, and you may wish to change the bounds for the 
parameter to something other than 0 to 10. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
x(t) := sin(t/3.2)$
y(t) := exp(-t)*t^2$
r(t) := [x(t),y(t)]$
print("")$
print("The position vector is: ")$
print(r(t))$
wxdraw2d(
    nticks=200,
    color=blue,
    parametric(x(t), y(t), t, 0, 10),
    xlabel="x",
    ylabel="y")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
COMPUTING THE VELOCITY AND ACCELERATION VECTORS
AND ANALYZING THE SPEED
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
xd1: diff(x(t),t,1)$
yd1: diff(y(t),t,1)$
vel: parametric(xd1,yd1,t,0,10)$
print("")$
print("The velocity function is: ")$
print([xd1, yd1])$
wxdraw2d(
    nticks=200,
    color=red,
    vel,
    xlabel="x",
    ylabel="y")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
xd2: diff(x(t),t,2)$
yd2: diff(y(t),t,2)$
acc: parametric(xd2,yd2,t,0,10)$
print("")$
print("The acceleration function is: ")$
print( [xd2, yd2])$
wxdraw2d(
    nticks=200,
    color=green,
    acc,
    xlabel="x",
    ylabel="y")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
speed(t) := sqrt((diff(r(t),t,1).diff(r(t),t)))$
print("The speed is ", speed(t))$
wxdraw2d(
    nticks=200,
    color=black,
    explicit(speed(t), t, 0, 2*%pi),
    color=orange,
    explicit(x(t), t, 0, 2*%pi),
    color=magenta,
    explicit(y(t), t, 0, 2*%pi));
print("speed(t): black, x(t): orange, y(t): magenta")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
print("")$
print("The position vector is: ")$
print(r(t))$
wxdraw2d(
    nticks=200,
    color=blue,
    parametric(x(t), y(t), t, 0, 10),
    xlabel="x",
    ylabel="y")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Identify the places on your path where the speed is 0 and the 
speed is a maximum.
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
FIND THE DISTANCE ALONG THE CURVED PATH

Adjust the values of the parameter in the integration you 
did so earlier.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
distance: quad_qags(speed(t), t, 0, 10)[1]$
wxdraw2d(
    nticks=200,
    color=blue,
    parametric(x(t), y(t), t, 0, 10),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=black,
    explicit(speed(t), t, 0, 10),
    xlabel="t",
    ylabel="speed");
print("")$
print("The distance traveled around the closed path is ")$
print(distance, " units")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Part II: A Figure Skater Tracing a Polar Plot
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
FOUR-PETAL PATTERN

Think of a figure skater who is tracing out a four-petal 
flower on the ice. The first set of commands gives the 
parametric equations of the figure skater in terms of a path 
that would be traced in the x-y plane. It is easiest to start 
with the equations in polar form.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(all)$
load(draw)$
ratprint:false$
r(t) := 16*sin(t)^2;
theta(t) := t/2;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
p1: polar(r(2*theta),theta,0,2*%pi)$
wxdraw2d(
    nticks=200,
    color=blue,
    p1,
    xlabel="East/West",
    ylabel="North/South");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
parx(t) := r(t) * cos(theta(t))$
pary(t) := r(t) * sin(theta(t))$
position: [parx(t),pary(t)]$
print("")$
print("The position vector is: ")$
print(position)$
velocity: diff(position,t,1)$

print("")$
print("The velocity vector is: ")$
print(velocity)$
speed(t) := sqrt(velocity.velocity)$
arr1: vector([parx(1), pary(1)], [parx(1.1)-parx(1),pary(1.1)-pary(1)])$
arr2: vector([parx(2.5), pary(2.5)], [parx(2.6)-parx(2.5),pary(2.6)-pary(2.5)])$
arr3: vector([parx(4), pary(4)], [parx(4.1)-parx(4),pary(4.1)-pary(4)])$
arr4: vector([parx(5.5), pary(5.5)], [parx(5.6)-parx(5.5),pary(5.6)-pary(5.5)])$
arr5: vector([parx(7), pary(7)], [parx(7.1)-parx(7),pary(7.1)-pary(7)])$
arr6: vector([parx(8.5), pary(8.5)], [parx(8.6)-parx(8.5),pary(8.6)-pary(8.5)])$
arr7: vector([parx(10), pary(10)], [parx(10.1)-parx(10),pary(10.1)-pary(10)])$
arr8: vector([parx(11.5), pary(11.5)], [parx(11.6)-parx(11.5),pary(11.6)-pary(11.5)])$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=blue,
    p1,
    head_angle=15,
    color=black,
    arr1,arr2,arr3,arr4,arr5,arr6,arr7,arr8);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Can you tell in what direction the skater is moving right at 
the origin?

Let's look at the speed function and see what it tells us. The
following plot shows the speed in black, the x-coordinate in 
orange, and the y-coordinate in  violet. Contrasting that to 
your parametric plot, identify the places where the speed 
function is 0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
tickst: [0,1,15]$
ticksf: [-15,5,20]$
print("")$
print("The speed is: ")$
print(speed(t))$
wxdraw2d(
    nticks=200,
    color=black,
    explicit(speed(t), t, 0, 4*%pi),
    color=orange,
    explicit(parx(t), t, 0, 4*%pi),
    color=magenta,
    explicit(pary(t), t, 0, 4*%pi),
    xtics=tickst,
    ytics=ticksf);
print("speed(t): black, x(t): orange, y(t): magenta")$
print("")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=blue,
    p1,
    head_angle=15,
    color=black,
    arr1,arr2,arr3,arr4,arr5,arr6,arr7,arr8);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Where are the places on your four-petal plot that the speed is
0?
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
VELOCTIY AND ACCELERATION: WHEN ARE THEY PERPENDICULAR?

Suppose we wish to determine for which values of t certain 
vectors describing the equations of motion are orthogonal. To 
do this, we will use the dot product, since perpendicular 
vectors yield a dot product of 0. Here we will examine when 
the velocity and acceleration vectors are perpendicular to one 
another. We begin by computing the dot product of the two 
vectors and we plot the resulting function of t to get an idea
of when the dot product might be 0.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
acceleration: diff(velocity,t,1)$
vdota: velocity.acceleration$
print("")$
print("velocity=")$
print(velocity)$
print("")$
print("acceleration=")$
print(acceleration)$
print("")$
print("velocity dotted into acceleration=")$
print(vdota)$
wxdraw2d(
    nticks=200,
    color=black,
    explicit(vdota,t,0,4*%pi),
    xlabel="t",
    ylabel="vdota");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
As you can see from the graph, there are many times when the 
velocity and acceleration vectors are perpendicular to each 
other.

The following commands find some of the places where the 
velocity and acceleration vectors are perpendicular. Notice 
that we use seed values in solve that seem close to some of 
the places where our function crosses the horizontal axis.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
sol1: []$
for i: 0.785 thru 12 step 0.785 do block(
    temp: find_root(vdota, t, i-0.3925, i+0.3925),
    sol1: append(sol1, [temp]))$
sol1;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Now we evaluate x and y at the values of t we have found.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
sol2: makelist([parx(sol1[i]),pary(sol1[i])],i,1,length(sol1));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
We can now see where those points are relative to our petals.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
psol: points(sol2)$
wxdraw2d(
    nticks=200,
    color=blue,
    p1,
    color=red,
    point_type=circle,
    psol);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The dots show where the velocity and acceleration vectors are 
perpendicular to each other. Can you describe what is 
happening to the figure skater at those points?
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
You Try It: Part II
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
Try your own functions for r(t) and theta(t). Remember to 
solve for t as a function of theta before you attempt to do a 
polar plot in the form of r as a function of theta. Replace 
the terms in r(t) and theta(t).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(t,x,y)$
r(t) := cos(t)^3$
theta(t) := t^2/8$
pp1: polar(cos(sqrt(8*theta))^3,theta,0,3)$
wxdraw2d(
    nticks=200,
    color=blue,
    pp1,
    xlabel="East/West",
    ylabel="North/South",
    xrange=[-1.5,1.5],
    yrange=[-1.5,1.5]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
parx(t) := r(t)*cos(theta(t))$
pary(t) := r(t)*sin(theta(t))$
position: [parx(t), pary(t)]$
velocity: diff(position, t, 1)$
speed: sqrt(velocity. velocity)$
print("Position Vector: ", position)$
print("Speed: ", speed)$
wxdraw2d(
    nticks=200,
    color=black,
    explicit(speed,t,0,5),
    color=orange,
    explicit(parx(t),t,0,5),
    color=magenta,
    explicit(pary(t),t,0,5),
    xlabel="t",
    ylabel="function")$
print("speed: black")$
print("x-coordinate of the path of motion: orange")$
print("y-coordinate of the path of motion: magenta")$
print("")$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=blue,
    pp1,
    xlabel="East/West",
    ylabel="North/South",
    xrange=[-1.5,1.5],
    yrange=[-1.5,1.5])$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
What is happening to your speed as t increases?
   [wxMaxima: comment end   ] */

/* Maxima can't load/batch files which end with a comment! */
"Created with wxMaxima"$