Introduction to Winplot
Richland College Spring 2012
John Ganci
jganci@dcccd.edu
Presentation Outline
•
•
•
•
•
•
•
•
•
What is Winplot?
Obtaining and installing Winplot
Learning about Winplot
Drawing 2-dimensional graphs
Drawing 3-dimensional graphs
Copying Winplot graphs into other applications
Sources of documentation
Summary
Appendix
2
What is Winplot?
• A Windows application that draws graphs
– 2-dimensional curves
– 3-dimensional curves and surfaces
• A lot more than a graphing calculator
• A tool to illustrate mathematical concepts
– Slopes of lines, areas, volumes, vectors, to name a few
– Animation allows one to show values as they change
• Best of all, it’s free!
– The author is a faculty member at Phillips Exeter Academy
3
Obtaining and Installing Winplot
• http://math.exeter.edu/rparris/winplot.html
• Download the self-extracting zip file wp32z.exe
• Run the program
– The default installation directory is c:\peanut
– Recommend that you change it to something else
– Resulting unzipped file is winplot.exe
• The next few slides show parts of this process
4
Obtaining Winplot
Download
link
Lots of
good info!
5
Installing Winplot
• Run the self-extracting zip file wp32z.exe
Recommend
that you
change this
6
Learning About Winplot
• Search the web for “winplot tutorial”
These two are
very good
7
Learning About Winplot
• Supplemental materials on Winplot home page
8
Learning About Winplot
• Al Lehnen’s home page; good stuff!!; scroll down
9
Learning About Winplot
Tutorial
and
examples
Tutorial
10
Learning About Winplot
Tutorial,
examples,
links, and a
Power Point
Introduction
to Winplot
11
Using Winplot
• Enough about learning!
• It’s time to fire up Winplot and take it out for a spin
• 2-dimensional graphs
– Draw some simple ones
– Show the various ways to draw graphs
– Show how to show and see additional data about them
• Adding labels
• Viewing table of values
• 3-dimensional graphs
• Copying Winplot data to other applications
• Okay, start Winplot
12
Winplot Initial Screen
• Close tip box
• Resize the Winplot screen the first time you invoke it
13
Winplot Initial Screen
• Everything is accessed via the Window menu item
• Next slide shows the two menu items
14
Winplot Initial Screen
• Note Window values; we’ll start with 2-dim
• Recommend checking Use defaults
15
2-dimensional Initial Screen
• Before we graph anything we’ll add grid lines; click View
16
Adding Grid Lines …
• Click Grid
17
… Adding Grid Lines
•
•
•
•
Dialog box appears
Axes is checked, as is both, so the x- and y-axes are shown
Ticks, arrows, and labels are checked
In the grid sub-box
– Check rectangular …
then check dotted
– Then Apply, Close
• We’ll look at polar a little later
18
2-dimensional Plotting
• Now we want to plot something (an equation); click Equa
19
2-dimensional Plotting
• Note values; we’ll do 1-4; click Explicit …
20
2-dimensional Plotting
•
•
•
•
Dialog box appears
Set the f(x)= value to 2*x+1
Leave the low and high x values at -5 and 5
Click color to select a color for the graph
21
2-dimensional Plotting
• Click on one of the colored squares (blue); then click Close
22
2-dimensional Plotting
• Back to the equation dialog box
• Change pen width for thicker line
• Click ok
23
2-dimensional Plotting
• A graph is displayed along with an inventory box
• If you want to change the color or line thickness, click Edit
• Click View
24
2-dimensional Plotting
• We’ve already looked at Grid; note values; click View …
25
2-dimensional Plotting
• Here’s how you can set the displayed bounds of the graph
• Click “set corners” to set the x and y bounds
• Click “set center” to set the center point and width
26
2-dimensional Plotting
•
•
•
•
•
•
Next we will draw a graph several different ways
First we delete the current graph so that we begin “fresh”
The first graph will again use Explicit
We enter sqrt(16-x^2) for the function
We take the default range, [-5,5], for x (wrong, but ok)
We choose a color and click ok
27
2-dimensional Plotting
• A graph is displayed along with an inventory box
• We want the other half of the semicircle; click dupl(icate)
28
2-dimensional Plotting
• Another explicit equation dialog box
• Enter minus sign to the left of sqrt(16-x^2); click ok
29
2-dimensional Plotting
• Now have entire circle; note the two items in the inventory
30
2-dimensional Plotting: Parametric
• Graph using parametric equations; click Equa, then Parametric
31
2-dimensional Plotting: Parametric
•
•
•
•
•
•
The equation dialog box for parametric is displayed
This time two functions must be entered; x=f(t), y=g(t)
Enter 3cos(t) for f(t) and 3sin(t) for g(t)
Note that t ranges from 0 to 1
Change high t to 2pi
As before, click color; click a color; click close; click ok
f(t)=3cos(t)
g(t)=3sin(t)
2pi
32
2-dimensional Plotting: Parametric
• Note the new graph and the new inventory entry
33
2-dimensional Plotting: Implicit
• Next is implicit; click Equa; click Implicit …
34
2-dimensional Plotting: Implicit
• Another equation dialog box, but for implicit
• Fill in x^2+y^2 = 4 (or xx+yy=4)
• Choose a color; click ok
x^2+y^2=4
35
2-dimensional Plotting: Implicit
• Note the new graph and the new inventory entry
36
2-dimensional Plotting: Polar
• Lastly we graph a polar equation; click Equa; click Polar …
37
2-dimensional Plotting: Polar
•
•
•
•
•
The equation dialog box for polar is displayed
The t value is theta; f(t) is the r value; that is, r=f(t)
Enter 1 for the f(t) value
Note that the t values range from 0 to 2pi
Choose a color; click ok
38
2-dimensional Plotting: Polar
• Note the fourth graph and the fourth inventory entry
39
2-dimensional Plotting: Table
•
•
•
•
•
Let’s look at the table of values for a few of the equations
All tables are accessed via the inventory button “table”
Highlight the inventory entry then click table
Each is shown on the following slides (no table for implicit)
Close the table by clicking Close
40
2-dimensional Plotting: Table
• y=sqrt(16-x^2); note undefined values; click Close when done
41
2-dimensional Plotting: Table
• Table for parametric equations x=3cos(t), y=3sin(t)
42
2-dimensional Plotting: Table
• Table for polar equation r=1
43
2-dimensional Plotting: Polar, Part 2
•
•
•
•
•
We now graph a cardioid
Start with a clean 2-dim screen
We enter the polar equation r = 2 + 3 sin θ
Add grid lines, but use a polar grid
The next slide shows Winplot after Equa->Polar
44
2-dimensional Plotting: Polar, Part 2
• Note: t instead of θ; t ranges from 0 to 2pi
• Change pen width to 2; choose a color
• Click ok
45
2-dimensional Plotting: Polar, Part 2
• Zoom out: PgDn a few times; then View->Grid…
46
2-dimensional Plotting: Polar, Part 2
• Click polar (axis); polar sectors; 24; apply; close
47
2-dimensional Plotting: Polar, Part 2
• Polar graph paper!
48
2-dimensional Plotting: Calculus I
• The next example illustrates how an integral
is approximated using Riemann sums
• The approximating rectangles are shown
• The number of rectangles can be changed
• The example also illustrates how to graph the
antiderivative and add explanatory text
• Start with a new 2-dim screen
• Begin by entering y=x^2 on Equa->Explicit
49
2-dimensional Plotting: Calculus I
• One->Measurement->Integrate
50
2-dimensional Plotting: Calculus I
•
•
•
•
Set lower limit to 1, upper limit to 2
Set subintervals to 5
Check left endpoint; check overlay; choose a color
Click definite
51
2-dimensional Plotting: Calculus I
• Note new rectangles and new approximate value
• Click indefinite
52
2-dimensional Plotting: Calculus I
• Note new graph and new inventory entry
• Highlight new inventory entry; click edit
53
2-dimensional Plotting: Calculus I
• Note that the f(x) value cannot be edited
• Change the color; click ok
54
2-dimensional Plotting: Calculus I
• Not bad! Let’s add a few descriptive labels
• First close the two dialog boxes by clicking close
55
2-dimensional Plotting: Calculus I
• Click Btns; note values; highlight or click Text
• Position cursor to left of blue graph; right-click
56
2-dimensional Plotting: Calculus I
• Add what text you want to display
• Change font/color; click font
• The “tie text to” radio buttons associate the text
with one of three possibilities
• If you don’t want the text to move when you
zoom in and out, check the frame button
57
2-dimensional Plotting: Calculus I
• Change font/font style/size
• Change color of font
• Click OK; Font dialog ends; click ok; edit text dialog ends
58
2-dimensional Plotting: Calculus I
• After adding a second label for the antiderivative
• Left-click and drag text box to fine-tune position
59
2-dimensional Plotting: Animation
•
•
•
•
•
•
•
•
•
•
Next we will see the power of Winplot
We construct an example that illustrates slope
The example is dynamic
The dynamics are done using Winplot’s animation
Animation is done with parameters A-W
X, Y, and Z are reserved for functions
We start with a clean 2-dim screen
We add the grid lines as before
We enter the explicit function m*x+b
Example taken from Steve Simonds’ videos
60
2-dimensional Plotting: Animation
• The initial parameter values are all 0, so y=0
y=m*x+b = 0
61
2-dimensional Plotting: Animation
• Anim->Individual->B …
• Repeat for M
62
2-dimensional Plotting: Animation
• Note initial M and B values; then slide M right
63
2-dimensional Plotting: Animation
• Note new M and new line; slide B left (down)
64
2-dimensional Plotting: Animation
• Note new B and new line position
65
2-dimensional Plotting: Animation
•
•
•
•
Now we have an animated line
Next we want to illustrate slope
Add two points to the line (P and P+ΔP)
Add the “rise” and “run” segments
– That is, we add the “slope triangle”
• Points are added via the Equa menu item
• Segments are added the same way
• The points and segments will be animated
66
2-dimensional Plotting: Animation
•
•
•
•
The first point is (p,m*p+b)
The second point is (p+d, m*(p+d)+b)
The parameter values P and D animate the points
Plot the points and open the animation boxes
67
2-dimensional Plotting: Animation
• Equa->Point->(x,y) …
68
2-dimensional Plotting: Animation
•
•
•
•
•
Set x to p; set y to m*p+b
Select solid
Set dot size to 4
Choose a color for the point
Click ok when done
x=p
y=m*p+b
69
2-dimensional Plotting: Animation
• Note inventory; P=0, so the point is (0,B)
70
2-dimensional Plotting: Animation
•
•
•
•
•
•
Follow the same steps to add the second point
The x-coordinate of the point is p+d
The y-coordinate of the point is m*(p+d)+b
Use a different color for this point
Finally, display the animation boxes for P and D
Resulting graph is shown on the next slide
71
2-dimensional Plotting: Animation
• Note inventory; since D=0, the points coincide
Both points
72
2-dimensional Plotting: Animation
• Slide D to the right so points don’t coincide
(p+d, m*(p+d)+b)
(p, m*p+b)
73
2-dimensional Plotting: Animation
• Now the “rise” and “run” must be added
• The “run” is the horizontal segment joining
(p, m*p+b) to (p+d, m*p+b)
• The “rise” is the vertical segment joining
(p+d, m*p+b) to (p+d, m*(p+d)+b)
• Segments are added via Equa
• The adding of the first segment is shown on
the next few slides
74
2-dimensional Plotting: Animation
• Equa->Segment->(x,y) …
75
2-dimensional Plotting: Animation
•
•
•
•
•
•
•
Dialog box for the segment is displayed
Set x1 = p, y1 = m*p+b
Set x2 = p+d, y2 = m*p+b
Set pen width to 3 (thicker line segment)
Choose a color
Click ok
Similar for “rise”
– x1 = p+d, y1 = m*p+b
– x2 = p+d, y2 = m*(p+d)+b
76
2-dimensional Plotting: Animation
• We have one small item to fix
77
2-dimensional Plotting: Animation
•
•
•
•
The “rise” and “run” terminate at the points
Because of the colors, we can see the segments
We must delete and add back the two points
Use the inventory dupl button to duplicate each
point; the duplicated point appears at the bottom
of the inventory
• Use the inventory delete button to delete the
original two points
78
2-dimensional Plotting: Animation
• Note the points are now on top of the segments
79
2-dimensional Plotting: Animation
• Now “animate” several of the values
– Make M a little smaller
– Make B a little larger
– Slide P to the left (move first point down)
• Result shown on next slide
80
2-dimensional Plotting: Animation
• How about that!!!
81
2-dimensional Plotting: Calculus I
• The next 2-dimensional example is optional
• The example illustrates “epsilon-delta” for the
“limit of f(x) as x approaches a” for a
continuous function f
• Explicit and implicit shading is illustrated
• The example uses e for epsilon and d for delta
• Animation is used to independently change a,
e, and d
• Appendix A1 contains the instructions to build
the example
• The next slide shows the example
82
2-dimensional Plotting: Calculus I
83
2-dimensional Plotting: Calculus II
• The final 2-dimensional example is optional
• The example illustrates the polar equation of a
conic
• The example uses e for the eccentricity and d
for the directrix
• Animation is used to independently change e
and d
• A point on the conic is auto-animated using u
to show how the conic is drawn
• Appendix A2 contains the instructions to build
the example
• The next slide shows the example
84
2-dimensional Plotting: Calculus II
85
3-dimensional Plotting
• Start at the Winplot main screen; click Window; click 3-dim
86
3-dimensional Plotting
• Note similarities to 2-dim; click Equa
87
3-dimensional Plotting
• Note similarities to 2-dim
88
3-dimensional Plotting
• The initial two screens are similar to 2-dim
• Equa and Anim are present
• Equa has some additional values
– cylindrical and spherical
– curve
– plane
• Some 3-dim graphs produce undesirable
results; must redraw using one of the other
options
• Our first example uses Equa->Explicit to draw
a hemisphere
89
3-dimensional Plotting
• Equa->Explicit …
90
3-dimensional Plotting
• Note the dialog box is for a function in two
variables: z=f(x,y)
• Want to draw the upper hemisphere
• The sphere is x^2+y^2+z^2=4
• Set z=sqrt(4-x^2-y^2)
• Set x lo = -2 = y lo, x hi = 2 = y hi
• Choose a color; click ok
91
3-dimensional Plotting
• No axes; what is the planar part?
92
3-dimensional Plotting
• Display the axes: View->Axes->Axes
93
3-dimensional Plotting
• The axes are shown; slightly hidden
94
3-dimensional Plotting
• View; uncheck Hide segments
95
3-dimensional Plotting
• Hidden lines are now visible
96
3-dimensional Plotting
• The planar “tags” are due to what Winplot
uses for the domain of the function
• The Winplot domain is [-2,2] x [-2,2]
– Look back at the Equa->Explicit dialog box
• There are points in the Winplot domain that
are not in the actual domain
• Winplot sets z to 0 for these points
• These points make up the planar “tags”
97
3-dimensional Plotting
• The sphere can be drawn using spherical
coordinates; ρ=2 is x^2+y^2+z^2=4
• Delete the first attempt from the inventory
98
3-dimensional Plotting
• Equa->Spherical …
99
3-dimensional Plotting
• The equation dialog box for spherical is
displayed
• The r value is the ρ value; enter 2 there
• The t value is the θ value; note that it ranges
from 0 to 6.28319 = 2π
• The u value is the Φ value; note that it
ranges from 0 to 3.14159 = π
• Choose a color
• Click ok
100
3-dimensional Plotting
• Display the axes again; much better!
101
3-dimensional Plotting
• All the quadric surfaces can be drawn
–
–
–
–
Ellipsoid
Elliptic paraboloid
Hyperbolic paraboloid
Hyperboloids with one and two sheets
• Equa->Explicit doesn’t always yield good
results
– Use parametric or cylindrical instead
• Cross-sections can be added using planes
– Animation can be used to show the level curves
• The next example shows a hyperboloid with
one sheet and its three cross-sections
102
3-dimensional Plotting
• Note inventory; animate on B,C,D; rotate graph
Anim
Parameters A-W
103
3-dimensional Plotting
• We next look at a space curve and its four
related vectors: r=position vector, T=unit
tangent vector, N=unit normal vector, and
B=unit binormal vector
• To make things interesting, we animate a
point, showing all the values as the point
moves along the curve
• The space curve is a variation of the “twisted
cubic”
• The animation gives visual feedback about
why it is called “twisted”
104
3-dimensional Plotting
• An initial view of the graph; includes labels
Slide notes area
shows the inventory
105
3-dimensional Plotting
• A final 3-dimensional graph is taken from a
problem in James Stewert’s “Calculus, Early
Transcendentals” text. It graphically shows a
3-dimensional solid bounded by several
curves and planes.
• You can use the arrow keys to rotate the solid
to see it from just about any angle.
106
3-dimensional Plotting
• Rotate the graph to “look inside”
107
Copying Graphs to Other Applications
• As a final illustration, we show how easy it is to copy
a Winplot graph into a Word document
• We copy the 3-dimensional graph on the previous
slide into a Word document
• In Winplot click on File
• Click on Copy to clipboard (or Control-C)
• Switch to your open Word document
• Position where you want the graph
• Click Edit; Paste (or Control-V)
• That’s it! Result shown on next slide
• The only recommendation is to do any fixing up in
Winplot before doing the copy and paste
108
Copying Graphs to Other Applications
109
Sources of Documentation
• The web
– Use your favorite search engine
• The Winplot home page Supplemental link
– Tutorials and examples in many languages
– The two highlighted ones are especially good
• The Help menu items found throughout Winplot
– While somewhat terse, there is good information there
• The tips shown when Winplot is started
– Reading through these provides lots of useful information
110
Summary
• Winplot is a free tool used to graph functions
• Both 2-dimensional and 3-dimensional graphs
• In this introduction we’ve only touched on some of the
many functions provided by Winplot
• With some thought, a lot of helpful animations can be
created to illustrate concepts to your students
• The Winplot author is very receptive to feedback and fixes
problems almost as soon as they’re reported
• Check the Winplot home page often to be sure your
version is current
– Always backup your current version before you replace it
• A special thanks to Richard Parris, the author of Winplot,
and Al Lehnen, a contributor to the Winplot supplemental
materials, for their help with my many questions
111
Appendix
• Appendix 1 (A1) contains the 2-dimensional
“epsilon-delta” example
• Appendix 2 (A2) contains the 2-dimensional polar
conic example
112
A1: Epsilon-Delta
• The example illustrates how epsilon and delta
interact with respect to a fixed function f and
an x value of a
• The function f is defined as a user function
• Epsilon is represented by e, delta by d
• Point P (a,f(a)) is on the graph of f
• Points Q (a,0) and R (0,f(a)) are on the axes
• Dashed lines connect P to the axis points
• The epsilon and delta “bands” are shaded
• The a, e, and d values are animated
• Next slide shows the end result
113
A1: Epsilon-Delta
114
A1: Epsilon-Delta
•
•
•
•
•
•
•
•
•
Overview of steps follows
Start with a new 2-dim graph
Add the function f as a user function
Add the a, d, and e animate dialog boxes
Add the points P, Q, and R
Add the dashed lines PQ and PR
Add “band” horizontal and vertical lines
Add shading
Add labels
115
A1: Epsilon-Delta
• Equa->User functions …
116
A1: Epsilon-Delta
• Fill in function name; fill in function
• Click Enter; note function; click close
• Note: name must be at least 2 characters
117
A1: Epsilon-Delta
• Equa->Explicit …
118
A1: Epsilon-Delta
• Set f(x)=ff(x)
• Set pen width=2; set color; click ok
119
A1: Epsilon-Delta
• Graph is displayed; next add animation boxes
120
A1: Epsilon-Delta
• Anim->Individual->A …
121
A1: Epsilon-Delta
• Note value is zero; note scroll bar in middle
• Set left (lower) and right (upper) bounds for a
• Enter -10; click set L (left bound); scroll bar at left
• Enter 10; click set R (right bound); scroll bar at right
• Scroll A to 1
122
A1: Epsilon-Delta
•
•
•
•
•
Repeat the steps for d and e
Set both lower bounds to zero
Set both upper bounds to 2
Scroll both so that the values are 1
Next slide shows results
123
A1: Epsilon-Delta
• We add P, Q, and R next
124
A1: Epsilon-Delta
•
•
•
•
•
P is (a,ff(a))
Q is (a,0)
R is (0,ff(a))
Details for P follow
Details for Q and R are not shown; similar to P
125
A1: Epsilon-Delta
• Equa->Point->(x,y) …
126
A1: Epsilon-Delta
•
•
•
•
•
Set x to a; set y to ff(a)
Select solid
Choose a color for the point
Click ok when done
Next slide shows P, Q, and R
127
A1: Epsilon-Delta
• Add dashed lines PQ and PR next
R
P
Q
128
A1: Epsilon-Delta
• Equa->Segment->(x,y) …
129
A1: Epsilon-Delta
•
•
•
•
•
•
•
Instructions for PQ follow
Set x1=0, y1=ff(a)
Set x2=a, y2=ff(a)
Set color; click dotted; click ok
PR is similar; use dupl, then edit
Dupl P then delete first P
Next slide shows both segments; P is above both
130
A1: Epsilon-Delta
• Next add the lines that bound the “bands”
131
A1: Epsilon-Delta
• The “epsilon band” lines are y=ff(a)-e and
y=ff(a)+e
• These are added as explicit functions
• The “delta band” lines are x=a-d and x=a+d
• These are added as lines
• The addition of one of each is shown on the
next few slides
132
A1: Epsilon-Delta
• Equa->Explicit …
133
A1: Epsilon-Delta
•
•
•
•
•
Add the bottom line
Set f(x) to ff(a)-e; set color; click ok
Repeat for top line; not shown (use dupl)
Set f(x) to ff(a)+e; click ok
Next slide shows the two lines
134
A1: Epsilon-Delta
• The vertical lines are done next
135
A1: Epsilon-Delta
• Equa->Line …
136
A1: Epsilon-Delta
•
•
•
•
•
Add the left vertical line
Set a=1, b=0, c=a-d; change color; click ok
Use dupl for right vertical line (not shown)
Set a=1, b=0, c=a+d; click ok
Next slide shows the two lines
137
A1: Epsilon-Delta
• One more item before we shade the “bands”
138
A1: Epsilon-Delta
• Equa; note “Shade explicit inequalities …” is
available but “Shade implicit inequalities …” is
grayed out
139
A1: Epsilon-Delta
• The “epsilon band” is shaded explicitly
• The “delta band” is shaded implicitly
• We need to add two implicit values so that
“Shade implicit inequalities …” is available
• The next few slides do this
140
A1: Epsilon-Delta
• Equa->Implicit …
141
A1: Epsilon-Delta
•
•
•
•
Fill in x=a-d; set color; click ok
Use dupl to add second implicit (not shown)
Fill in x=a+d; click ok
Click the graph button for these two items in
the inventory so that they are hidden
• Next slide shows the results
142
A1: Epsilon-Delta
• We’re ready to do the shading now
143
A1: Epsilon-Delta
• Equa->Shade explicit inequalities …
144
A1: Epsilon-Delta
•
•
•
•
First dropdown; select y=ff(a)-e
Click between radio button
Second dropdown; select y=f(a)+e
Select color; click shade; note values; click close
145
A1: Epsilon-Delta
• Equa->Shade implicit inequalities …
146
A1: Epsilon-Delta
• Click x=a-d; click change = to >; change color
• Click x=a+d; click change = to <
• Insure shading is correct; click close
147
A1: Epsilon-Delta
• Almost done! (Now is a good time to save)
148
A1: Epsilon-Delta
• The last items to add are the labels
• The addition of one label is shown
• The remaining labels are added in a similar
manner
149
A1: Epsilon-Delta
• Btns->Text
150
A1: Epsilon-Delta
•
•
•
•
•
•
•
Right click near where you want the label
Fill in text; optionally change font
Click tie text to frame; click ok
Repeat process for all the labels
Right click inside one to edit it
Click and drag them to fine-tune position
Final result shown on next slide
151
A1: Epsilon-Delta
• Note the labels; (save the graph again)
152
A1: Epsilon-Delta
•
•
•
•
Time to use the example
Set a to a particular value, say 1
Set e to a particular value, say 1
Scroll d until the graph, restricted to the
vertical band, is bounded by the horizontal
band (within the intersection rectangle)
• Setting e to a particular value corresponds to
“for every epsilon …”
• Scrolling d until the graph lies within the
intersection rectangle corresponds to “there is
a delta …”
• One possibility is shown on the next slide
153
A1: Epsilon-Delta
• Looks like delta=0.36 works for epsilon=1
154
A1: Epsilon-Delta
• Play some more
• Change e; does d need to change?
• Change a, leaving e as before; does d need
to change?
• Try changing the user-defined function; does
anything else need to change?
• This concludes the example
155
A2: Polar Conic
• Recall that a conic is the set of points P
whose distance from a fixed point F (the
focus) are a constant multiple (the
eccentricity) of the distance from P to a fixed
line L (the directrix); that is, |PF| = e|PL|
• The polar equation r = ed/(1+e*cos(θ)) is a
conic with focus F at the pole and directrix L
a vertical line that intersects the polar axis
• Our next example illustrates F, L, P, PF, PL,
and how P changes as the parameter value θ
changes
• The next slide shows the final result
156
A2: Polar Conic
157
A2: Polar Conic
• The example illustrates how a graph can do
an “active” animation
• The eccentricity and directrix are animated
• The example also illustrates how to use a
“User Function”, thus making the Inventory
somewhat “dynamic”
• Start with a new 2-dim screen
• Building this example takes some work, but is
worth it
158
A2: Polar Conic
• Equa->User functions …
159
A2: Polar Conic
• Type conic in name, ed/(1+ecos(x)) in name(x)
• Click Enter; note value; click close
160
A2: Polar Conic
• Equa->Polar …
161
A2: Polar Conic
•
•
•
•
•
Replace f(t) with conic(t)
Note low, high t are correctly set to 0, 2pi
Set pen width to 2
Set color
Click ok
162
A2: Polar Conic
• Can’t see graph because d and e are zero!!!
• Next open the d and e animate boxes
163
A2: Polar Conic
• Anim->Individual->D …
164
A2: Polar Conic
• Note value is zero; note scroll bar in middle
• Set left (lower) and right (upper) bounds for d
• Click set L (left bound); scroll bar now at left
• Enter 10; click set R (right bound); scroll bar at right
165
A2: Polar Conic
• Still no graph, this time because d is 10
• Lower the d value to 2; click or drag
166
A2: Polar Conic
• Hyperbolas appear!
• Open and set animation for e
167
A2: Polar Conic
• Note value is 2.71828; note scroll bar
• Set left (lower) and right (upper) bounds for e
• Enter 0; click set L; scroll bar at left
• Enter 10; click set R (right bound); scroll bar at right
• No change in graph; lower the e value to 2
168
A2: Polar Conic
• Still a hyperbola since e > 1
169
A2: Polar Conic
• Directrix is next; Equa->Line …
170
A2: Polar Conic
• Note a, b, c values
• Set a=1, b=0, c=d; change pen width to 2;
change color; click ok
x=d
171
A2: Polar Conic
• Now we see the directrix; next is the focus
x=d
172
A2: Polar Conic
• Equa->Point->(x,y) …
173
A2: Polar Conic
•
•
•
•
•
Set x=0, y=0
Click solid
Set dot size to 3
Set color
Click ok
174
A2: Polar Conic
• Now we see the focus
175
A2: Polar Conic
• The point P on the conic is now added
• Anticipating that we want to animate P to see
it move along the conic, we define it using a
parameter, u
– P has polar coordinates (conic(u),u)
• A second point, D (on the directrix), is added;
the distance from P to the directrix is |PD|;
the point D is also defined in terms of u
– D has rectangular coordinates (d, conic(u)sin(u))
• The next three slides show how P is added
• We do not show the addition of D since it is
similar to what was done for the focus
176
A2: Polar Conic
• Add point P: Equa->Point->(r,t) …
177
A2: Polar Conic
•
•
•
•
•
•
Set r=conic(u), t=u
Click solid
Set dot size to 3
Set color
Click ok
Next slide shows both P and D
178
A2: Polar Conic
• Red point is P; green point on directrix is D
• Note that u is initially zero
P
D
179
A2: Polar Conic
•
•
•
•
•
Add the animation dialog box for u
Proceed as we did for d and e
Set lower bound to 0 and upper bound to 2pi
Move the scroll bar so that u is greater than zero
Result shown on next slide
180
A2: Polar Conic
• Note that P and D have moved
181
A2: Polar Conic
• Almost done!
• What’s left is to add the line segment from the
focus, F, to P and the line segment from P to D
• We use rectangular coordinates for both
• The endpoints of PF are
(conic(u)cos(u),conic(u)sin(u)) and(0,0)
• The endpoints of PD are
(conic(u)cos(u),conic(u)sin(u)) and
(d,conic(u)sin(u))
• The next four slides show the addition of PD
• The addition of PF is similar so is not shown
182
A2: Polar Conic
• Equa->Segment->(x,y) …
183
A2: Polar Conic
•
•
•
•
•
•
Set x1=conic(u)cos(u), y1=conic(u)sin(u)
Set x2=d, y2=conic(u)sin(u)
Click dotted
Set color
Click ok
Next slide shows both segments
184
A2: Polar Conic
• Highlight P in inventory; dupl; delete original
185
A2: Polar Conic
• Graph is done! (Now is a good time to save it)
186
A2: Polar Conic
•
•
•
•
•
Time to play!
The next few slides simulate the playing
You can do better
We show e=1 (parabola) and e<1 (ellipse)
It’s much more fun to watch the graph change
as you move the scroll bar for e
• We wrap up the example by showing how you
can auto-animate P
187
A2: Polar Conic
• Parabola (e=1)
188
A2: Polar Conic
• Ellipse (e<1)
189
A2: Polar Conic
• Hyperbola (e>1)
190
A2: Polar Conic
• Our last illustration with this example is
actually why the example was created
• When the graph is drawn, it is drawn so fast
that you can’t tell the direction drawn
• We get around this by auto-animating u
• Close or move the e and d dialog boxes
• Zoom out (PgDn a few times)
• Close or move the inventory dialog box
• Reset the u value to zero
191
A2: Polar Conic
• Click autocyc
192
A2: Polar Conic
• Click Q to quit, F to speed up, S to slow down
193
A2: Polar Conic
• Do the same auto-animation for a parabola
• Do the same auto-animation for an ellipse
• This completes the polar conic example
194