USACAS 8
𝐓 𝟑 International Conference
Philadelphia, PA, USA
March 8-10, 2013
Overview
 About ETS: our students, our tools.
 What’s New with TI-Npsire CAS?
 Different Objects in the Same 2D Plot
Window.
 Why is Parametric 3D Plotting so
Important?
 Some Examples of Parametric 3D
Plotting.
 Conclusion.
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About ETS: our Students, our Tools
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Engineering school in Montréal, Québec, Canada.
Our students come from college technical programs.
“Engineering for Industry”.
More than 7000 students, 1700 new students each year.
All math teachers and students have the same calculator
and textbook.
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About ETS: Our Tools
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1999: TI-92 Plus CAS handheld.
2002 : TI Voyage 200.
2011 : TI-nspire CX CAS.
Different softwares (Derive, Maple,
Matlab, DPGraph, Geogebra).
 Only CAS calculators are allowed
during exams.
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What’s New with TI-Nspire CAS?
Compared to Voyage 200
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What’s New with TI-Nspire CAS?
Compared to Voyage 200
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2 platforms
managing documents
list and spreadsheet
faster processor
(better for solving, Taylor polynomials, special functions, …)
• some CAS improvements
• new graphical capabilities
• animations : powerful tool for teaching
• interactive geometry : « experimental mathematics »
• multiple 2D plots window
(functions, parametric, scatter plot, etc.)
• 3D parametric surfaces and curves (OS 3.2)
Today, we will mainly focus on parametric 3D plotting.
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Different Objects in the Same 2D Plot
Window
 (In OS 3.2) The 2D plot window graph
Entry/Edit accepts up to 7 different types
but 2D implicit plots are still missing.
 Slider bars, animations, dynamic
geometry, styles and colors make each of
these 2D plot windows very attractive
and useful for teaching mathematics and
sciences.
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Different Objects in the Same 2D Plot
Window
 Slider bars/animations, dynamic
geometry and calculus are now used by
teachers for showing many concepts:
this was never done before. Here is a
simple example.
 Two objects are moving in the plane,
with respective given position at time t.
When will the distance between them be
the smallest?
 The next slide shows the situation.
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Different Objects in the Same 2D Plot
Window
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Different Objects in the Same 2D Plot
Window
 A word on implicit plotting: it is a
must for multiple variable calculus
(but probably not so important for
college level).
 Here is a trick to “fake” implicit 2D
plotting in Nspire CAS.
 To plot the graph of f(x, y) = 0, use
f1(x) = zeros(f(x, y), y). Of course, the
equation should be solvable for y.
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Different Objects in the Same 2D Plot
Window
 Moreover, the “complex” format must
be selected.
 Here are some examples.
 Let us take a look at this circle:
 And at this cubic polynomial equation:
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Different Objects in the Same 2D Plot
Window
 Let us show, using Nspire CAS, how
to plot the circle
using different 2D plot windows.
 And let’s also show how to plot the
implicit curve
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Why is Parametric 3D Plotting so Important?
 First of all, where do I find the 3D plot
window? And what can I plot?
 How can I plot a surface defined by an
equation of the form x = f(y, z) or by
an equation of the form y = f(x, z)?
 How can I plot a circular cylinder? A
sphere?
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Why is Parametric 3D Plotting so Important?
 For the sphere, the “trick” using
“zeros” can be used but the graph
won’t be attractive…
 Since OS 3.2, a new 3D plot type is
available: PARAMETRIC.
 One or 2 parameters? A space curve
or a surface?
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Why is Parametric 3D Plotting so Important?
 Parametric 3D plotting is important
because:
 It yields nice graphs!
 We need to use mathematics in order
to get something.
 This shows students the importance of
some trig identities, some formulae,
some techniques.
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Why is Parametric 3D Plotting so Important?
 Parametric 3D plotting is important
but students will never master it if they
never use it by themselves!
 This is why, at ETS, we want our
students to use computer algebra in the
classroom (with their handheld).
 Of course, 3D plotting is easier on the
software version. But students can
learn the basics with the handheld.
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Some Examples of Parametric 3D Plotting
 Planes: the general equation is of the
form Ax+ By + Cz = D.
 If we can solve for z, we can use the
“classic” editor “z1(x, y) = ”. If C = 0,
then parametric plotting is used: x can
be t, z can be u and y depends on x (t).
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Some Examples of Parametric 3D Plotting
 We will show the graph of the 2
intersecting planes:
 The first plane can be defined as:
xp1(t, u) = t
yp1(t, u) = 2  t
zp1(t, u) = u
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Some Examples of Parametric 3D Plotting
 The second can also be described in
parametric form (replacing x by t and y
by u) or we can use the standard editor
z1(x, y) = 2x  3y + 1.
 Here is the result.
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Some Examples of Parametric 3D Plotting
 To plot a circular cylinder like
 The first trig identity can be used:
xp1(t, u) = u
yp1(t, u) =  1+ 2cos(t)
zp1(t, u) =  2+ 2sin(t)
Some Examples of Parametric 3D Plotting
 We want a nice plot of the sphere
 We need to use spherical coordinates:
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Some Examples of parametric 3D plotting
 Students can produce animations: a
“bumpy sphere” for example:
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Some Examples of Parametric 3D Plotting
 Here are some plots of space curves.
 Suppose an object moves along the
helix
 Here are the helix and the position
vector (a line segment).
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Some Examples of Parametric 3D Plotting
 Let us generate a torus: we will rotate
a circle around the z-axis (a “rotation
matrix” will be used).
 The circle is centered at (3, 0, 0), in
the plane y = 0, and has a radius of 1.
 We should obtain this:
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Some Examples of Parametric 3D Plotting
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Some Examples of Parametric 3D Plotting
 Let us obtain a space curve by
intersecting 2 surfaces, a sphere and a
plane.
 The sphere will be
and the plane will be
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Some Examples of Parametric 3D Plotting
 We will use the (very useful) function
“Complete the Square”.
 Moreover, students need to understand
what an equation like “f(x, y) = 0”
means in 2D and in 3D.
 We should obtain this:
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Some Examples of Parametric 3D Plotting
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Some Examples of Parametric 3D Plotting
 Let us obtain a space curve by
intersecting 2 surfaces (no need of
parametric plotting for the surfaces).
 The surfaces will be a parabolic
cylinder and a paraboloïd:
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Some Examples of Parametric 3D Plotting
 Before Nspire CAS, all of these last
examples were done (by me) using
Derive (and students could not do it
themselves before Nspire CAS CX).
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Conclusion
 When “implicit 3D plotting” will be
available, it will be easier to plot any
surface defined by an equation of the
form f(x, y, z) = 0.
 But this is not so urgent because many
important surfaces can be
parametrized.
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Conclusion
 Also, when time comes to solve
optimization problems (2 variables),
the functions we are dealing with are
of the form z = f(x, y).
 Since OS 3.2, my students are much
more familiar with cylindrical and
spherical coordinates than before.
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Conclusion
 They plot more 3D graphs than before
and are proud to do it themselves.
 Mathematics ─ especially for future
engineers ─ is a tool they will always
need; and if it can have a little taste of
“experimental sciences”, they don’t
disagree!
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Conclusion
 The 3D plot window of Nspire CAS
represents a big progress compared to
what we used to have with V200.
 But, it would be nice if TI could add
the possibility to plot space curves
with different styles (line weight: thin,
medium, thick as we have in 2D).
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Conclusion
 Also, if someone plots a point in space
(using parametric plotting), the point
is almost invisible!
 Unless you use the trick of plotting a
“small” sphere!
 But, in any case, this does not affect
the pleasure we have using Nspire
CAS!!!
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Conclusion
 Website for Nspire CAS at ETS (in
French):
http://www.seg.etsmtl.ca/nspire/
 Many documents on using Nspire
CAS in teaching can be found.
 My personal homepage:
https://cours.etsmtl.ca/seg/MBEAUDIN/
 A library of functions is useful for
engineering mathematics.
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