parametric 3d plotting with ti- nspire cas technology

Parametric 3D Plotting with TI-Nspire CAS

TechnologyMichel BeaudinÉTS, Montréal, Canada

USACAS 8 International Conference

Philadelphia, PA, USAMarch 8-10, 2013

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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.

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.3

About ETS: our Students, our Tools

• 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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About ETS: Our Tools

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Compared to Voyage 200

What’s New with TI-Nspire CAS?

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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.

Compared to Voyage 200

What’s New with TI-Nspire CAS?

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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:

2 2 16x y

3 2 2 33 16x y x y

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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

2 2 16x y

3 2 2 33 16x y x y

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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) = typ1(t, u) = 2 t zp1(t, u) = u

22 3 1

x yz x y

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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.

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)

2 2( 1) ( 2) 4y z

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Some Examples of Parametric 3D Plotting

We want a nice plot of the sphere

We need to use spherical coordinates:

2 2 2 25x y z

5cos( )sin( )

0 25sin( )sin( ) " and "

05cos( )

x t ut

y t u t uu

z u

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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).

cos( )sin( ) 4 4

3

x ty t t

z t

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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

2 2 2( 1) ( 3) 16x y z 2 2x y z

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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:

2 542 4x xz

22 5 6

4yz x

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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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Thank You!