Microworld
Title Page:
3 Dimensional Graphics
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Software environments that can bring the player "under the spell" of an imaginary
world they create, and can make the player an active participant in that microworld
can touch and stir the imagination of the player. And if the microworld mirrors
and illuminates the conventions that we teach in the schools, then the player
also becomes a learner, and the author a teacher.
In this way, software environments can provide entirely new learning experiences
and possibilities for students. Mathwright combines a number of features that
have these goals in mind:
Mathscript
combines both decimal and symbolic mathematics with a flexible and expressive
idiom for creating and managing complex and interesting simulations through
user-defined object hierarchies.
Mathematics
(and especially geometry) will always play an important role in constructing
3D simulations, simply because scripts must move the actors correctly. Mathscript
and OpenGL will "do the math" for you at the lowest levels. Mathwright32 provides
a high-level language environment in which you may design your scenes and
interactions abstractly, and from the top down, and then tell the lower levels
of the language what "math" to do in those abstract terms.
We
have not yet learned how to translate the hypnotic allure of video games (Myst,
Riven, even Super Mario Brothers 3!) into effective learning experiences for
mathematics and science, but we know that the key is through interactive simulations.
These have been successful in teaching since the early days, in environments
like Tom Snyder's Halley Project, and the Logo Turtle environments,
and these would never be mistaken for 'edutainment'.
This
Microworld is designed to illustrate how our new OpenGL based 3D Graphics
Objects may be used to support visualization. For Mathwright authors, it provides
a few examples that may guide them in the construction of their own dynamic
3D Microworlds. And for students, there are some utilities that they can use
to visualize a few interesting constructions in geometry. The nine page Microworld
contains the following 7explorations:
The
Microworld features a new capability of Mathwright (available since version
2.10, May 12, 2003) that makes use of Windows Help to give pop-up information
about each page if the reader desires it.
Requires
the Java MathwrightWeb ActiveX Control to read in your Browser.
For
proper viewing, be sure to use Version
2.10 or later, dated May 12, 2003
Download free MathwrightWeb to view Microworlds
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For proper viewing, be sure to use Version 2.10 or later, dated May 12, 2003
Microworld:
3
Dimensional Graphics
Click the Hyperlink above
to visit the Microworld.
Author:
James
White
In
Exploring Conic Sections is a simulation that allows readers to define a plane,
then to view, in three dimensions, the way the plane intersects a standard
cone. The entire scene may be rotated and translated, and you may "fly"
through it in the first person. See the Navigation section below in the Interaction
Instructions. In addition to this, you may view the "Conic Section"
itself as a plane curve. For each plane you define, the 3 dimensional intersection
is mapped isometrically onto a 2 dimensional plane. In this way, you may study
the passage from ellipse to parabola to hyperbola as you vary the plane parameters,
both in 2 dimensions and in 3. In this way, the idea of "Conic section" is
reinforced.
In
Strange Attractors, is a differential equation solver for ordinary differential
equations in 3 dimensionl space. The Lorenz equations are entered when it
comes up, but the reader may change the equation at will. When the reader
solves the Lorenz equations, he sees a "Lorenz Attractor" in R(3). He may
"walk around" the attractor, zoom in on any part, and so on using the built-in
navigational tools (See Below).
Orbit
the Shuttle shows how a realistic 3 dimensional model of a "Space Shuttle"
may be moved along a smooth path by a simple script. In a 3D scene, "actors"
may be moved independently of each other in support of simulations. These
objects may be arbitrary geometry created using our OpenGL interactive commands
(see also the next example) surfaces, space curves, or imported models from
3D Studio or DirectX format. We move the actors using Euclidean (rigid) transformations
based in translations and quaternionic rotations. If this sounds formidable,
we guarantee that is easier by far than the old-fashioned direct matrix manipulation
that one finds in most 3D environments. It is quite simple, as this example,
with its scripts, will show.
In
Quaternions in Motion, we take a closer look at how quaternions are used to
manage rotations of objects in space, and how they interact with translations.
We do not actually use quaternions directly in our scripts. We specify them
by selecting an axis of rotation in space (a vector) and an angle to rotate
about it using the right hand rule. Using OpenGL geometry, we create a "cube"
named "ted" and some other objects also. Then we place ted in a
loop where, for gradually increasing angles: theta, we first translate ted
to a point on the x axis, and then rotate ted through theta about the x axis,
then rotate ted through the same theta about the y axis. The overall effect
is a smooth somersaulting motion that is easy to understand. The net rotation
is obtained by multiplying the corresponding motions (thought of as unit quaternions).
Now the order of unit quaternionic multiplication matters, since the group
is not commutative. So when we put ted through the same paces, but multiply
the corresponding quaternions in the reverse order (rotate ted through
theta about the y axis, then rotate ted through the same theta about the x
axis after the translation) we see an entirely different smooth motion. The
important point is that from the point of view of scripting, the only difference
in the scripts is that we multiplied two objects in the opposite order.
Graphing
Functions is a simple utility that a student may use to draw the graph of
any function: z = f(x,y) that they care to define. These graphs are drawn
in different colors so that their intersections are easy to see and understand.
Again, the aim is to provide visualization here. An author may easily extend
this utility for dedicated purposes, providing for example, a command line
in which arbitrarily many functions may be created, graphed and compared.
Or the author may allow the reader to draw parametric surfaces (i.e. tori,
spheres, immersed Klein bottles, etc.) or to define implicit surfaces defined
by equations of the form f(x,y,z) = 0. Or a reader might simply use the utility
as a general purpose function grapher, and take snapshots of interesting graphs
so obtained. One direction in which this may be extended is in the next and
final example in this Microworld.
Implicit
Surfaces is another utility that a student may use to draw the solution of
an equation: f(x,y,z) = 0. The student defines the function f(x,y,z) by writing
any expression in x,y and z. The system needs two more pieces of information
and is then ready to draw. First, it needs a refinement, a positive integer
between 2 and 39. It divides the original box into a lattice of equal boxes
numbering refinement in each of the three dimensions. Of couse. It also needs
to know the dimensions of the original box. Given that information, the system
creates the surface. The larger the refinement, the more accurately it represents
the surface, and the longer it takes.
In
the Implicit Surface utility, one may define two surfaces. One is colored
red by default, and the other yellow. These surfaces are Actors and may (in
other contexts) be directly manipulated by Mathscript-OpenGL commands, just
as Ted and Ed and the gang were in the "quaternions in motion: example.
You may also edit their scripts to change their properties, such as size,
location and orientation, but you will have to read the online Mathscript
help at the Library to learn how to do that.
Gravitational
Potential Wells extends the function grapher in an interesting direction.
In this exercise, the reader defines a gravitational potential function. The
graph is drawn of course. But such a function defines, by its gradient, a
second order differential equation -- the equation of motion of a point that
is allowed to move without friction on the surface under the influence of
gravity alone. The reader sets the initial conditions (initial position of
the point on the surface, and initial velocity) and may then watch the simulation
as the point moves under the influence of gravity, illustrating in its smooth
motion the principle of conservation of energy.
Author: James E. White
Author Email:mathwrig@gte.net
Topics: Lorenz attractor, ODEs in three space, conic sections, second order differential equations in the plane, chaos, tutorial on quaternionic motion, simulations with 3D models
Suggested Use: Visualization of various geometric constructions
Number of Pages: 9
Animation:Yes
Grade Level:10-16
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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