v r m l . c a l c u l u s . n e t



§1.1 - Experiencing Three Dimensions

by Robert Curtis, Bill Davis, and Lee Wayand.

As the three-dimensional space in which we live is all too familiar to any reader, it is rather amazing that it was not until the 16th century when Renee Descartes presented an obvious way to measure and analyze two- and three-dimensional space.

His idea was to start with a reference point, called the origin, and describe the distances (or units) one would travel in "each direction" to get to the objective point. Of course this is just map coordinates, although the official title is the Cartesian Coordinate System.

All multivariable calculus courses (and sometimes in high school algebra) begin with a description of three dimensions, just to make sure everyone uses the same method of description of three-space. (It wouldn't be good if someone used "y" as the "up" coordinate, for example.)

In mathematics, there are two types of mathematicians, and the difference between them is best described by the answers to the following question: What is the interpretation of the point (2, -4, 3)?

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  • For the Type A mathematician, their response is: "The point (2, -4, 3) requires no further interpretation. This is a point in three-dimensional space." (They may go on and spout a variety of abstract concepts to impress you with their logical prowess.)
  • For the Type G mathematician, their response is the picture above.
Of course this text will be a horror for the Type A mathematician, and hopefully be well received by the Type G mathematician.

Visualization via a CAS and VRML

Until about 1987, the above picture would have been drawn on a chalk board, with a noble attempt and the best intensions of the amateur artist/mathematician. Some calculus teachers could make the chalk sing, and other would make pictures clear only to themselves. The goal of chalkboard drawings was always to give your mind some help in creating an internal picture in your brain of the relationships of the geometric objects.

This was hard/impossible to do for many students. They didn't "see" the same pictures we mathematicians were seeing in our heads.

Around 1987 the desktop computer algebra systems (CAS) in the form of Mathematica, Maple, MathCad, Theorist (now MathView), and MatLab made stronger visualization possible, like the spinning picture above.

This changed the multivariable calculus book as we know it.

If anyone is teaching three-dimensional calculus without using a graphing tool to visualize the concepts, then I hope they are using an abacus, quill and pen, and teaching in latin to complete their classical approach to the subject.

The Progression of Technology in the Visualization of 3D Geometry

The visualization technology ladder may be roughly described as follows: Our current goal is to move from the CAS "flat" tools into VRML visualization, where you may move through the picture, now called a world, and examine relationships as if the world were real and tactile.

The sequel book to this one will be when we may climb into a sensory suit with VR glasses/headset, and walk through the world with not just our eyes. Coming soon: full holographic scanners similar to the holodeck from StarTrek -- they are really just around the corner.

Experience 1 - "Walking The Plot Process"

In the following experience, we are presented with a VRML world along with a CAS flat graph. In this world you may:

Experience 2- "Using Only Geometric Tools to Find Coordinates"

This VRExperience will present to you a point plotted in MathView and VRML, and you be have to determine the coordinates of this point by using the visualization tools. You may check your answer in the given form. There are three points to practice your VR skills on.

Road Map

by Robert Curtis, Bill Davis, and Lee Wayand.

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