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Huh? A Model of Space, Infinity and Flow |
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Working Draft Copyright (c) 2005 - 2007 Jim Imboden |
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Chapter 15 How to Create a 3D Model Using Derive
To really get a feel of this
model you need to watch it in motion on a computer. The program
I used was Texas Instruments’ Derive 6.1 Note: Texas Instruments no longer sells the program. I’m certain other programs like Mathmatica will work.
What follows are instructions to get the Huh model functioning in Derive 6.1. If you are using another program you should note the settings of s: and t: as any program will have something similar when plotting.
Instructions: Run the setup program to install Derive on your computer.
After the installation run Derive and will see:
Figure 15-1
Click on yes to use the Default Settings
Figure 15-2
In Figure 15-2 notice I Have made a small change, I’ve added ‘s *’ after the ‘+’ sign and the equation we will be using is now y = a^3/(a^2 + s * (x + d)^2). The ‘s *’ will give us the ability to change from ‘compressed’ to ‘released’ mode without the need to keep changing the equation.
Once you have entered the equation press Enter and you will see the #1 equation above and also the equation will be highlighted on the input line.
Figure 15-3 Click on Edit, Copy to place the equation on the clipboard (see Figure 15-3)
Figure 15-4
Next, click on Insert, 3D-plot Object as shown in Figure 15-4. This creates a place to plot the model, see figure 15-5
Figure 15-5
The next thing we need is a way to change the variables we are using ‘a’, ’d’ and ‘s’. Derive has Slider Bars for this purpose, to create these slider bars; on the menu select Insert, Slider Bar as shown in Figure 15-6.
Figure 15-6
You will see a small Window for Slider Bar Properties:
Figure 15-7
The Slider Bar Properties will have default values in the settings; we need to change these to match what we will be doing. Figure 15-7 shows the settings for variable ‘a’, replace the default values with the values shown in Figure 15-7. Then click on OK and you will see a slider bar on the screen.
We need to Insert two more slider bars for the variables ‘s’ and ‘d’.
Figure 15-8
Figure 15-8 shows the values to put into the two new slider bars.
Next we need to set the Coordinate System for our plot. To do this select Set, Coordinate System on the menu as shown in Figure 15-9.
15-9
When you select the Set, Coordinate System option you will see a small window with three different coordinate systems as shown in Figure 15-10. Select Spherical.
Figure 15-10
We can now insert our plot of the model, to do this select Insert, Plot from the menu as shown in Figure 15-11.
Figure 15-11
You should now see a smaller window as shown in Figure 15-12.
15-12
The ‘s:’ values (Minimum, Maximum) should be units of Pi like described in chapter 2, and usually the minimum ‘s:’ value will be the same as the maximum, only minus. You can type numbers in for the minimum and maximum or use units of pi like pi, 2pi, 5pi, -4pi or 1.5pi.
The ‘t:’ values are the latitude values and are usually specified in pi units.
The Number of Panels is basically the resolution of the plot. The bigger the number the finer the resolution and the smaller the more course it will be.
Don’t worry about making mistakes because it is easy to come back and change these values.
When finished click on the ‘Next’ button and you will see the window shown in Figure 15-13
Figure 15-13
This window allows you to select the colors used on the plots. I usually select ‘Custom’ on the drop down box and set the colors to: Top – Minimum - Dark Blue and Maximum – Light Blue Bottom Minimum – Orange and Maximum – Red Click Finish to view the plot.
Figure 15-14 Figure 15-14 shows our plot, but it doesn’t look like anything we’ve seen. We need to tune it in a little. The biggest problem is we have is ‘s’ or the sign set to -1 and this means our plot will be in the released or relaxed mode. If we move the ‘s’ slider from ‘-1’ to ‘1’ will put the model in a compressed mode as shown in Figure 15-15. I also changed ‘a’ or the size from -10 to 4. The reason was to keep the same viewing angle.
Figure 15-15
Now this looks much better. If I wanted to change the viewing angle I could change ‘a’ from ‘4’ to ‘-4’ as shown in Figure 15-16.
Figure 15-16 You can also remove the Box, Legend and change Background colors by clicking on ‘Options, Display’ and making the changes shown in Figure 15-17.
Figure 15-17
The one checkbox you need to make sure and uncheck is on the Display Options, Color window. The program defaults to automatically changing colors on new plots and you will find yourself constantly needing to undo these changes. Unchecking the box on the Color options corrects this nuisance.
Figure 5-18
The Slider bars we added shown in Figure 5-18 allow you to manipulate the variables in the model. I should warn you that these variables are very interactive; changing a variable by one number can change the look and ability to display the conditions.
The following is an explanation of what each does and what can happen when changing them.
Slider Bar ‘a’ This controls the size of ‘a in our basic y=a^3/(a^2+x^2). I look at it as the amount of space we have to insert amounts of ‘x’ into that space. In the compression mode (where s=+1) it is the volume of a container with sides of the length ‘a’.
In the released or relaxed mode (where s=-1) it is the space needed to hold what was in the compressed container except it is not compressed and doesn’t include the container ‘a^3’, it is outside of it. The space needed is #units of pi * pi.
Slider Bar ‘s’ This sets the mode of the model to be either ‘compressed’ or ‘released’ in numeric terms compressed equals ‘+1’ and released equals ‘-1’. (See chapter 8 on the magnetosphere).
The area needed to plot the model if s=-1 can be extremely large and depending on your viewing angle you may see nothing or a real mess.
Slider Bar ‘d’ This is spacetime distance, when set to zero there is no spacetime effect calculated into the model. The bigger ‘d’ is the greater the spacetime effect.
Figure 15-19 shows a few views that can be generated and if you click on a view you will see the information for ‘s:’, ‘t:’ and variables ‘a,s,d’ using the sliders.
Figure 15-19 17 different views of the Huh model. Click on a view to see the parameters that created it.
Some plots also have a Length parameter given; this is the size of the box it is plotted in. You can change the box size by clicking on the Zoom In and Zoom Out buttons to the left of the Auto Rotate button.
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