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GRAPH User's Manual Graph V3.1 - An experiment in digital waveform synthesis. Authored by: Pacific Laser Art; President: Thomas C. Poff #9 Tunzi Parkway Petaluma, CA 94952 Release date : 18 September 1989 Version 3.1 (c) 1989 by Pacific Laser Art Permission is granted to copy this document and related software. GRAPH User's Manual Page 2 "GRAPH" is a program designed to allow you, the user, to generate a myriad of two-dimensional shapes. By plotting two individual waveforms on a cartesian, (or two dimensional) plane, you will be able to create a variety of beautiful shapes. Most of the shapes that this program can produce are by nature abstract. However, with some practice it is possible to gain a talent for creating shapes that are easily recognizable to us as familiar objects. We much prefer the abstract shapes, personally, and for this reason, the program has been developed from this standpoint. In the next several pages, We will discuss the varying levels of sophistication that have been written into "Graph." We will start by explaining the simplest functions and will gradually introduce you to it's fancier accouterments! "Graph" has been written to be one of the easiest of it's kind to operate due to it's unique user interface. We hope that you will find agreeable! Before we begin the first lesson, it is necessary to bring the following information to your attention. First of all, whether or not you like the program, we would very much like to hear your thoughts on it. Your input is needed! The authors are anxious to find ways of improving the existing software as well as incorporating new ideas into it! If you do like the program, we would appreciate receiving a registration fee from you. The registration cost for the first two versions of "GRAPH" is $25 and will provide you with the following items: (1) A 100% free update to the next major update of GRAPH (with updated user documentation on the disk). (2) A bi-annual newsletter with tips about how to make the most of programs published by Pacific Laser Art. (3) Notification of new software releases. (See the appendix for upcoming developments.) Whew, that's enough of the sales pitch! Please write to us, regardless of whether or not you wish to become registered. If nothing else, send a postcard with a one-liner with your strongest impression about "Graph." Registrations and suggesstions should be sent to: Pacific Laser Art #9 Tunzi Parkway Petaluma, CA 94952 GRAPH User's Manual Page 3 Hardware Recommendations: The first action you will want to take with your new program will be to make sure that it will run correctly on your equipment. Of particular concern is the use of display adapters. Any of the following displays will function well with "Graph." 1) All types of EGA adapters, (including monochrome EGA), will be perfect for executing GRAPH. 2) VGA and MCGA adapters will work equally well. 3) Believe it or not, this program will also work well with the Hercules Graphics adapters and the AT&T400 display adapters. The CGA displays will not function with "Graph" as they simply do not have the amount of resolution required to operate properly. Of secondary importance to you as a user will be the speed of the machine that you are using. Although it is unknown to many users of personal computers, a math coprocessor can be most crucial in terms of performance. Six megahertz PCs can outperform sixteen megahertz machines when a coprocessor is present. This is common with programs like GRAPH that must perform a great deal of floating point arithmetic. For this reason, I recommend that you look first for an available machine with a math coprocessor, before seeking sheer microprocessor speed. If you do have a slower machine, that is okay too. You will find, however, that some really high-resolution plots can take hours instead of minutes on the slower 8088 machines. To be fair, I should note that most of your waveforms will take only seconds to plot. Lastly, there must exist 512K of RAM on the motherboard of the computer in which you are to use. "Getting down to business" Now that you are aware of the necessary hardware to use with GRAPH, you can begin our first "session" with the program itself. To begin, insert your diskette in the computer. If you look at the directory, you will find several executable files. For the moment, we will only look at the most important one. At the dos prompt, type in the word GRAPH and press the <Enter> key. In a few seconds, a happy but slightly intimidating screen with some text will appear. Now then, at the upper left hand corner you will please notice a large green box. The box has a horizontal dotted-blue line in the middle. You will be able to plot individual waveforms in this area. GRAPH User's Manual Page 4 On the lower right hand side, you will notice that there exists a series of four small boxes containing text. The boxes are used for choosing basic trigonometric waveforms, their durations and amplitudes. In fact, each one of these boxes makes up one user controlled "oscillator." People have pointed out that GRAPH looks, (and acts), a lot like a video game, because the "controls" are easy to operate and the program in itself is extremely interactive. At the bottom of the screen you will see two message boxes. The first message tells you how to get a list of basic commands, while the second mentions that it is possible to leave the program by pressing ctrl-x. I mention this now, as you will soon see that when you give the program invalid input, the resulting error message will scroll the current messages from the screen. The message area, you see, is set up to operate as a queue. At this point, you will want to act upon one of these messages! If you are a wise-guy (person), and you choose the ctrl-x option, just remember that you'll have start to over! All joking aside, you should now press the 'L' key. This will provide you with a list of available commands which will be your key to operating GRAPH until you become familiar with it. You may get to the list of commands at any time, except during a plot, or when you are entering data from the keyboard. Lesson 1: Plotting simple waveforms. Section 1.1 - Sine wave plot. Our first objective will be to plot a sinewave on the display. To create this sinewave, we will do the following exercise: Exercise: Press the key on your keyboard after || reading the associated comments. || || || || || \/ \/ < 1 > : This keystroke ensures that our current "oscillator" will be designated as oscillator #1. <F1 > : This keystroke will mark the sinewave function for plotting in the "Oscillator #1" box. A small "blurb" should appear in the sinewave oval as soon as the <F1> key is pressed. If it does not appear, try pressing the <F1> key several times until you get a feeling for how waveforms are enabled. Note that the <F1> key acts as a "toggle" switch to turn the GRAPH User's Manual Page 5 waveform on or off. <F10> : When you have successfully enabled the sinewave oscillator, push the <F10> key to plot the waveform. A beautiful sinewave of length one cycle should now grace your computer screen. <ESC> : When you are through admiring your first waveform, clear the waveform display box by pressing the Escape key. <F2 > : Now press <F2>, <F3>, <F4>, <F5> and <F6> in order to enable <F3 > : each of the trigonometric waveforms in oscillator #1. <F4 > (These keys work in the same manner as the <F1> key, but <F5 > for cosine, cotangent, cosecant, secant, & cotangent <F6 > respectively.) <F10> : Please press the <F10> key to plot all six basic trigonometric waveforms simultaneously. Pretty neat, huh? <ESC> : Now, press the Escape key to clear the waveform display box. < D > : Press <D> to reset all oscillator values to their original state. This returns all oscillator parameters to the original settings present when you first ran the program. Section 1.2: Plotting a sine wave with different AMPLITUDES & CYCLES. In this lesson, we will learn how to plot waveforms with amplitudes and cycles specified by the user. Let us now plot the following waveforms: Exercise: <F1 > : Press the <F1> key to enable the sine wave. < A > : Press the <A> key to change the amplitude of the sine wave. When you press the key, you will see a cursor at the right hand side of the screen just beckoning to you for data-input. Please enter the number 100 on your keyboard and press <Enter>. <F10> : Plot the waveform. Notice how the sinewave is larger from top to bottom than the last time that we plotted it. < A > : Next, change the <A>mplitude again, but enter the number 20 on your keyboard and press <Enter> this time. < C > : Now were cooking! Press the <C> key and enter the number 5 followed by the <Enter> key. This will tell the computer that you want a sinewave with a duration of 5 cycles to be plotted on the screen. <F10> : Plot the waveform. Now you will see a new small sinewave that goes up and down several times. The waveform is short, because we changed the amplitude to a smaller value (20). It "oscillates," because we have told it to cycle up and down GRAPH User's Manual Page 6 five times. Can you dig it? <ESC> : Clear the screen. < D > : Reset oscillators back to the default values again. Before you pushed escape, you should have had two waveforms displayed in the waveform display box. If you did not, you should probably go back to the the beginning of the lesson and start again. Don't be discouraged. As you may know, sometimes it can take a little time to get used to operating a new program. Section 1.3: Controlling the PHASE SHIFT of a trigonometric waveform. The last of the individual oscillator control options is the phase shift offset control. If you are familiar with trigonometry, you will remember that every sinewave has a phaseshift that can range from 0PI to 2PI. Since it is most important to think in terms of PI for this parameter, we will specify a number between 0 and 2 upon input. To see how this affects a cosine wave, perform the following operations: Exercise: <F3 > : Enable the cosine waveform. <F10> : Plot the cosine waveform. < S > : Change the phase <S>hift by entering the <S> key. Type in the number 0.1 and push the <Enter> key. <F10> : Plot the phaseshifted cosine waveform. Note that it's position has shifted slightly to the right. < S > : Now Change the phase <S>hift to 0.2 <F10> : Plot the phaseshifted cosine waveform. <ESC> : Clear the screen. < D > : Reinitialize default values. If you keep incrementing the phase <S>hift value in 0.1 increments, you will eventually approach 2PI. Note that this parameter is entered in terms of PI. Another way of looking at this is to say that we can increment the phaseshift of the waveform by one-tenth of PI, (.1 PI = 1/10 PI = ), until we reached 2PI. When we reached the 2PI phase shift offset, we will noticed that the waveform was write back where it started from. If you wish to see this, try repeating the exercise and keep incrementing the phaseshift offset to 0.3, 0.4, 0.5... until you approach 2. In the last several pages we have covered the concepts of oscillator control. If you did not feel comfortable with the exercises you have performed, you should probably review the exercises again. Lesson 2: Waveform RESOLUTION/ Producing a UNIT CIRCLE. GRAPH User's Manual Page 7 In the second lesson, you will discover how to modulate one waveform with another. That is, you will be taught how to take one waveform, such as a sine wave, and sum it with another waveform, perhaps a cosine wave. Before we begin the explanation of how to modulate waveforms, however, you will first be shown how to alter the resolution of any plotted waveform. Section 2.1 - Controlling the resolution of a waveform. You may have noticed that when you choose to plot a waveform at two cycles, there are no apparent breaks or separations in the waveform itself. However, if you plot a waveform at ten cycles, it is a different matter. At twenty-five cycles, the waveform becomes unrecognizable. Try the following exercise to demonstrate this aspect of the program: Exercise A: What do you think will happen? <F1 > : Enable the sinewave <F10> : Plot the waveform at 1 complete cycle. < C > : Change the waveform cycle count to 10. <F10> : Plot the waveform at 10 complete cycles. <ESC> : Clear screen. < C > : Change the waveform cycle count to 25. <F10> : Plot the waveform at 25 complete cycles. <ESC> : Clear screen. < D > : Reinitialize default values. Can you see how the waveform becomes disconnected as the cycle count increases? If you cannot, please try Exercise A again! This behavior occurs because the waveforms are forced to have the same number of "samples" per plot. This means that no matter what oscillator(s) you assign, the waveform will always be plotted with the same number of elements in terms of resolution. So, if you plot a sine curve at one cycle as opposed to a curve at ten cycles, the same number of dots are plotted on the screen. In order to acheive a more continuous waveform at higher cycle settings, we must increase the resolution. This is done by changing the resolution posted at the middle right hand section of the screen. Let's try it, shall we: GRAPH User's Manual Page 8 Exercise B: <F1 > : Enable the sinewave < C > : Change the waveform cycle count to 10. <F10> : Plot the waveform at 10 complete cycles. < R > : Change the waveform resolution to 0.25 <F10> : Replot the waveform. <ESC> : Clear the screen ( Do not reinitialize default values! ) If all went well, you should be able to see how the new waveform is more continuous than it was before. Note that in order to increase the resolution, we had to decrease the input value (from 1.5 to 0.25) The advantage with greater resolution, is that we can see more of the waveform. If you increase the number of cycles, you must certaintly beware of the resolution setting and readjust it accordingly. The drawback to increasing the resolution of a waveform is that it takes more time to plot than a waveform plotted at low resolution. This fact will become more apparent as we venture into modulated waveforms. Section 2.2 - The infamous unit circle. Before we get into an explanation of the "heavier" part of this program, let us digress to something easy to understand. Some of you may have been wondering whether or not you were ever going to see a unit circle. The answer is most certaintly, "Yes!" It is not a terribly useful option, but since it can be a comforting thing to see a unit-circle, it has been included in the program. Using the previous settings, perform the following task: Exercise: < R > : Change <R>esolution to 0.1 : Enable the unitcircle grid. <F10> : Plot the waveform. (You can press <ESC> to stop the plotting in "mid-stream.") <ESC> : Clear the screen. : Disable the unitcircle grid. < D > : Reinitialize default values. As you can see, as the waveform cycles from 0 to 2PI, a pointer moves around the unit circle accordingly. It should be noted that this option slows down the plotting of the waveform and thus should be executed with care during plots at high resolution. GRAPH User's Manual Page 9 Lesson 3: WAVEFORM MODULATION. This section will tell you about how to obtain modulated or complex waveforms. For those of you in the dark, a modulated/complex waveform is a waveform that consists of several simple waveforms that are mixed together in one way or another. An example of a complex waveform could be two sinewaves that are added to each other. The sinewaves could very well be different in terms of the number of cycles within a given interval. Their amplitudes could also differ from one another. You can also mix waveforms with different phaseshift offset, as we'll discuss in the theoretical example below. Section 3.1 - A theoretical look at simple and complex sinewaves. To help you imagine this "mixing" of waveforms together, you might note the following example. Some children take great pleasure in hitting long pipes planted on playgrounds with their hands. This causes the pipe to fiercly vibrate. You may have done this yourself when you were a child. Now take that example one step further. Imagine a very long metal pipe with its two ends planted in the ground. These are often found at school playgrounds. Now then, imagine that you have a baseball bat. If you were to pound on the center of the pipe, you would see it oscillate back and forth. Through this action, you would create a pair of sinewaves that travel back and forth through the pipe. If you were to inspect the pipe from several careful angles, you might even be able to see or feel the oscillations moving through the pipe. Finally, if you really got into and started hitting the pipe repeatedly with the bat, you would notice that afterwards, when you touched the pipe, it would oscillate in a seemingly erratic manner. In actuality, these oscillations would represent a complex waveform composed of a lot of sinewaves traveling at the same rate of oscillation, but at a different phaseshift settings. Note: Please do not try this experiment at home! Section 3.2 - Modulating one waveform with another. Ahhh, finally what we've been patiently waiting for. This is the challenge that separates the alligators from the crocodiles! Seriously, it's not that difficult, but you may need to take some time covering this material and repeating various exercises. Make sure that you have reinitialized the default values. All waveforms should be disabled (or turned off) before performing the following instructions. GRAPH User's Manual Page 10 Exercise A: < R > : Change <R>esolution to 1.5 < 1 > : Assign oscillator #1. <F1 > : Enable the sinewave in oscillator #1. < 2 > : Assign oscillator #2. (The oscillator pointer will move.) <F1 > : Enable the sinewave in oscillator #2. < C > : Change the cycle count in oscillator #2 to 3. < M > : Enter the following formula EXACTLY: (O1+O2) Press <Enter> to enter this formula. Make sure that you typed in O's and NOT zeroes. This is most important. Zeroes will not work! This command tells the program that you want to add the first Oscillator to the second Oscillator during the plotting of the waveforms. This is how you specify specific waveforms that you want to "mix" together while the waveforms are being plotted. <F7 > : Enable modulation mode so that "Graph" will know to modulate the waveforms specified in the modulation formula. <F10> : Plot the waveforms. When you hit the <F10> key, three waveforms should appear on the screen. Two of the waveforms are your component waveforms, and the other is the modulated waveform. In The modulated waveform represents the "sum" of the two other waveforms. Let's try another one: < R > : Change <R>esolution to 0.5 < 1 > : Assign oscillator #1. < C > : Change cycle count to 12. <F10> : Plot the waveform. <ESC> : Clear screen. Pretty darn neat huh! Now, you might have noticed that things are getting pretty crowded now. In order to alleviate this problem, we can keep the component waveforms from displayingon the screen during a plot: <F9 > : Specify only the modulation waveform. <F10> : Replot the modulation waveform only. <ESC> : Clear screen < D > : Reinitialize default parameters. It should be stated that <F7> and <F9> act as toggle switches. Thus, if you press them once, they become active, and if you press them again, they become inactive. GRAPH User's Manual Page 11 Exercise B: Plot a complex waveform consisting of four component waveforms. < R > : Change <R>esolution to 1. < 1 > : Specify Oscillator #1. <F1 > : Specify sinewave. < 2 > : Specify Oscillator #2. <F1 > : Assign sinewave. < C > : Set cycles to 2. < 3 > : Specify Oscillator #3. <F1 > : Assign sinewave. < C > : Set cycles to 4. < 4 > : Specify Oscillator #4. <F1 > : Assign sinewave. < C > : Set cycles to 8. < M > : The old modulation formula was only good for two waveforms. Try the following one to mix all four waveforms together: (O1+O2+O3+O4)/4 Remember to type in "Os" and NOT "0s" in the modulation formula. This new formula will sum the four waveforms to and divide the total by four to create the new complex waveform. <F7 > : Enable waveform modulation. <F10> : Plot the component and modulation waveforms. <ESC> : Clear the screen. <F9 > : Disable component waveforms. <F10> : Plot modulation waveform only. <ESC> : Clear screen < D > : Reinitialize default parameters. You may have noticed that with all four waveforms assigned, the graphing does become slower. You can of course try other modulation formulae, such as: < M > : (O1+O2+O3)/3 ( This adds oscillators 1,2 & 3 together and divides the result by three. ) < M > : (O1+O2)/2 ( This adds oscillators 1 & 2 together and divides the result by two. ) < M > : (O1) ( This plots only the sum of all waveforms enabled in oscillator #1. ) < M > : (O1+O3)/5 ( This adds oscillators 1 & 3 together & divides the result by five. ) GRAPH User's Manual Page 12 The modulation formula decoding is kept quite simple in order to increase graphing performance in terms of time. The tradeoff is that there is very little error checking in this option. You should avoid the following kinds of mistakes when creating waveform modulation formulae: 1) Do not try to add an oscillator where no waveforms are enabled. 2) Do not try to divide a set of waveforms [e.g. (O1+O3) ] by a number exceeding nine. 3) Do not try to multiply or divide oscillators. Performing any one of the above "no-no's" will cause the program to ignore you when you hit the plot key. It will try to plot, but will find an inconsistency in the modulation waveform, causing nothing to be plotted. Hint: if you are having problems with a waveform that won't plot, check to make sure that you have at least one waveforms enabled in each oscillator that appears in the modulation formula! Section 3.3 - Complex waveforms constructed from one oscillator. Thus far, we have specified only one waveform per oscillator in the complex waveform plots. However, we can easily choose any number of the basic waveforms included in each oscillator. To illustrate this, you may wish to perform the following exercise: Exercise: <F1 > : Enable the sinewave and cosecant function in oscillator #1. <F2 > < M > : Sum the enabled waveforms within oscillator #1 by the formula: (O1) <F7 > : Enable waveform modulation. <F10> : Plot the modulated waveform. Different, isn't it? < A > : Change the amplitude to 10. <F10> : Replot the modulated waveform. <F3 > : Enable all other waveforms in the oscillator. <F4 > <F5 > <F6 > <F9 > : Display the modulation waveform only. ( optional ) <ESC> : Clear the screen. GRAPH User's Manual Page 13 <F10> : Plot the waveform again. <ESC> : Clear screen < D > : Reinitialize default parameters. Note that this exercise modulates off of one oscillator only. If you wish, you may perform this exercise with all 4 oscillators combined. This will give you a total of 24 waveforms combined in the final modulated waveform! You will need to keep the amplitude values small in order to see the modulated waveform. Section 3.4 - Multiple plots Multiple plots can useful as well as beautiful. By changing values phase shift values in one or more oscillators, you can have a group of different waveforms on the screen at one time. The waveforms can be related to each other in a visual manner. To illustrate this phenomena, please try the following exercise: Exercise A: <F3 > : Enable the cosine in oscillator #1. < 2 > : Assign oscillator #2. <F3 > : Enable the cosine. < C > : Change the cycle count to 2. < 3 > : Assign oscillator #3. <F3 > : Enable the cosine. < C > : Change the cycle count to 3. < 4 > : Assign oscillator #4. <F3 > : Enable the cosine. < C > : Change the cycle count to 4. < M > : Enter the following modulation formula: (O1+O2+O3+O4) <F7 > : Allow waveform modulation. <F9 > : Display modulation waveform only. <F10> : Plot the waveform. < 2 > : Assign oscillator #2. < S > : Change the phaseshift value to 0.1 <F10> : Plot the waveform. < S > : Change the phaseshift value to 0.2 <F10> : Plot the waveform. < S > : Change the phaseshift value to 0.3 <F10> : Plot the waveform. GRAPH User's Manual Page 14 < S > : Change the phaseshift value to 0.4 <F10> : Plot the waveform. ------> Keep incrementing the phaseshift value by 0.1 and plotting the waveform until after you have plotted the waveform with the value 1.9 as the phaseshift offset. This is a sample of the kind of graphing patterns that you can produce with GRAPH. With a little practice and experimenting, you can create a wide variety of complex graphs without ever having to know the equations behind them. When you are done, clear the screen and reset defaults as we have done before: <ESC> : Clear screen < D > : Reinitialize default parameters. Lesson 4 - TWO-DIMENSIONAL waveshaping. Arguably the most interesting aspect of GRAPH is it's capability to create an array of two-dimensional objects. This is really what the program is designed to do best. We will start with a simple circle and quickly work up to more complex shapes. If you feel fairly comfortable with Lesson Three, you will not probably not have any problems with this section. If you only did marginally well in the last section, you should probably go back and perform the exercises in Lesson Three again. If you do go back to repeat those exercises, try substituting some of your own values. The secret behind this program is to be creative, not necessarily mathematical. You should develop a feeling for what complex waveforms that might be neat with some practice. Section 4.1 - A second bank of oscillators. Two dimensional objects are created by plotting two "equations" against each other. You can think of an "equation" as being the represented by any complex or modulated waveform that we studied in Lesson Three. We already have access to the one modulated waveform. In order for us to get the second dimension, we must have access to another, completely independent complex waveform. To reiterate, two dimensional objects can be created by plotting two complex waveforms against each other on a cartesian plane (or x,y axis). In other words, one of the complex waveforms is plotted on the x axis, whilst the other is plotted on the y axis. GRAPH User's Manual Page 15 This can be a difficult concept to accept for some. We are used to simply drawing objects with our pencils, or if we are on a computer, some of us use a "mouse" to draw shapes. We don't even really think about what we're doing when we draw. The fact that we usually base our drawings on a horizontal and vertical axis is mostly taken for granted. However, that is all a cartesian plane is. The x axis in the cartesian coordinate plane represents the horizontal, while the y axis represents the vertical. So it is with graph. One complex waveform is plotted on the horizontal, while another is plotted on the vertical. The only problem is that we have only one complex waveform available to us. For this reason, GRAPH has been given an alternate set of oscillators. These oscillators are called "Bank 1" and can be addressed by pushing the following key: : Change to Oscillator Bank #1. You may have noticed in the middle right hand side of the screen that there is a prompt that says the word "Bank." This refers to the current bank setting. In the past, the bank has always been set to zero. From this point on, however, you will need to use both banks of oscillators together to generate the shapes that we want to see. : Change the Oscillator Bank back to #0. (Note that the bank change option is simply a "toggle" switch that can be used to quickly choose either bank 0 or bank 1). Section 4.2 - Plotting a circle. For our first two-dimensional graph, we will need plot a sine wave against a cosine wave. For those of you who are not terribly familiar with trigonometry, you will want to know that a sine wave on the horizontal (x) axis, plotted against a cosine wave on the vertical (y) axis will generate a circle. To create the circle you will need to do the following: Exercise: < M > : Enter the following modulation formula: (O1) <F7 > : Enable modulation. <F9 > : Enable Plot for the modulation waveform only. <F1 > : Enable the sinewave in oscillator #0. <F10> : Plot the sinewave. GRAPH User's Manual Page 16 : Change to Oscillator Bank #1. <F3 > : Enable the cosinewave in oscillator #0. <F10> : Plot the cosinewave. <Alt-F10> : This command plots a circle in the upper right hand corner of the display. ( This is not the unit circle. ) Note that the modulation formula and the modulation enable controls are associated with both banks of oscillators. Section 4.3 - "Rotating" the circle. Due to a convenient special case in trigonometry, we can create a sequence of shapes that appear much like an ellipse rotating in three dimensions. This can be done by simply incrementing the phaseshift offset value of either oscillator by 0.1 repeatedly after plotting. To illustrate this, try the following exercise: Exercise: Plot a set of objects that appear as a rotating ellipse. < S > : Set the phaseshift to 0.1 <F10> : Plot the waveform. <Alt-F10> : Draw a new graph. < S > : Set the phaseshift to 0.2 <F10> : Plot the waveform. <Alt-F10> : Draw a new graph. < S > : Set the phaseshift to 0.3 <F10> : Plot the waveform. <Alt-F10> : Draw a new graph. ----------> keep incrementing the phaseshift in 0.1 increments and repeating the Plot/Draw sequence as many times as you like. : When you are done, press the nit circle enable twice to clear the area where the graphs are drawn. <ESC> : Clear the main screen. < D > : Return to default values. As you can see, the phase shift can be used to "massaged" two- dimensional shapes. GRAPH User's Manual Page 17 Exercise: Drawing a "Limacon." < M > : Enter the following modulation formula: (O1+O2)/2 <F7 > : Enable modulation waveform plotting. <F9 > : Disable component waveforms. <F1 > : Enable sinewave in oscillator #1. < 2 > : Assign oscillator #2. <F1 > : Enable sinewave. < C > : Set cycles to 2. <F10> : Plot the waveform. : Change to alternate bank of oscillators < 1 > : Assign oscillator #1. <F3 > : Enable cosinewave. < C > : Set cycles to 2. < 2 > : Assign oscillator #2. <F3 > : Enable cosinewave. <F10> : Plot the waveform. <Alt-F10> : Draw the limacon. Now then, this Limacon is pretty cute, but it's painfully tiny to look at. For this reason, you can draw the object in larger size in the box where the waveforms are usually plotted. To do this, execute the following keystrokes. <Alt- F9> : Draw the limacon, but first... After you press <Alt-F9>, the prompt "Scale: " will appear at the lower left hand part of the large display area. If we want to observe the object at 3 times the size it was plotted at before, we can simply type a <3> followed by <Enter> to scale the object to a factor of 3. Be careful not to make the scaling factor too large, or you will not be able to see all of the object! Now then, you probably noticed that the limacon looks very disconnected when it is displayed in the large box. To fix this problem, we need to increase the resolution, and replot both complex waveforms. < R > : Change the resolution to 0.25 <ESC> : Clear the large waveform display area. <F10> : Replot the waveform. : Change banks. <F10> : Replot the other waveform. <Alt- F9> : Redraw the limacon in the large waveform display area. (Do not forget to enter the scaling factor for the object). Try the <Alt- F9> option several times at different scaling factors, to get the feel for it. Sometimes the two-dimensional shapes look much different at particular sizes. GRAPH User's Manual Page 18 Note: If you would like to plot both complex waveforms via one keystroke, you may use the following keystrokes to do so: <ESC> : Clear the screen. : When you are done, press the nit circle enable twice to clear the area where the graphs are drawn. < 1 > : Assign first oscillator. < S > : Change phase<S>hift to .15 : Change banks. < 2 > : Assign second oscillator. < S > : Change Phase<S>hift to .25 < R > : Change the resolution to 0.75 <Alt- F1> : Plot BOTH complex waveforms. <Alt- F10>: Sketch the two-dimensional waveform Section 4.4 - Printing a copy of your graph. If you have an EPSON compatible printer, you can print your graphs as you create them. This is done by pressing the following key: < O > : Output the contents of the display to the printer. Lesson 5 - A diverse collection of two-dimensional shapes. You may wish to exercise these keystrokes between each plot: < D > : Assign defaults parameters. <ESC> : Clear the screen : When you are done, press the nit circle enable twice to clear the area where the graphs are drawn. Object #1 : Flower pattern ------------------------------------------------------- < R > : Set resolution to 0.5 (Recommended only!) < 1 > : Assign oscillator #1. <F1 > : Enable Sine. GRAPH User's Manual Page 19 < S > : Set phase<S>hift to 0.5 < 2 > : Assign oscillator #2. <F3 > : Enable Cosine. < A > : Set <A>mplitude to 25. < C > : Set <C>ycles to 4. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F3 > : Enable Cosine. < S > : set phase<S>hift to 0.5 < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < A > : Set <A>mplitude to 25. < C > : Set <C>ycles to 4. < M > : Enter modulation formula: (O1+O2)/2 <F7 > : Enable modulation. <F9 > : Plot Modulation only. <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. ( Don't forget to clear all defaults before you try the next object! ) Object #2 : Hang-glider. ------------------------------------------------------- < R > : Set resolution to 0.2 (Recommended only!) < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < C > : Set <C>ycles to 3. < 3 > : Assign oscillator #3. <F1 > : Enable Sine. < C > : Set <C>ycles to 5. < 4 > : Assign oscillator #4. <F1 > : Enable Sine. < C > : Set <C>ycles to 7. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F3 > : Enable Cosine. < S > : Set phase<S>hift to 0.25 GRAPH User's Manual Page 20 < 2 > : Assign oscillator #2. <F3 > : Enable Cosine. < C > : Set <C>ycles to 2. < S > : Set phase<S>hift to 0.25 < 3 > : Assign oscillator #3. <F3 > : Enable Cosine. < C > : Set <C>ycles to 3. < S > : Set phase<S>hift to 0.25 < 4 > : Assign oscillator #4. <F3 > : Enable Cosine. < C > : Set <C>ycles to 4. < S > : Set phase<S>hift to 0.25 < M > : Enter modulation formula: (O1+O2+O3+O4)/4 <F7 > : Enable modulation. <F9 > : Plot Modulation only. <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. ( Don't forget to clear all defaults before you try the next object! ) Object #3: Fat squiggly thing. ------------------------------------------------------- < R > : Set resolution to 0.2 (Recommended only!) < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < C > : Set <C>ycles to 2. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < C > : Set <C>ycles to 8. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < C > : Set <C>ycles to 6. < M > : Enter modulation formula: (O1+O2+O3+O4)/3 <F7 > : Enable modulation. <F9 > : Plot Modulation only. GRAPH User's Manual Page 21 <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. Object #4: Fast squiggly thing. ------------------------------------------------------- < R > : Set resolution to 0.2 (Recommended only!) < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < C > : Set <C>ycles to 4. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < C > : Set <C>ycles to 10. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < C > : Set <C>ycles to 20. < M > : Enter modulation formula: (O1+O2)/2 <F7 > : Enable modulation. <F9 > : Plot Modulation only. <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. Object #5: Floating Three-Dimensional object. ------------------------------------------------------- < R > : Set resolution to 0.4 (Recommended only!) < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < C > : Set <C>ycles to 4. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. GRAPH User's Manual Page 22 < C > : Set <C>ycles to 3. < M > : Enter modulation formula: (O1+O2)/2 <F7 > : Enable modulation. <F9 > : Plot Modulation only. <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. Object #6: Elegant Three-Dimensional squiggle. ------------------------------------------------------- < R > : Set resolution to 0.2 (Recommended only!) < 1 > : Assign oscillator #1. <F1 > : Enable Sine. < A > : Set <A>mplitude to 10. < C > : Set <C>ycles to 2. < 2 > : Assign oscillator #2. <F3 > : Enable Cosine. < A > : Set <A>mplitude to 100 < C > : Set <C>ycles to 2. < 3 > : Assign oscillator #3. <F1 > : Enable Sine. < C > : Set <C>ycles to 3. < 4 > : Assign oscillator #4. <F1 > : Enable Sine. < C > : Set <C>ycles to 9. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F3 > : Enable Cosine. < C > : Set <C>ycles to 4. < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < 3 > : Assign oscillator #3. <F1 > : Enable Sine. < 4 > : Assign oscillator #4. <F1 > : Enable Sine. < M > : Enter modulation formula: (O1+O2+O3+O4)/3 <F7 > : Enable modulation. <F9 > : Plot Modulation only. GRAPH User's Manual Page 23 <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. Object #7: A circle travelling in three dimensions. ------------------------------------------------------- < R > : Set resolution to 0.011 < 1 > : Assign oscillator #1. <F3 > : Enable Cosine. < S > : Set Phase<S>hift to 0.3 < 2 > : Assign oscillator #2. <F1 > : Enable Sine. < A > : Set <A>mplitude to 30. < C > : Set <C>ycles to 99 < 3 > : Assign oscillator #3. <F1 > : Enable Sine. < A > : Set <A>mplitude to 5. < C > : Set <C>ycles to 4. : Switch to alternate bank of oscillators. < 1 > : Assign oscillator #1. <F3 > : Enable Cosine. < 2 > : Assign oscillator #2. <F3 > : Enable Cosine. < A > : Set <A>mplitude to 30. < C > : Set <C>ycles to 100. < 3 > : Assign oscillator #3. <F3 > : Enable Cosine. < A > : Set <A>mplitude to 5. < C > : Set <C>ycles to 3. < M > : Enter modulation formula: (O1+O2+O3)/2 <F7 > : Enable modulation. <F9 > : Plot Modulation only. <Alt-F1 > : Plot both banks of complex waveforms. <Alt-F9 > : Plot two-dimensional shape with Scaling factor of 3. Note: This "object" will take a long time to plot! GRAPH User's Manual Page 24 Appendix - Feedback, Registration, Program developments. Now we have covered all of the commands available to you in the program "GRAPH." Again, we would very much like to hear from you! If you like, you may send the following two-page questionnaire back to us. This will give us a feeling for where program developments and improvements need to be made, as well as the order in which we should take care of them. If you would like to register, we welcome this of course. As stated before, you will receive a free update to the next version of GRAPH, which will be quite substantial. You will also receive the newsletter, which contains settings for some exceptionally beautiful two-dimensional plots as well as information about various interesting articles about various applications concerning our programs. Emphasis of articles will be on PACIFIC LASER ART endeavors which involve exploration of mathematics, and hardware/software design. As of this date, our latest endeavor is a project tentatively named WIREANIM. This package is a shape animator/sequencer designed to create and manipulate two-dimensional objects. The main reason why we are referring to it here is that it is designed largely to expand the capabilities of GRAPH. Those who make up PACIFIC LASER ART loathe vaporware, but we thought it would be healthy to let people know what is in the works. We hope that you enjoy exploring our products. Questionnaire (optional) This questionnaire is included to give us feedback about your feelings about "GRAPH." By returning this questionnaire to us, you will help to shape the program into you want it to be. You may answer one, or several, or all of the questions as you feel is necessary. 1. Please indicate the types of computers that you use on a regular basis? ___ a) 8088/8086 IBM PC or clone ___ e) Apple Macintosh SE ___ b) 80286 IBM PC or clone ___ f) Apple Macintosh ][ ___ c) 80386 IBM PC or clone ___ g) Commodore Amiga 500 ___ d) Apple Macintosh Plus ___ h) Commodore Amiga 2000/2500 2. What speed does your PC or PC clone run at? ___ a) 4.77 megahertz ___ f) 16 megahertz ___ b) 6 megahertz ___ g) 20 megahertz ___ c) 8 megahertz ___ h) 25 megahertz ___ d) 10 megahertz ___ i) 33 megahertz ___ e) 12 megahertz ___ j) More than 33 megahertz 3. Do you have a math coprocessor in your PC or PC clone? ___ a) Yes ___ b) No 4. What type of display adapter are you using in your PC or PC clone? ___ a) Hercules Monochrome Graphics Adapter ___ b) AT&T400 Monochrome Graphics Adapter ___ c) Enhanced Graphics Adapter (EGA) ___ d) Video Graphics Array (VGA) ___ e) Other 5. How did you find out about "GRAPH?" ___ a) Word of mouth ___ b) Advertisement ___ c) Software distributor 6. Are you satisfied with the speed at which "GRAPH" plots simple and complex trigonometric waveforms? ___ a) Yes ___ b) No 7. Did you like the overall idea and implementation of "GRAPH?" ___ a) Yes ___ b) No 8. Was the program useful to you in terms of: ___ a) Mathematical learning ___ b) Artistic enjoyment ___ c) Educational tool ___ d) Entertainment ___ e) Other (Please specify) ______________________________________ 9. Which of the following capabilities would you like to see incorporated into the program "GRAPH:" ___Yes ___No (a) more oscillators for each bank. ___Yes ___No (b) availability of non-trigonometric waveforms such as triangle, saw-tooth, & digital waveforms. ___Yes ___No (c) a shape/waveform editor. ___Yes ___No (d) a shape sequencer/animator. ___Yes ___No (e) 3D shape generation capabilities. ___Yes ___No (f) A mouse driven user interface. ___Yes ___No (g) Shape rotation ___Yes ___No (h) more extensive documentation. ___Yes ___No (i) A documented file format for loading and saving shapes to your disk. ___Yes ___No (j) Digital to Analogue or Analogue to digital conversion for waveforms. ___Yes ___No (k) availability of source code. ___Yes ___No (l) more extensive documentation. ___Yes ___No (m) Polar coordinate plotting capabilities. ___Yes ___No (n) a formula generator for each plot. ___Yes ___No (o) a more sophisticated modulation formula interpreter so that multiplication, division and exponentiation of functions can be plotted on the screen 10. If you could magically change one thing about the program itself, what would it be. 11. What is your area of focus in your line of occupation: ___ a) Physics ___ b) Mathematics ___ c) Engineering ___ d) Computer Scientist ___ e) Artist ___ f) other (Please specify) _______________________________________ 12. If you consider yourself to be artistically inclined, could you please briefly tell us where your artistic interests lie. If your artistic interests relate to this program, could you please tell us about this relation. 13. Additional comments: