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This makes a great gift, t-shirt, mug, etc. for any computer or math geek, friend, child or baby! Write you name or message in up to 100,000 of the infinitely non-repeating digits of pi. Originally published: Mar 21, 2006.

Introduction

This makes a great gift, t-shirt, mug, etc. for any computer or math geek, friend, child or baby! Write you name or message in up to 100,000 of the infinitely non-repeating digits of pi. PiShop is a free online application that allows you to generate customized image to use as a t-shirt transfer, banner, or whatever else you want to use it for. It’s free for personal use, but if you like it, I encourage you to make a little donation for my kids’ college funds! =) 

What You’ll Need

For t-shirts, onesies, bibs, and other garments:

  • Printable iron-on transfers
  • A onesie, hat, bodysuit, or other garment to receive the transfer
  • A printer to print the transfer (suggestion below)
  • The image to print (create it below)

Writing with Pi

Simply fill out the short form below and put a check mark in the box by “Mirror Image” if you want the image to be reversed for transfer paper. (Many printers do this automatically when you indicate that you’re printing on transfer paper.)

Type in your name or message; select a font, size and color.

Next click “Make Image.” The page will refresh, and the image will appear (sized to fit) below the Make Transfer button. You can save the full size version of the image by simply right-clicking on the thumbnail, then selecting “Save Picture As…” or “Save Image As…”A note about privacyI don’t keep (or even see) any of this information. It’s sent directly to a computer program that generates a transfer image and then immediately discards the data once the image is sent to you.

Making the Shirt

Depending upon the size of the shirt, onesie, hat, bib, sweater or other garment you want to make, you may need to resize the image using a word processing or photo editing program before you print it to the transfer paper. Once you get the size about right, print out a test page to make sure it will fit the garment.

Need a good printer? I use a Canon Pixma iP6600D photo printer shown below, and I’m continually astounded by the superb prints I get — all the way up to 8×10, which cost a fortune if you buy them online or at the photo store. Plus it’s quite fast and it prints of both sides of the paper, so it basically use it to print everything. (Use the “Pro” paper for photos you won’t regret it!)After you’ve printed your test page on cheap paper, load in the expensive stuff (the iron-on transfer paper) and stqart printing. Once it’s printed, I typically like to cut around the image and cut out large sections of unprinted transfer. This tends to make the final product look a little cleaner and more professional.

The transfer paper should come with instructions on how best to transfer the image from the paper to the garment.

Licensing & Registration

This application is Copyright (c) 2006-2008 by Alex Franke, and it offered as what I like to call CollegeFundware. This means you’re free to try it out for a reasonable period of time, but if you like it and want to keep using it, you need to contribute to my kids’ college funds =).

Visit the Software Registration page on this site for details on how to legally use this software after a reasonable evaluation period. Important: By downloading and/or running this software, you agree to the terms and conditions set forth on the the Software Registration page page. (It’s not that bad, really =)

When you register the product by contributing, any limitations the software may have will be removed, including registration reminder pop-ups windows. Registration may also “unlock” special features not available to unregistered users.

It’s always difficult to suggest a donation amount for registration. I recommend considering the value of the time you save by using the tool, and contribute an amount you feel is reasonable and appropriate. Below is a general guideline that may also help you decide.

  • $15-20 for private, personal use (e.g. hobbyist)
  • $20-50 for individual professional use (e.g. contractor)
  • $50+ if your company is writing the check

I suppose this doesn’t *need* the ATMega, but they’re only a few bucks, so what the hey. This uses an old GI SP0256-AL2 speech synthesizer IC containing all 59 English phenomes. It sounds very computer-like (even though the original documentation boasts “natural sounding speech”. The DIP switches allow the user to specify a unique device ID for the network. Because it makes noise, I added brown-out detection — something I probably should add to all the devices eventually. The PWM speech output is filtered though a 5KHz low-pass, then sent through a volume control pot, then on to a low-power amplifier. The same 4-pin input is at the upper left, and the 2-pin speaker connection is at the lower right (above the big cap). Bonus: A programmable LED nestled between the ICs! (Oooo!)
An even tinier 4-amp dual stepper motor controller and driver, providing 500mA per phase for two 4-phase unipolar steppers, or 1A per phase for 1 motor when signals are combined (separate input for motor power). Four-position DIP switches assign a unique device number for addressing within the network, and various motor commands received via the 4-pin interface (+5V, GND, SCL, SDA) are translated and clocked out by the onboard microprocessor, completely freeing the controlling device of this tedious task. Quite small — less than 1.75″ x 1.5″ including mounting area.

I should say that there’s another version of this as a low-end PIC application (PIC12C505, and alternatively the 8-pin 12C508). The 14-pin version takes a 6-pin input, with power, ground and direction & step bits for each motor. This offers the user very simple motor control with just a few pins and the PIC just acts as a translator. Then I got the itch make it smaller. The 8-pin version takes a cable with +5V, GND, Device, Direction and Step, leaving only three pins for all the dirty work. I used a 74LS595 shift register with latch to maintain the motor states. The PIC waits to receive commands for both motors then updates the motor phases. (e.g. Sending pins 3-5 a “011” followed by a “101” will set motor 0 to step one phase forward, and motor 1 will step one phase back. Note that the last bit may be 0, meaning no step will be taken.) The microcontroller then goes into low power sleep while the shift register holds the motor.

I ended up wanting a bit more flexibility with an I2C interface. But, since a 4MHz PIC runs at only 1 MIPS (It uses four clocks per instruction cycle.), it’s a little difficult to implement even the “slow” I2C mode at 100kbps. I do want to go back and see if this is doable by continually forcing the master into wait states. Regardless, it turned out to be more trouble than it was worth — especially if I could get an AVR micro for a fraction of the cost, use it to replace the shift register (it’s only 4 pins larger), and crank out 12 MIPS. This gave me room on the board for the DIP switches and even a flashing LED! Anyway, the PCB is no-frills and still needs some work.

P.S. If you were wondering why, in paragraph 2, I felt the “Device” pin was necessary when the controller waits for two commands anyway, it’s because the user doesn’t have to send both commands. The PIC will time out waiting for the second command and simply update the motor state as appropriate. So then why is the “Step” pin necessary?? Now THAT’s a good question!
A tiny I2C master controller designed to monitor and handle sensor loop and control  operations. Conditioned power (+5V) is brought into this board, where it is coupled with the TWI for twelve external devices on three independent clock and data lines. I2C need not be implemented on each bus. Additionally power is combined with three weak-high pins, and three floating for other control applications. Includes an analog comparator (ANC) and a programmable LED.


This is how I chose to solve the problem of making trig calculations with the PICmicro MCU.

Note that this does not use the CORDIC method because I imagine that would hog too much valuable computing time. This solution probably should not be used to try to get someone to Mars or anything — I’ll work on a more complete solution at a later date. Until then, I hope this will do.

First of all, I attacked this problem with a few basic assumptions:

  • 256-degree circle – It makes sense, when using a computer, to base everything on powers of two. Here I’m going to consider circle in terms of only 256 degrees. This is because 256 can be represented nicely by one byte and it’s close enough to 360 degrees that results shouldn’t be too far off base. Also, common angles can be easily represented as well, with a half-circle, quarter, eighth, sixteenth, etc, all settling nicely at a power of two. If it’s difficult to think of a circle in 256 degrees, just remember that the number 360 was completely arbitrary in the first place. (You can thank the ancient Babylonians for it, as well as for the number of minutes in an hour and seconds in a minute. They used a base-60 counting system.)
  • Unsigned magnitude, zero to 255 – Sines and cosines are usually represented by fractions. Since computers can’t tolerate fractions very easily, I chose to return results as a positive number from zero to 255, where zero is, well, zero, and 255 represents 1. Why only positive numbers? Because when you take the sine of an angle in a 256-degree circle, the most significant bit of the angle you’re looking for will quickly tell you if the result should be positive or negative. This is because quadrants 3 and 4 (angles 128-255) generate negative sine values. (As it turns out, the MSB for all these numbers is 1.) This allows me to use the full byte for sine values, effectively doubling the granularity of the result. 
  • One-quadrant sine-only look-up table – The sine values for any one quadrant can be used to determine any of the other trig values for any other quadrant. I’ll provide a refresher on trigonometric identities below. Suffice it to say that one quadrant of sine values is all we need.
  • Non-CORDIC Solution – CORDIC would take too long to implement with a negligable increase in accuracy for typical applications. This solution is a look-up table that essentially digitizes the first quadrant of a sine wave as shown below.

The code listed below was written for mid-range PIC MPU’s like the 16f84, but can be ported down with a few minor changes. (For example, you may need to use comf and incf to negate values in the lower-end divices.) It finds the unsigned sine of an angle in eight instruction cycles (32 clock cycles), or 8us on a 4MHz microprocessor. With a few extra instructions (about 13 total) you can determine the sign as well as the magnitude. Determining the not-as-accurate signed sine requires 17 instruction cycles and returns a value ranging from -127 to 127. It takes about 15 cycles to determine the 8-bit cosine magnitude and discover its sign, while the signed cosine requires 18 instruction cycles. If you see a more efficient way of doing this, please let me know. Because instruction cycles vary, I don’t recommend using this for bit-banging routines where timing is critical. Maybe later I’ll do a version that takes the same number of cycles regardless of what function its performing.Before you forget, please help spread the word about this site by referring friends and associates!

The sinw subroutine does not call any other routines. This should be helpful if you’re working with a device that has a short call stack. If I were to return a signed value for the sine, I would have needed to make an additional call on the stack. I believe the way I’ve done it works out nicer. As it is, only ten or so instructions are required to determine both the sine of the angle and it’s sign (positive or negative).

The code below is well commented and includes examples of how to compute both sine and cosine (signed and unsigned) using the sine look-up table. Also, in the comments I’ve listed all the other trig equations I can think of — you can use these to figure out tangents, cosecants, etc.

It might be helpful to remember when sines and cosines are positive or negative. In our 256-degree circle, each quadrant is represented by 64 degrees. So Q1 is angles 0-63, Q2 is 64-127, etc.

  • Q1 (0-63 deg) – Sine and cosine are both positive. Bits 6 and 7 are both clear.
  • Q2 (64-127 deg) – Sine is positive, cosine is negative. Bit 6 is set and 7 is clear.
  • Q3 (128-191 deg) – Sine and cosine are both negative. Bit 6 is clear and 7 is set.
  • Q4 (192-255 deg) – Sine is negative, cosine is positive. Bits 6 and 7 are both set.

Bit seven (the MSB) of the 8-bit angle indicates if the sine is positive or negative (zero or one, respectively).  Bit six indicates whether or not the sine wave is a mirror image of the preceeding quadrant. (Q2 is a mirror image of Q1 and Q4 is a mirror image of Q3.) This bit can therefor be used to indicate whether or not we need to read from the start or the end of the look-up table. The following rules always apply:

  • If bit 7 (MSB) is set, the angle is in Q3 or Q4, where sine is always negative.
  • If bit 7 (MSB) is clear, the angle is in Q1 or Q2, where sine is always positive.
  • If bit 6 is set, the angle is in Q2 or Q4, which are mirror-images of Q1 and Q3, so read the table backwards.
  • If bit 6 is clear, the angle is in Q1 or Q3, which can be read directly from the look-up table.

We only use the table to look up a sine value. Rather than trying to use the table to also look up a cosine, figure out what angle’s sine returns the same value and go to the table for that. This will ensure you can determine the correct sign (positive or negative) of the magnitude returned. To really drive this point home, let’s take a look at each case for the two most significant bits in an 8-bit unsigned angle:

  • 00 – Angle is in Q1, the sine is positive, and it can be looked up directly from the table
  • 01 – Angle is in Q2, the sine is positive, and it can be looked up by indexing the table backwards
  • 10 – Angle is in Q3, the sine is negative, and it can be looked up directly from the table
  • 11 – Angle is in Q4, the sine is negative, and it can be looked up by indexing the table backwards

Skip this paragraph if you’re not a beginner. Why does rolling (shifting) bits to the left or right magically double or halve whatever number your shifting? It works in a similar way for any counting system — just think about the numbers you’re used to using. If you “shift” the digits in the decimal number 123 to the left you get 1230 — effectively multiplying the original number by ten, which is the base of the decimal counting system. Likewise, shifting the hexadecimal value F3 to the left you get F30. F3 in decimal notation is 243. 243 times 16 (the base of hex) is 3,888, which in hexadecimal is F30.

Feel free to use this code, but be sure to leave the copyright information intact.

#define _version "0.01"

; Author: 		Alex Franke
; Copyright: 		(c)2002 by Alex Franke. All rights reserved. 
;
; Title: 		sin
; Description: 		Computes the sine of an angle given in degrees of a 
;			256-degree circle.
;
; Device: 		Microchip PIC MCU, p16f84
;			or other mid-range PIC MCU's
;
; Update History:	6/11/02	Created
;

 LIST 		R=DEC
 INCLUDE 	"p16f84.inc"

; Registers
 CBLOCK	0x020
sinTemp, theta, temp, temp2
 ENDC

 __CONFIG _CP_OFF & _WDT_OFF & _RC_OSC & _PWRTE_ON


; Main

 org 0

Start

  ; Set up
  movlw		84		; specify an angle value
  nop				; pause here to change value if desired for testing
  movwf		theta		; save that angle for use later

  ; now start the examples...
  ; Assume the angle you're trying to find is in theta and you want to find...
  ; ...its sine (0 to 255) and its sign (positive or negative)
  movf		theta, w	; copy to w so we can work with it
  call		sinw		; we now have the sine in w
  btfsc		theta, 7	; test the original angle to see if it's in Q3 or Q4
    nop				; ... if so, we know it's a negative, so handle that here

  ; ...its sine (-127 to 127)
  movf		theta, w	; copy to w so we can work with it
  call		sinw		; we now have the sine in w
  movwf		temp		; store result in temp register
  bcf		STATUS, C	; clear the carry bit if it's set -- we're about to roll
  rrf		temp, w		; roll temp result right to cut in half, store in w
  btfsc		theta, 7	; test the original angle to see if it's in Q3 or Q4
   sublw 	0		; ...it is, so negate w 
  nop				; sine (-127 to 127) is now in w


  ; ...its cosine (0 to 255) and its sign (positive or negative)
  ; A cosine is basically a sine, but shifted to the left one quadrant. This means
  ; that a sine is a cosine shifted to the right. So the cosine of any angle is 
  ; equal to the sine at the same position one quadrant to the right, or 
  ; cos(x)=sin(90+x). So to convert to sine we add 90 deg (64 degrees in our 256-deg circle)
  ; to the original angle.  
  movf		theta, w	; copy to w so we can work with it
  addlw		64		; add pi/2 so we can just take the sine
  movwf		temp		; move new angle to temp because we test this later
  call		sinw		; we now have the sine in w
  btfsc		temp, 7		; test the original angle to see if it's in Q3 or Q4
    nop				; ... if so, we know it's a negative, so handle that here


  ; ...its cosine (-127 to 127)
  movf		theta, w	; copy to w so we can work with it
  addlw		64		; convert cos to sin
  movwf		temp2		; store in temp2 register so we can compare for negative
  call		sinw		; we now have the sine in w
  movwf		temp		; store result in temp register
  bcf		STATUS, C	; clear the carry bit if it's set -- we're about to roll
  rrf		temp, w		; roll temp result right to cut in half, store in w
  btfsc		temp2, 7	; test the original angle to see if it's in Q3 or Q4
   sublw 	0		; ...it is, so negate w 
  nop				; sine (-127 to 127) is now in w



  ; ...its tangent, cotangent, secant, cosecant, etc
  ; Rather than going into a dissertation on how computers divide numbers, 
  ; let me just recommend using a good divide macro and any of the following 
  ; equations... 
  ;
  ;   tan( x ) = sin( x ) / cos( x )
  ;   cot( x ) = 1 / tan( x )		= cos( x ) / sin( x )
  ;   csc( x ) = 1 / sin( x )
  ;   sec( x ) = 1 / cos( x )
  ; 
  ; These Pythagorean identities may also be useful
  ; 
  ;   sin^2( x ) + cos^2( x ) = 1
  ;   sec^2( x ) = 1 + tan^2( x )
  ;   csc^2( x ) = 1 + cot^2( x )
  ;
  ; These will help with adding and subtracting
  ;
  ;   sin( x + y ) = ( sin( x ) * cos( y ) ) + ( cos( x ) * sin( y ) )
  ;   sin( x - y ) = ( sin( x ) * cos( y ) ) - ( cos( x ) * sin( y ) )
  ;   cos( x + y ) = ( cos( x ) * cos( y ) ) - ( sin( x ) * sin( y ) )
  ;   cos( x - y ) = ( cos( x ) * cos( y ) ) + ( sin( x ) * sin( y ) )
  ;   tan( x + y ) = ( tan( x ) + tan( y ) ) / ( 1 - ( tan( x ) * tan( y ) ) )
  ;   tan( x - y ) = ( tan( x ) - tan( y ) ) / ( 1 + ( tan( x ) * tan( y ) ) )
  ; 
  ; And for doubled angles...
  ;
  ;   sin( 2x ) = 2 * sin( x ) * cos( x )
  ;   cos( 2x ) = cos^2( x ) - sin^2( x ) = ( 2 * cos^2( x ) ) - 1 = 1 - ( 2 * sin^2( x ) )
  ;   tan( 2x ) = ( 2 * tan( x ) ) / ( 1 - tan^2( x ) )
  ;
  ; And as we've used above... What's a cosine? It's on of several cofunctions:
  ;   
  ;   sin( 90 deg - x ) = cos( x )	cos ( x - 90 deg ) = sin( x )
  ;   sec( 90 deg - x ) = csc( x )	csc ( x - 90 deg ) = sec( x )
  ;   tan( 90 deg - x ) = cot( x )	cot ( x - 90 deg ) = tan( x )
  ;
  ; And finally... What's an arcsine, arccosecant, arccosine, arcsecant, arctangent, 
  ; and arccotangent? They're simply the inverses...
  ;   
  ;   arcsin ( sin(x) ) = x		arccos ( cos(x) ) = x
  ;   arcsec ( sec(x) ) = x		arccsc ( csc(x) ) = x
  ;   arctan ( tan(x) ) = x		arccot ( cot(x) ) = x
  ; 

  goto 	Start

Finished
  goto $



sinw
; If bit 7 is clear, we're in quadrant 1 or 3
; If bit 8 is clear, we're in quadrant 1 or 2
;		
;		bit 8 	bit 7
; Quadrant 1	  0	  0	read from begining of table
; Quadrant 2	  0	  1   	read from end of table
; Quadrant 3   	  1	  0	read from begining of table and negate
; Quadrant 4	  1	  1	read from end of table and negate
;
; Angles in quadrants 3 and 4 result in negative sines. Because returning
; a negative value from this subroutine would require an additional entry
; on the call stack (a call to the sine table to get the value before 
; negating it), we'll just leave it up to the user to negate the result
; if necessary. This saves us an additional call. 
; This table returns a value sclaed 0 to 255

  movwf		sinTemp		; Move input to temp variable to test it
  andlw		0x03F		; Clear the first two bits of input - number needs to be <= 64
  btfsc 	sinTemp, 6	; read from the begining or the end of the table?
   sublw	64		; ...read from end, so w = TableSize - Input
  addwf		PCL, f		; Position program counter to appropriate result
  dt 	  0,   6,  13,  19,  25,  31,  37,  44
  dt	 50,  56,  62,  68,  74,  80,  86,  92
  dt	 98, 103, 109, 115, 120, 126, 131, 136
  dt	142, 147, 152, 157, 162, 167, 171, 176
  dt 	180, 185, 189, 193, 197, 201, 205, 208
  dt	212, 215, 219, 222, 225, 228, 231, 233
  dt	236, 238, 240, 242, 244, 246, 247, 249
  dt	250, 251, 252, 253, 254, 254, 255, 255, 255

  end
  

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