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Volume Number:3
Issue Number:1
Column Tag:Assembly Language Lab

Rose Curve Generator

By Victor Barger, Waunakee, WI

An application of mathematics which produces fascinating designs is the graphing of trigonometric functions using polar coordinates. Each function leads to a beautiful design which bears a strong resemblance to a flower, such as the rose. With its excellent graphics capabilities and superior user interface, the Macintosh is perfect for quickly plotting “rose” functions on the screen with a program simple enough that anyone can use it. The application described in this article takes full advantage of these capabilities to allow the user to experience the graphic beauty of trigonometry.

Using Rose Curves

After launching Rose Curves, a window will open in the center of the screen and the default function will be drawn inside. (See Figure 1) Both the standard apple menu and a Control menu will be in the menu bar. The apple menu contains the item “About Rose Curves...” and the names of all desk accessories available to the user from the System file. Under the Control menu are two options: Change Parameters and Quit. When the Change Parameters item is selected a window similar to the one in Figure 2 is opened in the center of the screen. With this window the user can select the function to be drawn, choose the size of the drawing and indicate whether the drawing should be multilayered. Function and size are selected by clicking on the appropriate function name or size. For example, selecting cos (sin 100r) gives the design in Figure 3. If multilayered is selected, the function will be drawn at the original size and then shrunk repeatedly to create a “layered” effect, making the design more interesting: See Figure 4.

Entering the Program

Start up the Macintosh 68000 Development System. The startup disk should be titled ‘MDS1’ and the external disk ‘MDS2.’ The applications Edit, Asm, Link, Exec and RMaker should be on MDS1 and a folder entitled ‘.D Files’ with SysEqu.D, ToolEqu.D, QuickEqu.D, and MacTraps.D inside on MDS2. To make Rose Curves work on a Macintosh 512 with the old ROM, FixMath.Txt and FixMath.Rel must also be in the .D Files folder (The information on where to get these two files is under How Rose Curves Works). FixTraps.Txt and FixTraps.Rel are files which were distributed by Apple Computer under the Apple Software Supplement to aid assembly language programmers in calculating basic arithmetic and trigonometric functions. As the program is presented here, you have everything you need to type the program in and run it on a new ROM machine, since the FixTraps necessary for this program are explicitly defined with the ".TRAPS" statement at the start of the program. The program may also be entered into the new MPW assembler, or assembled with the Consulair C system, which includes the complete MDS assembler built-in.

To create the program, simply assemble the Rose.Asm listing. Then compile the resource file with RMaker. Finally execute the linker on the link file listing and the program object file and resource object file (both ".REL" files) will be linked together, an application generated, and the cute rose application icon should appear on the disk. Double click "Rose Curves" and your in business making fascinating polar coordinate art.

How Rose Curves Works

The program begins by initializing the standard managers, initializing its variables, setting up the menu bar and opening the window. The initialized variables are Equation, Scale, RealScale, Multilayered and drawn. Equation contains the item number of the selected function in the Change Parameters dialog. To obtain the real equation number, subtract 3 from Equation. Scale is the item number of the selected scale or “size” and RealScale is the actual scalar multiplied on each point before drawing it. Multilayered determines whether the rose will be layered. Finally, drawn is a boolean expression which determines whether the window needs to be updated by redrawing the rose (false) or simply by copying the rose from the offscreen bitmap (true). The offscreen bitmap is also created in this routine. To create a bitmap, create an area in memory with space for a pointer, a word and a rectangle. In Rose Curves, this appears at the end of the program under the label windowBitMap. Now select the rectangle of the bitmap and calculate the rowbytes with the equation

       rowbytes = int((xright - xleft)/8+1)

If rowbytes is odd, round it up. Suppose our rectangle is 0, 0, 50, 96 (remember that a QuickDraw rectangle is stored as y1, x1, y2, x2). Then rowbytes would be 14 and the bitmap would look like this:

myBitMapDC.L0  ;pointer to bitmap storage
 DC14 ;rowbytes
 DC0, 0, 50, 96  ;boundsRect

The size of the bitmap storage must be determined before it can be created. Use the equation

 size = rowbytes * (ybottom - ytop)

To create the bitmap, use the following routine.

 move.l #size,D0
 _NewPtr;allocate new pointer
 lea  myBitMap,A1
 move.l A0,(A1)  ;put the pointer to the
 ;storage in myBitMap

To copy a picture from the current window to myBitMap, use this routine

 pea  aPort(A5)  ;storage for the port
 _GetPort ;get the current port
 move.l aPort(A5),A0 ;put the port pointer in A0
 pea  portBits(A0) ;push ptr to port’s bitmap
 pea  myBitMap ;push ptr to our bitMap
 pea  srcRect  ;push srcRect
 pea  dstRect  ;push dstRect
 clr  -(SP) ;srcCopy
 clr.l  -(SP)  ;no mask region
 _CopyBits;copy the bits

Finally, remember to dispose of the bitmap’s storage after finishing with it. This is accomplished simply with

move.l  myBitMap,A0

The EventLoop allows desk accessories to perform their periodic events by calling _SystemTask and, if an event occurs, branches to HandleEvent which transfers execution to the appropriate routine: Activate, Update, KeyDown or MouseDown. The Activate routine activates or deactivates the Rose Curves window depending on what the user requested. Update updates the window by either drawing the curve or using _CopyBits, depending on whether the curve had been drawn previously. KeyDown checks if the command-key was down when the KeyDown event occurred by testing bit eight of the ModifyReg. If bit eight equals one, a command-key was pressed and _MenuKey is called to determine the appropriate menuNumber and itemNumber for Choices, the routine which handles menu selections. MouseDown finds the area the mouse was clicked in using _FindWindow and jumps to the appropriate routine to handle the click.

Several other important routines are SystemEvent, InMenu, UnhiliteMenu, NextEvent InAppleMenu and ParameterChange. SystemEvent passes mouse clicks in system windows to the system for it to handle. InMenu converts a mouse click in the menu bar to the correct menuNumber and itemNumber for Choices. Choices then calls InAppleMenu if the click was in the apple menu, calls ParameterChange if the first item in the Control menu was selected, or quits to the Finder if Quit was selected. UnhiliteMenu unhighlights all menus in the menu bar and NextEvent returns to the EventLoop. InAppleMenu either displays the InfoAlert About Rose Curves... or opens the selected desk accessory using _OpenDeskAccessory. Finally, ParameterChange opens the parameter dialog, hilights the buttons which correspond to the Equation, Scale and Multilayered variables and monitors the user’s actions until the dialog is dismissed with the OK or Cancel button. If OK was selected, ParameterChange updates the variables Equation, Scale, RealScale and Multilayered before closing the dialog and returning to the EventLoop.

The heart of the program lies in the DrawRose subroutine. Before outlining how it operates, an explanation of the fixed-point math routines is needed. Fixed-point math is not as accurate as floating-point math, but much faster. With graphics programs, speed is often much more important than accuracy, especially when there is the possibility of losing the user’s attention. Since Apple elected not to implement the fixed-point routines into the old toolbox, they released the file's FixTraps. Txt and FixMath.Rel in the Apple Software Supplement of May 6, 1985. These files are available from local user groups, bulletin boards, Apple dealerships or directly from Apple. FixTraps.Txt includes a nice description of the “toolbox additions” and FixMath.Rel contains the actual routines. To use the fixed-point routines on a Macintosh with the old ROM, type ‘INCLUDE FixTraps.Txt’ at the beginning of your .Asm file. Also include the line ‘FixTraps’ after the name of the application under development in the .Link file. To use the fixed-point routines on a Macintosh with the new ROM, simply define the trap using .TRAP at the beginning of the ASM file as we did in our listing here. The complete list of new fixed-point traps is as follows:

Long2Fix $A83F

FracSin $A848

FracSqrt $A849

FracMul $A84A

FracDiv $A84B

FixAtan2 $A818

FixDiv $A84D

Fix2Long $A840

Fix2Frac $A841

Frac2Fix $A842

Fix2X $A843

X2Fix $A844

Frac2X $A845

X2Frac $A846

FracCos $A847

The fixed-point routines support three types of numbers: longint (long integer), fixed, and fract (fractional). Type longint represents integers between ±2147483647, type fixed represents fractional quantities between ±32768 with about 5 digits of accuracy and type fract represents fractional quantities between ±2 with about 9 digits of accuracy. Rose Curves uses _FracSin, _FracCos, _FixMul and _FracMul to perform its arithmetic operations. These routines work like any other toolbox trap: space is cleared on the stack for the result, the parameters are pushed on the stack and the trap is called. The Pascal definitions for these routines are

function FracSin( x : Fixed ) : Fract
function FracCos( x : Fixed ) : Fract
function FixMul( x, y : Fixed ) : Fixed
function FracMul( x, y : Fract ) : Fract

With _FixMul and _FracMul it is possible to pass parameters of different types than specified. For example, one could pass x as type longint and y as type fixed to _FixMul and the result would be type longint. The following table contains the result types for multiplying different types using _FixMul and _FracMul.

_FixMul _FracMul

x y result x y result

fixed fixed fixed fract fract fract

longint fixed longint longint fract longint

fixed longint longint fract longint longint

fract fixed fract fixed fract fixed

fixed fract fract fract fixed fixed

Note: To convert a number to type Fract, multiply it by 65536. To convert a number to type Fixed, multiply it by 32768.

DrawRose begins by changing the cursor to a watch, setting the counter to 721 steps (1 + 360° x 1/increment), the increment to 572 (.5° in fixed-point radians) and the present angle to 0. It then loops until the counter is zero, evaluating the current function (bsr EvalFunction), converting the function result to rectangular coordinates and scaling and centering the points. After completing each layer, the program decreases the scale by 20 and redraws the curve if multilayered was selected. After drawing all of the layers, the variable drawn is set to true and the image in the window is copied onto the offscreen bitmap for future update events.

Possible Modifications

Rose Curves could be improved in several ways. The most obvious way would be to add more functions to the program. To accomplish this, add the “function” buttons in the Rose.R file and add the subroutines to evaluate the new functions to Rose.Asm. Note that the displacement to the function must be included in the equation table for the new subroutine to be executed. To do this, add the line DC functionLabel-EquationTable after the line DC Six-EquationTable. As more functions are added, keep adding the labels to the equation table. The second improvement would be to decrease the time required to draw the rose when multilayered is on. To do this, copy the image of the rose into an offscreen bitmap using _CopyBits after the first rose has been drawn. Then shrink CopyBits’ dstRect and copy the rose from the offscreen bitmap onto the screen. Although many times faster, this method greatly reduces the accuracy with which layers are drawn since they are being scaled by _CopyBits, not mathematically graphed. An option to this problem would be to include a checkbox in the Change Parameters dialog titled ‘Accurate Multilayers.’ If selected, Rose Curves would draw the layers using its normal drawing routine. If not, Rose Curves would use the copy-and-shrink method, thus giving the user the choice of accuracy or speed.

;File: Rose.Asm
;Rose Curves draws curves expressed as a
; function of r and theta.
;Written by Victor Barger on April 2, 1986.
 .trap  _FracMul $A84A  ;New Roms
 .trap  _FracCos $A847
 .trap  _FracSin $A848
Radians EQU D3
TopButton EQU  D3
BottomButtonEQU  D4
ModifyReg EQU  D4
Increment EQU  D4
MenuReg EQU D5
Counter EQU D5
CheckBoxEQU D6
MenuItemReg EQU  D6
FuncResultEQU  A4

AppleMenuID EQU  1
ControlMenuID  EQU 2

ParamDlgIDEQU  2000
DrawWindIDEQU  2001
InfoAlertID EQU  2002

InitialScaleEQU  13
RealInitScale  EQU 460
 bsr  InitManagers ;initialize managers
 bsr  InitVariables;initialize variables
 bsr  SetupMenu  ;setup the menu
 bsr  SetupWindow;draw the window

 _SystemTask;periodic actions for da's
 clr  -(SP)
 move #$0FFF,-(SP)
 pea  EventRecord
 _GetNextEvent   ;get the next event
 move (SP)+,D0
 beq.s  EventLoop;no event, loop back
 bsr  HandleEvent;handle the event
 beq.s  EventLoop;if 0, continue; else quit

 move #12,-(SP)
 pea  -4(A5)
 move.l #$0000FFFF,D0
 clr.l  -(SP)
 move #3,Equation(A5)
 move #InitialScale,Scale(A5)
 move.l #RealInitScale,RealScale(A5)
 clr  Multilayered(A5)  ;not multilayered
 move.l #11520,D0
 _NewPtr;create a new bit map
 lea  windowBitMap,A1
 move.l A0,(A1)  ;remember the pointer
 clr  drawn(A5)  ;not drawn yet
 clr.l  -(SP)
 move #AppleMenuID,-(SP)
 _GetRMenu;get the apple menu
 move.l (SP),AppleMenu(A5)
 move.l (SP),-(SP)
 clr  -(SP)
 _InsertMenu;insert the menu in the menu bar
 move.l #'DRVR',-(SP)
 _AddResMenu;add the DA's to the apple menu

 clr.l  -(SP)
 move #ControlMenuID,-(SP)
 _GetRMenu;get the control menu
 clr  -(SP)
 _InsertMenu;insert the menu in the menu bar

 _DrawMenuBar    ;draw the new menu bar
 clr.l  -(SP)
 move #DrawWindID,-(SP)
 pea  WindowStorage(A5)
 move.l #-1,-(SP)
 _GetNewWindow   ;get the RoseCurves Window
 move.l (SP),DrawPtr(A5)
 _SetPort ;set the new grafport
 move Modify,ModifyReg
 move What,D0
 add  D0,D0
 move EventTable(D0),D0
 jmp  EventTable(D0)

 move.l DrawPtr(A5),D0
 cmp.l  Message,D0 ;was it our window?
 bne  NextEvent  ;no, get next event
 btst #0,ModifyReg ;activate?
 beq  NextEvent  ;no, deactivate

 move.l DrawPtr(A5),-(SP) ;push pointer to draw window
 _SetPort ;set our port
 bra  NextEvent
 move.l Message,D0 ;get the window ptr
 cmp.l  DrawPtr(A5),D0
 bne  NextEvent

 tst  drawn(A5)
 bne.s  copyIt
 move.l DrawPtr(A5),-(SP) ;push window ptr
 _SetPort ;set the port
 move.l DrawPtr(A5),-(SP)
 move.l DrawPtr(A5),-(SP)
 pea  windowRect
 bsr  DrawRose   ;draw the rose
 bra  NextEvent

 move.l DrawPtr(A5),-(SP) ;push window ptr
 _SetPort ;set the port
 move.l DrawPtr(A5),-(SP)
 pea  windowBitMap ;get the rose from the bitMap
 move.l drawPtr(A5),A0
 pea  portBits(A0) ;put the rose in our window
 pea  windowRect ;same rect for src and dest
 pea  windowRect
 clr  -(SP) ;srcCopy
 clr.l  -(SP)    ;no maskRgn
 _CopyBits;copy It
 move.l DrawPtr(A5),-(SP)
 bra  NextEvent
 btst #8,ModifyReg ;is the command key down?
 beq  NextEvent  ;no, exit
 clr.l  -(SP)    ;space for menu and item
 move Message+2,-(SP);get character
 _MenuKey ;see if its a command
 move (SP)+,MenuReg;preserve menu
 move (SP)+,MenuItemReg ;and Menu item
 bra.s  Choices
 clr  -(SP) ;space for result
 move.l Point,-(SP);get mouse coordinates
 pea  WWindow    ;push event window
 _FindWindow;find the window
 move (SP)+,D0   ;get region number
 add  D0,D0 ;*2 for index into table
 move WindowTable(D0),D0  ;point to routine offset
 jmp  WindowTable(D0);jump to routine
 DCNextEvent-WindowTable  ;in desk (not used)
 DCInMenu-WindowTable;in menu bar
 DCSystemEvent-WindowTable;in system window
 DCNextEvent-WindowTable  ;in content (not used)
 DCNextEvent-WindowTable  ;in drag (not used)
 DCNextEvent-WindowTable  ;in grow (not used)
 DCQuitRoutine-WindowTable;in go away
 pea  EventRecord;ptr to the event record
 move.l WWindow,-(SP);push window ptr
 _SystemClick    ;let system take care of it
 bra  NextEvent  ;next event
 clr.l  -(SP)    ;space for menu choice
 move.l Point,-(SP);mouse at time of event
 _MenuSelect;menu select
 move (SP)+,MenuReg;preserve menu
 move (SP)+,MenuItemReg ;preserve menu item

 cmp  #2,MenuReg ;was it the control menu?
 bne.s  NotQuit  ;no
 cmp  #2,MenuItemReg ;Quit selected?
 bne.s  NotQuit  ;no, continue
 bsr  UnHiliteMenu ;unhilight the menu bar
 move.l DrawPtr(A5),-(SP) ;push ptr to draw window
 _CloseWindow    ;close the window
 move.l windowBitMap,A0
 _DisposPtr ;dispose of the bitmap
 move #-1,D0;say it was Quit

 cmp  #1,MenuReg ;in apple menu?
 beq.s  InAppleMenu;yes, go do apple menu
 cmp  #2,MenuReg ;in file menu?
 beq  ParameterChange;yes

 bsr  UnHiliteMenu ;unhilight the menu bar
 ;fall through get next event
 moveq  #0,D0    ;say it's not Quit
 rts    ;return to event loop
 clr  -(SP) ;all menus
 _HiLiteMenu;unhilite every one
 cmp  #1,MenuItemReg
 beq.s  Info

 move.l AppleMenu(A5),-(SP)
 move MenuItemReg,-(SP)
 pea  DeskName
 _GetItem ;get name of the desk accessory
 clr  -(SP)
 pea  DeskName
 _OpenDeskAcc    ;open the desk accessory
 move (SP)+,D0

 bsr  SetOurPort
 bra  ChoiceReturn

 clr  -(SP)
 move #InfoAlertID,-(SP)
 clr.l  -(SP)
 _Alert ;handle the infoAlert
 move (SP)+,D0
 bra  RestoreOurPort
 clr.l  -(SP)
 move #ParamDlgID,-(SP)
 pea  DStorage
 move.l #-1,-(SP)
 _GetNewDialog   ;get the dialog
 move.l (SP),ParamDlgPtr(A5);preserve the dialogPtr

 _SetPort ;set the port

 move Equation(A5),TopButton;TopButton = current button
 ;highlighted in top section
 move TopButton,D0
 bsr  FindButton ;sub. to return handle
 move.l itemHandle(A5),-(SP)
 move #1,-(SP)
 _SetCtlValue    ;highlight the button
 move Scale(A5),BottomButton;BottomButton = current
 ; button in bottom section
 move BottomButton,D0
 bsr  FindButton
 move.l itemHandle(A5),-(SP)
 move #1,-(SP)
 _SetCtlValue    ;highlight the button

 move Multilayered(A5),CheckBox
 move #17,D0
 bsr  FindButton ;get handle to check-box
 move.l itemHandle(A5),-(SP)
 move CheckBox,-(SP)
 _SetCtlValue    ;check or uncheck it

 clr.l  -(SP)
 pea  ItemHit
 _ModalDialog    ;modal dialog

 move ItemHit,D1
 cmp  #2,D1 ;was it Cancel?
 beq  Done;yes
 cmp  #1,D1 ;was it OK?
 beq  SetNewParams ;yes
 cmp  #9,D1 ;was it in the bottom set?
 bge.s  BottomSet;yes

 move TopButton,D0
 move D1,TopButton ;set new button number
 bsr  FindButton ;get handle to last button
 move.l itemHandle(A5),-(SP)
 clr  -(SP)
 _SetCtlValue    ;unhighlight last button

 move TopButton,D0
 bsr  FindButton
 move.l itemHandle(A5),-(SP)
 move #1,-(SP)
 _SetCtlValue    ;highlight new button
 bra.s  RepeatLoop

 cmp  #14,D1;in bottom set?
 bge.s  CheckForBox;no

 move BottomButton,D0
 move D1,BottomButton
 bsr  FindButton ;get handle to last button
 move.l itemHandle(A5),-(SP)
 clr  -(SP)
 _SetCtlValue    ;unhighlight last button

 move BottomButton,D0
 bsr  FindButton
 move.l itemHandle(A5),-(SP)
 move #1,-(SP)
 _SetCtlValue    ;highlight new button
 bra  RepeatLoop

 cmp  #17,D1;click in the check box?
 bne  RepeatLoop
 cmp  #1,CheckBox
 bne.s  NotOne
 move #0,CheckBox
 bra.s  SetCheck

 move #1,CheckBox
 move #17,D0
 bsr  FindButton ;get handle to check-box
 move.l itemHandle(A5),-(SP)
 move CheckBox,-(SP)
 _SetCtlValue    ;check or uncheck it
 bra  RepeatLoop

 clr  drawn(A5)
 move TopButton,Equation(A5);set new equation number
 move BottomButton,Scale(A5);set new scale
 move CheckBox,Multilayered(A5)  ;set multilayered
 move Scale(A5),D0
 sub  #9,D0
 asl.l  #2,D0
 move.l ScaleTable(D0),RealScale(A5) ;look up real scale

 move.l ParamDlgPtr(A5),-(SP)
 _CloseDialog    ;close the dialog
 bra  RestoreOurPort

 DC.L 140,220,300,380,460
 move.l ParamDlgPtr(A5),-(SP)
 move D0,-(SP)
 pea  itemType
 pea  itemHandle(A5)
 pea  dispRect
 _GetDItem;get handle to itemnumber in D0
;to convert a fract number to type fract, multipy by 65536
 clr.l  -(SP)
 move #4,-(SP)
 _GetCursor ;get the watch cursor (ID=4)
 move.l (SP)+,D0
 move.l D0,A0
 move.l (A0),-(SP)
 _SetCursor ;set the new cursor

 move.l RealScale(A5),-(SP) ;preserve the realScale
 move #720,Counter ;721 steps (360 x 2)
 move.l #572,Increment    ;572 = .5° increment in radians
 moveq  #0,Radians ;present degrees

 bsr  EvalFunction ;evaluate fn r(Radians)

 sub  #14,SP
 move.l Radians,-(SP)
 move.l FuncResult,-(SP)
 _FracMul ;y = fn r(Radians) * sin(Radians)
 move.l RealScale(A5),-(SP)
 _FixMul;y = y * realScale
 move (SP)+,y(A5);y = int(y)

 sub  #14,SP
 move.l Radians,-(SP)
 move.l FuncResult,-(SP)
 _FracMul ;x = fn r(Radians) * cos(Radians)
 move.l RealScale(A5),-(SP)
 _FixMul;x = x * realScale
 move (SP)+,x(A5);x = int(x)

 move x(A5),D0
 add  #154,D0
 move D0,-(SP)   ;x = x + 154

 move y(A5),D0
 add  #144,D0
 move D0,-(SP)   ;y = y + 144

 tst  Radians
 bne.s  NotFirst
 _MoveTo;moveto x,y if first point
 bra.s  Skip

 _LineTo;else lineto x,y
 add.l  Increment,Radians ;Radians = Radians + Increment
 dbra Counter,nextDegree  ;loop back
 cmp  #1,Multilayered(A5) ;another layer?
 bne.s  NoExtraLayers;no
 move.l #20,D0
 sub.l  D0,RealScale(A5)  ;decrease the real scale by 20
 cmp.l  RealScale(A5),D0  ;is the real scale = 20?
 bne  DrawRose1  ;no, do next layer

 move #1,drawn(A5) ;the rose has been drawn
 move.l drawPtr(A5),A0
 pea  portBits(A0) ;put the rose in our window
 pea  windowBitMap ;get from offscreen bitmap
 pea  windowRect ;same size windowRect
 pea  windowRect
 clr  -(SP) ;srcCopy
 clr.l  -(SP)    ;no mask region
 _CopyBits;copy the bits
 move.l (SP)+,RealScale(A5)
 move Equation(A5),D0
 sub  #3,D0
 add  D0,D0
 move EquationTable(D0),D0
 jmp  EquationTable(D0)

 DCOne-EquationTable ;sin 4r
 DCTwo-EquationTable ;cos (2 sin r)
 DCThree-EquationTable  ;cos (2 sin 2r)
 DCFour-EquationTable;cos (sin 100r)
 DCFive-EquationTable;cos (sin 8 r)
 DCSix-EquationTable ;cos (4 sin 2r)
 move.l #0,FuncResult
 ;sin 4r
 clr.l  -(SP)
 move.l Radians,D0
 asl.l  #2,D0
 move.l D0,-(SP)
 move.l (SP)+,FuncResult ;fn r(Radians) = sin (Radians * 4)
 ;cos (2 sin r)
 subq #8,SP
 move.l Radians,-(SP)
 move.l (SP),D0
 asr.l  #7,D0
 asr.l  #6,D0    ;convert to fixed x 2
 move.l D0,(SP)
 move.l (SP)+,FuncResult  ;fn r(Radians) = cos (2 sin r)
 ;cos (2 sin 2r)
 subq #8,SP
 move.l Radians,D0
 asl.l  #1,D0
 move.l D0,-(SP)
 move.l (SP),D0
 asr.l  #7,D0
 asr.l  #6,D0    ;convert to fixed x 2
 move.l D0,(SP)
 move.l (SP)+,FuncResult  ;fn r(Radians) = cos (2 sin 2r)
 ;cos (sin 100r)
 sub  #12,SP
 move.l Radians,-(SP)
 move.l #3276800,-(SP)
 move.l (SP),D0
 asr.l  #7,D0
 asr.l  #7,D0    ;convert to fixed
 move.l D0,(SP)
 move.l (SP)+,FuncResult  ;fn r(Radians) = cos (sin 100r)
 ;cos (sin 8 r)
 subq #8,SP
 move.l Radians,D0
 asl.l  #3,D0
 move.l D0,-(SP)
 move.l (SP),D0
 asr.l  #7,D0
 asr.l  #7,D0
 move.l D0,(SP)
 move.l (SP)+,FuncResult  ;fn r(Radians) = cos (sin 8r)
 ;cos (4 sin 2r)
 subq #8,SP
 move.l Radians,D0
 asl.l  #1,D0
 move.l D0,-(SP)
 move.l (SP),D0
 asr.l  #7,D0
 asr.l  #5,D0
 move.l D0,(SP)
 move.l (SP)+,FuncResult  ;fn r(Radians) = cos (4 sin 2r)
;------------------ Local Variables -----------------
 What:  DC0
 Message: DC.L 0
 When:  DC.L0
 Point: DC.L0
 WWindow: DC.L 0

windowBitMapDC.L 0

ItemHit DC0
DStorageDCB DWindLen,0
DeskNameDCB 16,0
;-------------- Application Globals ----------------
WindowStorage  DSWindowSize ;storage for draw window
DrawPtr DS.L1    ;pointer to draw window
AppleMenu DS.L 1 ;apple menu
EquationDS1 ;equation number
Scale   DS1 ;scale
RealScale DS.L 1 ;longint real scale
MultiLayeredDS 1 ;multilayered (0 = false, 1 = true)
ParamDlgPtr DS.L 1 ;pointer to the parameter dialog
itemHandleDS.L 1 ;itemHandle
yDS1    ;y-value
xDS1    ;x-value
drawn   DS1

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Animate CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Flash Professional customer). Animate CC 2018 (was Flash CC) lets you... Read more
Adobe After Effects CC 2018 15.0 - Creat...
After Effects CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous After Effects customer). The new, more connected After Effects CC... Read more
Adobe Premiere Pro CC 2018 12.0.0 - Digi...
Premiere Pro CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Premiere Pro customer). Adobe Premiere Pro CC 2018 lets you edit... Read more
Alarm Clock Pro 10.3 - $19.95
Alarm Clock Pro isn't just an ordinary alarm clock. Use it to wake you up in the morning, send and compose e-mails, remind you of appointments, randomize the iTunes selection, control an internet... Read more
Adobe Lightroom 20170919-1412-ccb76bd] -...
Adobe Lightroom is available as part of Adobe Creative Cloud for as little as $9.99/month bundled with Photoshop CC as part of the photography package. Lightroom 6 is also available for purchase as a... Read more
Adobe Illustrator CC 2018 22.0.0 - Profe...
Illustrator CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Illustrator customer). Adobe Illustrator CC 2018 is the industry... Read more
Hopper Disassembler 4.3.0- - Binary disa...
Hopper Disassembler is a binary disassembler, decompiler, and debugger for 32- and 64-bit executables. It will let you disassemble any binary you want, and provide you all the information about its... Read more
Adobe InDesign CC 2018 - Prof...
InDesign CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous InDesign customer). Adobe InDesign CC 2018 is part of Creative Cloud.... Read more

ICEY (Games)
ICEY 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: ICEY is a 2D side-scrolling action game. As you follow the narrator's omnipresent voice, you will see through ICEY's eyes and learn the... | Read more »
The best new games we played this week -...
We've made it, folks. Another weekend is upon us. It's time to sit back and relax with the best new releases of the week. Puzzles, strategy RPGs, and arcade games abound this week. There's a lot of quality stuff to unpack this week, so let's hop... | Read more »
Wheels of Aurelia (Games)
Wheels of Aurelia 1.0.1 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0.1 (iTunes) Description: | Read more »
Halcyon 6: Starbase Commander guide - ti...
Halcyon 6 is a well-loved indie RPG with stellar tactical combat and some pretty good writing, too. It's now landed on the App Store, so mobile fans, if you're itching for a good intergalactic adventure, here's your game. Being a strategy RPG, the... | Read more »
Game of Thrones: Conquest guide - how to...
Fans of base building games might be excited to know that yet another entry in the genre has materialized - Game of Thrones: Conquest. Yes, you can now join the many kingdoms of the famed book series, or create your own, as you try to conquer... | Read more »
Halcyon 6: Starbase Commander (Games)
Halcyon 6: Starbase Commander Device: iOS Universal Category: Games Price: $6.99, Version: (iTunes) Description: An epic space strategy RPG with base building, deep tactical combat, crew management, alien diplomacy,... | Read more »
Legacy of Discord celebrates its 1 year...
It’s been a thrilling first year for fans of Legacy of Discord, the stunning PvP dungeon-crawling ARPG from YOOZOO Games, and now it’s time to celebrate the game’s first anniversary. The developers are amping up the festivities with some exciting... | Read more »
3 reasons to play Thunder Armada - the n...
The bygone days of the Battleship board game might have past, but naval combat simulators still find an audience on mobile. Thunder Armada is Chinese developer Chyogames latest entry into the genre, drawing inspiration from the explosive exchanges... | Read more »
Experience a full 3D fantasy MMORPG, as...
Those hoping to sink their teeth into a meaty hack and slash RPG that encourages you to fight with others might want to check out EZFun’s new Eternity Guardians. Available to download for iOS and Android, Eternity Guardians is an MMORPG that lets... | Read more »
Warhammer Quest 2 (Games)
Warhammer Quest 2 1.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0 (iTunes) Description: Dungeon adventures in the Warhammer World are back! | Read more »

Price Scanner via

12″ iPad Pros on sale for $50 off MSRP, no ta...
Adorama has 12″ iPad Pros on sale today for $50 off MSRP. Shipping is free, and Adorama charges sales tax in NY & NJ only: – 12″ 64GB iPad Pro: $749, save $50 – 12″ 256GB iPad Pro: $899, save $50... Read more
9″ iPads on sale for $30 off, starting at $29...
MacMall has 9″ iPads on sale for $30 off including free shipping: – 9″ 32GB iPad: $299 – 9″ 128GB iPad: $399 Read more
Apple restocks full line of refurbished 13″ M...
Apple has restocked a full line of Apple Certified Refurbished 2017 13″ MacBook Pros for $200-$300 off MSRP. A standard Apple one-year warranty is included with each MacBook, and shipping is free.... Read more
13″ 3.1GHz/256GB MacBook Pro on sale for $167...
Amazon has the 2017 13″ 3.1GHz/256GB Space Gray MacBook Pro on sale today for $121 off MSRP including free shipping: – 13″ 3.1GHz/256GB Space Gray MacBook Pro (MPXV2LL/A): $1678 $121 off MSRP Keep an... Read more
13″ MacBook Pros on sale for up to $120 off M...
B&H Photo has 2017 13″ MacBook Pros in stock today and on sale for up to $120 off MSRP, each including free shipping plus NY & NJ sales tax only: – 13-inch 2.3GHz/128GB Space Gray MacBook... Read more
15″ MacBook Pros on sale for up to $200 off M...
B&H Photo has 15″ MacBook Pros on sale for up to $200 off MSRP. Shipping is free, and B&H charges sales tax in NY & NJ only: – 15″ 2.8GHz MacBook Pro Space Gray (MPTR2LL/A): $2249, $150... Read more
Roundup of Apple Certified Refurbished iMacs,...
Apple has a full line of Certified Refurbished 2017 21″ and 27″ iMacs available starting at $1019 and ranging up to $350 off original MSRP. Apple’s one-year warranty is standard, and shipping is free... Read more
Sale! 27″ 3.8GHz 5K iMac for $2098, save $201...
Amazon has the 27″ 3.8GHz 5K iMac (MNED2LL/A) on sale today for $2098 including free shipping. Their price is $201 off MSRP, and it’s the lowest price available for this model (Apple’s $1949... Read more
Sale! 10″ Apple WiFi iPad Pros for up to $100...
B&H Photo has 10.5″ WiFi iPad Pros in stock today and on sale for $50-$100 off MSRP. Each iPad includes free shipping, and B&H charges sales tax in NY & NJ only: – 10.5″ 64GB iPad Pro: $... Read more
Apple iMacs on sale for up to $130 off MSRP w...
B&H Photo has 21-inch and 27-inch iMacs in stock and on sale for up to $130 off MSRP including free shipping. B&H charges sales tax in NY & NJ only: – 27″ 3.8GHz iMac (MNED2LL/A): $2179 $... Read more

Jobs Board

Engineering Manager, *Apple* Retail Enginee...
# Engineering Manager, Apple Retail Engineering Job Number: 58139948 Santa Clara Valley, California, United States Posted: 20-Oct-2017 Weekly Hours: 40.00 **Job Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
Commerce Engineer, *Apple* Media Products -...
Commerce Engineer, Apple Media Products (New York City) Job Number: 113028813New York City, New York, United StatesPosted: Sep. 20, 2017Weekly Hours: 40.00 Job Read more
US- *Apple* Store Leader Program - Apple (Un...
US- Apple Store Leader Program Job Number: VariousUnited StatesPosted: Oct. 19, 2017Retail Store Job Summary Learn and grow as you explore the art of leadership at Read more
Product Manager - *Apple* Pay on the *Appl...
Job Summary Apple is looking for a talented product manager to drive the expansion of Apple Pay on the Apple Online Store. This position includes a unique Read more
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