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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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MacPilot gives you the power of UNIX and the simplicity of Macintosh, which means a phenomenal amount of untapped power in your hands! Use MacPilot to unlock over 1,200 features, and access them all... Read more
Transmission 3.00 - Popular BitTorrent c...
Transmission is a fast, easy, and free multi-platform BitTorrent client. Transmission sets initial preferences so things "just work", while advanced features like watch directories, bad peer blocking... Read more
Doom 3 1.3.1 - First-person shooter acti...
A massive demonic invasion has overwhelmed the Union Aerospace Corporation's (UAC) Mars Research Facility, leaving only chaos and horror in its wake. As one of only a few survivors, you must fight... Read more
Box Sync 4.0.8004 - Online synchronizati...
Box Sync gives you a hard-drive in the Cloud for online storage. Note: You must first sign up to use Box. What if the files you need are on your laptop -- but you're on the road with your iPhone? No... Read more
LibreOffice - Free, open-source...
LibreOffice is an office suite (word processor, spreadsheet, presentations, drawing tool) compatible with other major office suites. The Document Foundation is coordinating development and... Read more
Day One 4.14 - Maintain a daily journal.
Day One is an easy, great-looking way to use a journal / diary / text-logging application. Day One is well designed and extremely focused to encourage you to write more through quick Menu Bar entry,... Read more
Doxie 2.12.2 - Scan, share, and store do...
Introducing Doxie, the new, modern paper scanner that's so simple, it'll revolutionize the way you think about sharing and storing docs and photos forever. Doxie is ultra-portable, fully automatic,... Read more
Tinderbox 8.7.0 - Store and organize you...
Tinderbox is a personal content management assistant. It stores your notes, ideas, and plans. It can help you organize and understand them. And Tinderbox helps you share ideas through Web journals... Read more
Viber 13.0.0 - Send messages and make fr...
Viber lets you send free messages and make free calls to other Viber users, on any device and network, in any country! Viber syncs your contacts, messages and call history with your mobile device, so... Read more

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SINoALICE, Yoko Taro and Pokelabo's...
Yoko Taro and developer Pokelabo's SINoALICE has now opened for pre-registration over on the App Store. It's already amassed 1.5 million Android pre-registrations, and it's currently slated to launch on July 1st. [Read more] | Read more »
Masketeers: Idle Has Fallen's lates...
Masketeers: Idle Has Fallen is the latest endeavour from Appxplore, the folks behind Crab War, Thor: War of Tapnarok and Light A Way. It's an idle RPG that's currently available for Android in Early Access and will head to iOS at a later date. [... | Read more »
Evil Hunter Tycoon celebrates 2 million...
Evil Hunter Tycoon has proved to be quite the hit since launching back in March, with its most recent milestone being 2 million downloads. To celebrate the achievement, developer Super Planet has released a new updated called Darkness' Front Yard... | Read more »
Peak's Edge is an intriguing roguel...
Peak's Edge is an upcoming roguelike puzzle game from developer Kenny Sun that's heading for both iOS and Android on June 4th as a free-to-play title. It will see players rolling a pyramid shape through a variety of different levels. [Read more] | Read more »
Clash Royale: The Road to Legendary Aren...
Supercell recently celebrated its 10th anniversary and their best title, Clash Royale, is as good as it's ever been. Even for lapsed players, returning to the game is as easy as can be. If you want to join us in picking the game back up, we've put... | Read more »
The Magic Gladiator class arrives in MU...
The Magic Gladiator class is now available in MU Origin 2 following the most recent patch. It also marks the start of Abyss Season 11 and the introduction of Couple Skills and Couple Dungeons. [Read more] | Read more »
The 5 Best Racing Games
With KartRider Rush+ making a splash this past week, we figured it was high time we updated our list of the best mobile racing games out there. From realistic racing sims to futuristic arcade racers (and even racing management games!), check out... | Read more »
KartRider Rush+ Guide - Tips for new rac...
KartRider Rush+ continues to be a surprisingly refreshing and fun kart racer that's entirely free-to-play. The main reason for this is just how high its skill ceiling is. Check out the video above if you're curious to know what top level play looks... | Read more »
KartRider Rush+ might be good, actually?
It's hard to find good racing games on mobile. Most of them are free-to-play, and free-to-play racers generally suck. Even Nintendo couldn't put together a competent Mario Kart game, opting instead for a weird score chaser that resembles--but feels... | Read more »
LifeAfter, NetEase's popular surviv...
A new map will be making its way into NetEase's popular survival game LifeAfter. The map is set to arrive on May 28th and will introduce a volcano that's teetering on the verge of eruption, bringing a host of added challenges to the game. [Read... | Read more »

Price Scanner via

Memorial Day Weekend Sale: Take $300 off thes...
Apple resellers are offering $300 discounts on select 16″ MacBook Pros as part of their Memorial Day Weekend 2020 sales. Prices start at $2099: – 16″ 2.6GHz 6-Core Space Gray MacBook Pro: $2099 at... Read more
Best Memorial Day Weekend 2020 Apple AirPods...
Apple resellers are offering discounts ranging up to $50 off MSRP on AirPods as part of their Memorial Day Weekend 2020 sales. These are the best deals today on various AirPods models. See our... Read more
Memorial Day Weekend Sale: 10″ Apple iPads fo...
Amazon is offering new 10.2″ iPads for $80-$100 off Apple’s MSRP as part of their Memorial Day Weekend 2020 sale, with prices starting at only $249. These are the same iPads sold by Apple in their... Read more
Memorial Day Weekend Sale: 2020 Apple iPhone...
Sprint is offering Apple’s new 2020 64GB iPhone SE for $0 per month for 18 months as part of their Memorial Day Weekend 2020 sale. New line of service and trade-in required. Offer is valid from 5/22/... Read more
Amazon’s popular $100 Apple Watch Series 5 di...
Amazon has Apple Watch Series 5 GPS + Cellular models on sale for up to $100 off Apple’s MSRP today. Shipping is free. These are the same Apple Watch models sold by Apple in their retail and online... Read more
2020 13″ 4-Core MacBook Air on sale for $949,...
Apple reseller Adorama has the new 2020 13″ 1.1GHz 4-Core Space Gray MacBook Air on sale today for $949 shipped. Their price is $50 off Apple’s MSRP, and it’s the lowest price currently available for... Read more
Apple Retail Chief Announces Staggered Reopen...
NEWS: 05.20.20 – In the midst of a global pandemic, after its retail outlets were shuttered temporarily in mid-March as a mitigation measure enacted by Apple during the peak of the spread of COVID-19... Read more
Apple’s Pro Display XDR models in stock today...
Abt Electronics has Apple’s new 32″ Pro Display XDR models in stock and on sale today for up to $305 off MSRP. Shipping is free. Their prices are currently the lowest available for these new models... Read more
Apple restocks refurbished iPhone 8 for $339
Apple has restocked Apple Certified Refurbished 64GB iPhone 8 models for only $339. Apple dropped their price on this phone last month by $160, from $499 to $339. Each refurbished iPhone comes with a... Read more
New at AT&T: 50% off iPhone 11 for new cu...
AT&T is offering a 50% off the 64GB iPhone 11 for new customers who switch to AT&T and open a new line of service. Discount applied over a 30 month period The fine print: “iPhone 11 64GB for... Read more

Jobs Board

Cub Foods - *Apple* Valley - Now Hiring Par...
Cub Foods - Apple Valley - Now Hiring Part Time! United States of America, Minnesota, Apple Valley New Retail Operations Post Date 5 days ago Requisition # Read more
Senior Data Engineer - *Apple* - Theorem, L...
Job Summary Apple is seeking an experienced, detail-minded data engineeringconsultant to join our worldwide business development and strategy team. If you are Read more
Cub Foods - *Apple* Valley - Now Hiring Par...
Cub Foods - Apple Valley - Now Hiring Part Time! United States of America, Minnesota, Apple Valley New Retail Operations Post Date 4 days ago Requisition # Read more
Medical Screener - *Apple* Hill - Evenings...
Medical Screener - Apple Hill - Evenings Tracking Code D-MS-AH-E Job Description Medical Screener - Temporary We Are Hiring: WellSpan Health has a temporary Medical Read more
*Apple* Systems Administrator - Solidus Tech...
Solidus is searching for an Apple Systems Engineer. The engineer will be responsible for desktop and server infrastructure. This includes support for operating Read more
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