Sine Table Demo

One of the best ways to learn the fine points of Mops programming is to study existing programs and then work slowly to customize them by modifying methods, defining new subclasses, creating new Mops words and objects, and sending messages to the various objects in memory.

In the next few lessons, You'll be studying two programs whose source files are in the folder 'Demo folder' as plain document files. The first one is called Sin (the source file "Sin"), the second is called Turtle (the source file "Turtle"). Although we provide a listing for you in the next pages, you might also want to print out a copy of the source code to follow along as the discussion works its way into the lesson. Sin is an excellent example of how Mops array-type data structures work. Turtle reinforces the class-object relationship.

In the source code discussions in these lessons, the code will be shown with line numbers off to the left margin. These have been inserted here only to make it easier to refer to precise lines of code when explaining various operations. There are, of course, no line numbers in Mops code (although if you are using QuickEdit, you will have a general idea where you are in the file).

Building a Sine Table

Before we proceed, it's important that you understand what these programs were designed to do, just as you should clearly define the goal and operation of every Mops program you write.

Sin will actually be a general purpose building block for a great many programs, including some you may write later. Its purpose is to create a reference table of sine values plus a fast and simple way for later program parts to retrieve sine and cosine values.

If you're a little rusty on trigonometry, a sine value of an angle is a convenient way to work with angular measurement. Mathematically, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse of an imaginary right triangle having that angle in it. For example, if we have an angle labeled θ (theta):

The sine of θ equals the length of A (the opposite side of theta) divided by C (the hypotenuse).

If you were to calculate all possible values for sin θ, from 0 to 360 degrees and plot the results, you'll find that the values trend up and down throughout the circle, including two quadrants with negative values.

Notice, however, that two angular measurements can, for example, have sine values of 0.5. In the first quadrant, it's at 30 degrees. In the second quadrant, it's at 150 degrees, -30 degrees from the "zero" value (180 degrees). In other words, the sin values in the first and second quadrants, dividing them at the 90 degree mark, are mirror images of each other.

The same is true for quadrants three and four. And the relationship between one half (0-180 degrees) to the other (180-360 degrees) is that the second half mirrors the first, but as negative values. Therefore, if you have a table of sine values for -90 degrees, it is a relatively simple matter to calculate the corresponding values in each of the remaining quadrants. The Sin program takes care of both the table and calculations.

Some graphics programs will likely need to fetch sine or cosine values to draw sophisticated shapes on the screen. Sin (and its classes TrigTable and Angle) will probably come in handy for you in the future.

Sin will often be summoned from the second program, Turtle. The intent of Turtle is two-fold. First of all, it will create class definitions of a pen (PEN) and a polygon (POLY) that you'll use to experiment developing a Logo-like environment. Turtle will also use the pen and polygons it creates (along with definitions from Sin) to draw some sophisticated graphics on the screen. As it turns out, these graphics will be incorporated into yet a third demonstration program which will be the subject of the last lessons in this Tutorial.

This building block approach is a common tactic in designing a Mops program. Carefully, generically-designed building blocks, such as Sin and parts of Turtle, can be used in a wide variety of programs, making it easier and faster to assemble programs from your library of proven blocks.

How the Sine Table Works

Let's start with the Sin source code, which is numbered from lines 1 to 76:

1 \ These classes obtain the sine and cos of an angle by table lookup.
2 \ Modified from the original Neon version by Mike Hore.
4 \ The main class is ANGLE, which has SIN: and COS: methods that look
5 \ up a table defined with the TRIGTABLE class.
8 need struct1
11 :class TRIGTABLE super{ wArray }
13 4 wArray AXISVALS \ 90 degree values
15 :m SIN: { degree \ quadrant -- sin }
16 \ Looks up a sin * 10000 of an angle
18 degree 360 mod \ Put angle in range -359 to +359
19 dup 0< IF 360 + THEN \ Now 0 to +359
20 90 /mod \ Convert angle to range 0-89 and get quadrant
21 -> quadrant -> degree
22 degree \ Test for an axis
23 NIF quadrant at: axisVals \ If an axis, get value
24 ELSE quadrant 1 and \ True for "mirror" quadrants 1 and 3
25 IF 90 degree - \ Create mirror image
26 ELSE degree
28 at: self \ Get sin for this degree
29 quadrant 2 and \ True for "negative" quadrants 2 and 3
30 IF negate THEN
31 THEN ;m
33 :m COS: \ ( degree -- cos )
34 90 + sin: self ;m \ Cos is sin shifted by 90 degrees
37 0 0 to: axisvals
38 10000 1 to: axisvals
39 0 2 to: axisvals
40 -10000 3 to: axisvals ;m
42 ;class
44 90 TrigTable SINES \ system-wide table of sines
46 : 's \ ( val degree -- ) Fills a Sin table entry
47 to: sines ;
49 00000 00 's 00175 01 's 00349 02 's 00524 03 's 00698 04 's
50 00872 05 's 01045 06 's 01219 07 's 01392 08 's 01571 09 's
51 01736 10 's 01908 11 's 02079 12 's 02250 13 's 02419 14 's
52 02588 15 's 02756 16 's 02924 17 's 03090 18 's 03256 19 's
53 03420 20 's 03584 21 's 03746 22 's 03907 23 's 04067 24 's
54 04226 25 's 04384 26 's 04540 27 's 04695 28 's 04848 29 's
55 05000 30 's 05150 31 's 05299 32 's 05446 33 's 05592 34 's
56 05736 35 's 05878 36 's 06018 37 's 06157 38 's 06293 39 's
57 06428 40 's 06561 41 's 06691 42 's 06820 43 's 06947 44 's
58 07071 45 's 07193 46 's 07314 47 's 07431 48 's 07547 49 's
59 07660 50 's 07771 51 's 07880 52 's 07986 53 's 08090 54 's
60 08192 55 's 08290 56 's 08387 57 's 08480 58 's 08572 59 's
61 08660 60 's 08746 61 's 08829 62 's 08910 63 's 08988 64 's
62 09063 65 's 09135 66 's 09205 67 's 09272 68 's 09336 69 's
63 09397 70 's 09455 71 's 09511 72 's 09563 73 's 09613 74 's
64 09659 75 's 09703 76 's 09744 77 's 09781 78 's 09816 79 's
65 09848 80 's 09877 81 's 09903 82 's 09925 83 's 09945 84 's
66 09962 85 's 09976 86 's 09986 87 's 09994 88 's 09998 89 's
68 : SIN sin: sines ;
69 : COS cos: sines ;
71 :class ANGLE super{ int }
73 :m SIN: get: self sin ;m
74 :m COS: get: self cos ;m
76 ;class

Lines 1 to 5

These lines are comments that serves as a plain English heading for the source code, describing what this module does, who wrote it, and what its main features are. This particular module creates a table of sine values that Turtle will use to draw complex curves and graphics. We use the "backslash" (\) type of comment here, in which a word consisting of just a backslash causes Mops to ignore the rest of that line.

Line 8

This line causes Mops to load (compile) the file "Struct1" (in the folder 'Extras'), if it is not already loaded. This file contains the definitions for the class wArray, which we will need here. The use of the syntax 'need \<var\>myfilename\</var\>' means that you don't have to worry about whether the file <var>myfilename</var> is already loaded or not. In fact, it may sometimes be loaded, sometimes not, at different stages of your program development. Using the NEED syntax means that you are explicitly stating the requirements of this source file; Mops will then take care of the details.

Line 11

Here marks the beginning of a class definition for the class TrigTable. This class establishes the rules and procedures that will be followed for looking up sines in a sine table (the table is created in lines 44-66). Since the sine table will be a list of sine values in fixed-point arithmetic (in a range of 0 to 10,000), two bytes of data could be used for each entry (10000 decimal is 2710 hexadecimal, with each two-digit hexadecimal number taking up one byte of memory). Class TrigTable is defined as a subclass of class wArray.

If you look at the source code listing for the superclass wArray (in the file "Struct1"), You'll notice that wArray is defined as an indexed class:

`:class WARRAY  super{ indexed-obj }  2 indexed`\

When a class in indexed, it means that every object created of that class must explicitly state how big an area of memory is to be reserved for its private data (or, how many data slots should be reserved). The number 2 in the class wArray definition indicates that each slot is to be two bytes wide. When it comes time to create an object from an indexed class, the line of code must begin with the number of data slots that object will need (each slot has a unique index number associated with it). In line 44 of our Sin program, the object Sines created of class TrigTable is reserving 90 slots; each slot is two bytes wide because TrigTable inherits wArray's two byte wide indexed class behavior. Indexing should become more clear as we describe the rest of this class definition and see some practical examples.

Line 13

This line establishes the instance variable (ivar) for an object of class TrigTable. Every object created from class TrigTable will have space reserved for the array created here, as well as the indexed data noted above. The array is preceded by the number of elements that it will contain in every instance of class TrigTable (4 in this case).

This array, AxisVals, is a 4 element array of 2 byte cells. The range of values to be stored in this array is from -10,000 to +10,000 (the integer values the program will use to signify sine values). The values in these four cells will be the sine values (times 10000) of the 90 degree multiples (0, 90, 180, and 270 degrees), and will play a role in the calculation of the sine value later in this class definition.

Here is a summary of the four quadrants, their signs, and sine values:

+----------------------------------+----------------------------------+ | | Quadrant Sign | | | Degree Range Sine Value Range | | | ---------- ------ - | | | ------------- ------------------ | | | 0 | | | + 0 to 90 0 to 10000 | | | 1 | | | + 90 to 180 10000 to 0 | | | 2 \ | | | - 180 to 270 0 to -10000 | | | 3 \ | | | - 270 to 360 -10000 to 0 | +----------------------------------+----------------------------------+

In subsequent reference to our four quadrants, we'll refer to them as they are listed here (counting up from integer , not the natural number one).

Line 15

This begins the definition of the method SIN:. The notation '{ degree \\ quadrant \-- sin }' indicates that there is one named input parameter for this method, called DEGREE, and one local variable, QUADRANT. References to such values are made by their name, not by their stack location, thus eliminating much stack manipulation in the course of calculating sine values in the next several lines of code. Within the definition, QUADRANT will be used to store the value of the quadrant (0, 1, 2, 3) for which the sine is being calculated.

This curly braces notation also serves the purpose of documenting the stack effects of the execution of the method. This tells you that if you use this SIN: method as a selector in a message, and if you pass a degree figure as a parameter, (e.g., '90 SIN: \<var\>myobject\</var\>'), then the corresponding sine value would be left on the stack when the method's computations are completed.

Line 16

The comment tells you what is happening in this method: the program looks up the sine value of an angle (in degrees). In the calculations the actual sine values will be (implicitly) multiplied by a factor of 10000. All sine values in the sine table, therefore, will be integers.

Lines 18 to 34

Next comes the actual calculation and retrieval of the sine values. Because the math in this calculation is so tightly interwoven with IF¦THEN decision constructions, we will trace what happens to the stack at each step, as well as explain why various operations are performed.

As an overview, we can say that the math calculations first convert the degree value to be in the range -359. Allowance is made for degree values entered as negative numbers, or degrees of magnitude 360 or greater. Once the degree is "normalized", it is converted to the equivalent degree in the range -89 and the quadrant is saved for doing mirror image calculations and determining the sign. For degrees on an axis (, 90, 180, or 270) the sine is gotten from the ivar AxisVals. Otherwise a lookup is performed on the TrigTable array.

To best understand the operation of the decision processes in this section, we will follow what happens to the values on the stack when we try degree values less than 90 degrees, exactly 180 degrees, and a value in the third quadrant (somewhere between 270 and 360). But to do this properly, we should go on to explain how the arrays are filled with the values that the method SIN: will be retrieving, and what those values mean.

Line 36 to 40

The method CLASSINIT: is a special method that executes whenever an object of the current class is created. The operations in this particular CLASSINIT: are four messages, all of them TO: operations. The TO: selector of these messages is defined by a TO: method in the receiver's class. Since the receiver is the ivar AxisVals, the class in question is wArray.

For arrays, the methods AT: and TO: are the equivalents of GET: and PUT: for ordinary scalar objects. They expect an index on the stack at the start, to indicate which array element is to be accessed.

Since class TrigTable has now been defined (all the code from line 10 through line 42), we can now create an actual table in memory as an object of that class. The statement in line 44 does just that, establishing an indexed array object, called Sines, capable of storing 90 values in addition to the ivar AxisVals. At this point, no values have been entered into the 90 cells of the Sines array, but the space is there, ready for values to be plugged in. The array bears the characteristics of arrays defined in TrigTable's superclass, wArray.

Line 46 to 66

While the columns of numbers in lines 49 through 66 may look intimidating, they are really nothing more than the values of what becomes a computer version of a lookup table, like the kind at the end of a trigonometry text book. Lines 46 and 47 define a Mops word, 's (the apostrophe is pronounced "tick") that performs a similar kind of TO: storage operation as demonstrated in CLASSINIT:, but this time the storage is to an instance of TrigTable called Sines. When Sines receives a message with a TO: selector, Sines first looks in its own class (TrigTable) for a matching definition. Since there is none here, Sines then looks to its superclass wArray, where it finds a TO: method.

The stack notation for this definition uses round parentheses rather than braces, since it is a straight comment, not a definition for named input parameters or local variables. This definition makes no use of named parameters or locals. To make it quite clear, we have put a \\ first (don't forget the space!), which makes the remainder of the line a comment anyway.

The table was designed so that the values of the degrees to be looked up would range from to 89. That way, these very degree values can also be used as index numbers to the respective sine values in the table. Therefore, when it comes time (in the SIN: method, for instance) to lookup a sine value in the table, the degree value coming in as a parameter from a message will be used as the index value associated with the desired sine value. We'll see how that works in a moment, but that's why the stack notation in line 46 indicates that the parameters to be passed with each 's operation are the sine value and the angle in degrees, when in actuality, the TO: selector sees the degree figure literally as only an index number to a value within the array (our table).

The sine values, then, are added to the table by the long series of 's operations, each preceded by the sine value (times 10000) and the double-purpose index/degree value.

What Happens On the Stack

Now we can go back to method SIN: in lines 15 to 31 to see what happens when we send three different degree values and the SIN: selector to the Sines object. The three values will be 35, 180, and 293 degrees. In the listings below, the numbers next to each operation indicate the actual numbers on the stack at that instant of execution. When more than one number is on the stack, the topmost number in the listing is the number on the top of the stack.

Statement 35° 180° 293°

degree 35 180 293 360 360 360 360 35 180 293 mod 35 180 293 dup 35 180 293 35 180 293 0< 0 0 0 35 180 293 IF 35 180 293 360 : : : + : : : 90 90 90 90 35 180 293 /mod 0 2 3 35 0 23 -> quadrant 35 0 23 -> degree --- --- --- degree 35 0 23 NIF --- --- --- : : quadrant : 2 : : : at: axisVals : 0 : ELSE --- : --- quadrant 0 : 3 1 1 : 1 0 : 3 and 0 : 1 IF --- : --- : :
90 : : 90 : :
degree : : 23 : : 90 : :
- : : 67 : : : ELSE --- : : : : degree 35 : : : : THEN 35 : 67 :
at: self 5736 : 9205 :
quadrant 0 : 3 5736 : 9205 :
at: signs 0 : 1 5736 : 9205 :
IF 5736 : 9205 : :
negate : : -9205 : :
THEN 5736 : -9205 :
THEN 5736 0 -9205

Now for a description of what happens to each degree value.

The MOD operation in line 18 provides the stack with the remainder of dividing the degree entry by 360. If the entry was 360 or more, this will normalize the degree value to be between and 359. If the entry was negative, the MOD operation returns a negative value between -359 and , and further normalization is required. Line 19 tests the result to see if it was negative, and if it was negative, adds 360 to convert it to the equivalent positive angle (correctly in the range to 359).

The /MOD operation on line 20 takes the normalized degree value off the stack and returns a quotient and remainder. A quotient of zero indicates it is in the upper-right quadrant, a one places the degree in the second quadrant in the upper-left, and so on. The remainder becomes the degree value that will be checked against the sine table, since the table contains values for only a 90 degree chunk of the full 360 degree range. On line 21 these values are taken from the stack and put into local storage.

For the next operation on line 22 and 23, we recall the value from DEGREE (but this does not remove it from DEGREE, it only copies it onto the stack) and test to see if it is equal to zero.

If the value is zero, that means that the degree value is a multiple of 90 degrees, and therefore lies on a boundary between two quadrants. To save time and calculation, the sine values for those four boundaries have been stored in the AxisVals array. Since the degree value is zero, the operation after the NIF statement on line 23 is performed. The quadrant value saved earlier is placed on the stack and used as an index for the AT: selector. The AT: method in AxisVals' class, wArray, is the opposite of the TO: storage operator, which was used to place values in the arrays. The AT: operation instead fetches a value from an array object (in this case named AxisVals) according to the index number that is on the top of the stack. In our 180 degree example, a value of 2 was saved in quadrant and the put on the stack. The value in the AxisVals cell corresponding to the index 2 is then placed on the stack (it has only been copied from the array, not removed). At this point, the final sine value is in the stack, so there is no further operation needed. Following the rules of nested IF¦ELSE¦THEN statements, execution continues to the outermost THEN statement, which is at the end of the method.

But when the degree value is not zero, much more happens. The quadrant value is ANDed with 1 on line 24 and tested to see if is 1 or 3. If so, then the DEGREE value is recalled and has 90 degrees subtracted from it on line 25 (sine values increase to 90 degrees, then decrease to 180 in a reverse, mirror image). Otherwise, just the DEGREE value is placed on the stack again on line 26.

In line 28, the AT: selector takes the index value currently on the stack (it also happens to be the degree to be checked in the sine table) and fetches the value from the Sines array. The SELF notation tells Mops to perform the AT: fetch itself, in this case, the Sines object.

That AT: fetch operation places the sine value from the table on the stack. One last job remains�to determine if the sine value is positive or negative. The quadrant number is ANDed with 2 on line 29. If the quadrant is 2 or 3, which are the quadrants for which the sine is negative, the result of this AND will be non-zero. In this case the sine value, which is all that remains on the stack, is made negative (with the NEGATE operation of line 30), otherwise, it stands positive, and the method ends.

The COS: method in lines 33 and 34 uses the power of the SIN: method, but simply modifies it to take into account the mathematical relationship between a sine and cosine of an angle. A cosine can be calculated from a sine by phase-shifting 90 degrees.

At this point in the program (up to line 66), the kind of message you would send to calculate the sine of a given value in degrees would be:

125 sin: Sines

To simplify this even more, two Mops definitions are added (lines 68-69). Each word sends a message like the one above. Thereafter, the only code you need in a program to obtain the sine of an angle is:

125 sin

Lines 71 to 76

Class Angle provides an example of how the SIN and COS definitions in lines 68 and 69 can be used in other class definitions, even though those words are defined by messages to objects of a different class. This class, an integer class, has two methods, SIN: and COS:. They may appear to have the same method names as methods in class TrigTable, but there will be no interference between the two. That's because if you create an object of class Angle, that object looks up methods only in its own class hierarchy. It doesn't even know class TrigTable exists. When a method in class Angle uses the new Mops word SIN, it lets the word reach into memory to do what it has to, even if it means working in other classes�all without disturbing the integrity of class Angle and inheriting all the baggage of TrigTable.

The 'get: self' message (lines 73 and 74) retrieves the value of the integer stored in an object created from class Angle. To store a value in that object, you would need to look through class Angle's hierarchy until you found a PUT: method in the Int superclass that stores the value. For example, if you create an object like this:

angle Narrow

you are setting aside a cell in Narrow's memory for an integer, because class Angle is a subclass of the Int class. You would then need to send the message:

30 put: Narrow

to store the value, 30, in the object Narrow. After that, you can send the message:

sin: Narrow

which sets the SIN: method in class Angle to work. The value, 30, is retrieved by the 'get: self' operation, and then the sine value is calculated by the Mops word, SIN.

With Mops loaded up and running, try this out yourself. Use Mops' Load menu item (in the File menu) to load the file "Sin" (in 'Demo folder') into the dictionary, and then create an object of class Angle. Then, PUT: a value in the object, then send messages to that object to calculate the sine and cosine of the value.