Overview of Project Butterfly

3. Non-Object uses of the WMS-Library 3.1. The Three Plains of Drawing
3.2. Normalized Device Coordinates
3.3. One Theoretical Application 4. Downfalls
5. Project Motivation
6. Project Goals
7. Requirements and Assumptions
Bibliography

Abstract

Project BUTTERFLY is the design and development of an object-oriented C++ graphics library called the WMS-Library (World Modeling Scheme Library). It's not an application specific program, but an experimental application specific tool used by the application programmer to develop specific applications, or just simulate the reality of an image. The WMS-Library is unique in that it uses an object-oriented architecture and allows a programmer to graphically model reality using object orientation. The code name BUTTERFLY represents the butterflies polymorphic nature that has been said to be the essence of a truly object-oriented design.

This document begins by discussing the WMS-Library concepts as applied to an ideal called the World Modeling Scheme. Using an example to demonstrate the World Modeling Scheme we see how to simulate reality. We then move to a discussion of the world coordinate theory implemented by the World Modeling Scheme, not the entirety of the library, but the ideal. Following this is a more general discussion of the WMS-Library that reveals those applications not apparent while specifically discussing the World Modeling Scheme. Remaining parts of this chapter give reasons for project BUTTERFLY, what my goals are, the project requirements and assumptions, and some of the foreseen down falls.

Discussions delve heavily into the object-orient paradigm and those theories of the paradigm that are built into the C++ language. It is recommended, that if the reader is not familiar with these theories, for he/she to reference books pertaining to the subject of this report; Object Oriented development. Theories involving computer graphics are moderately exercised compared with that of the object-oriented paradigm. However, some material on computer graphics may be useful in the reading of this report.

Illustrations

Figure 1 George The Robot

Figure 2 The World Modeling Scheme Hierarchy

Figure 3 George's World Model

Figure 4 George's Pseudocode

Figure 5 Window To View Mapping of A Graphics Port

Figure 6 Pseudocode That Renders George

Figure 7 Window to View Geometry Map of a Graphics Port

Figure 8 Drawing Plains of Space and Their Modes of Access

Acronyms

*ACS*	Absolute Coordinate System
*CDC*	Current Draw Color
*CGP*	Current Graphics Port
GO	Graphic Object
GP	Graphics Port
*GPB*	Graphics Port Buffer
*WMS*	World Modeling Scheme
*ICS*	Intermediate Coordinate System
*PDM*	Primitive Drawing Method
*NDC*	Normalized Device Coordinates
*SCS*	Setup Coordinate System
*SAS*	Space Access Status
*SPS*	Space Mapping Status

Glossary

Abstract Class
A class having at least one pure virtual function, unable to create objects of its type, and used only as a base class in the derivation of another class that may define the function bodies of the inherited pure virtual functions.

Attachment
Process by which a graphic object is associated to and owned by a container graphic object and that which provides the translation of the graphic object out of its coordinate system into the coordinate system of the container graphic object.

Class
An Object-Oriented programming data type possessing special abilities (encapsulation, polymorphism, and inheritance) not present in traditional programming that defines a set of objects that share common structure and behavior.

Classification
World Modeling Scheme process by which the dissected parts of reality are classified to be one of the three graphic object types shape, scene, or world.

Composition
(1) the declaration within a class definition of a class type variable; (2) OOP container technology from which objects are allocated form the heap and added to a list of objects maintained by an already existing object; (3) the World Modeling construction technique that provides the containment of graphic objects by a container graphic via attachment.

Dissection
World Modeling Scheme process by which reality is simplified by the breaking of it up into smaller and smaller pieces that can be built individually.

Draw
Public method of a graphic object class that defines the drawable substance of a graphic object.

Graphic Container
World Modeling Scheme graphic object of scene or world class organ that contains other graphic objects via attachment.

Graphic Object
OOP object that graphically represents a specific aspect of reality, unique to the World Modeling Scheme, autonomous, and classified as either a shape, scene, or world.

Inheritance
(1) The property of all class types that allows a class to be defined which inherits all data and method definitions that were previously defined by another class; (2) world modeling construction technique by which graphic objects are given drawable substance and that which provides the creation of new graphic object types of shape, scene, or world class organ.

Instantiation
The declaration of a programming variable of class type in the source code of a software program.

Method
Terminology coming from Smalltalk, but used in C++ when referring to a member function.

Model Coding
World Modeling Scheme process by which a world model is transformed into C++ software instructions that create in computer memory a representation of its reality.

Model Rendering
World Modeling Scheme process by which different aspects of a world model are rendered on the graphics terminal through one or more graphics ports.

Object
(1) Any tangible and/or visible thing, something that may be apprehended intellectually, or something toward which thought or action is directed; (2) a variable having state, behavior, its own private data and set of operations that can access that data, and identity that's define by a class describing the structure and behavior of like things.

Object-Oriented Programming
A method of software implementation in which code is organized as cooperative collections of objects each representing an instance of some class and whose classes are members of a hierarchy of classes united by inheritance.

Primitive Drawing Method
A graphics primitive that's a member of a graphic object type class which defines a piece of the graphic objects drawable substance upon being placed within the graphic objects method draw.

Private
Member data and methods of a class that are private to the class object and only accessed by methods belonging to the class from which the object was instantiated.

Public
Member data and methods of a class that are public to the class object and accessible by entities outside of the class definition.

Pure Virtual Function
An undefined class methods that is specified pure using a pure-specifier in the function declaration of the class definition, thus resulting in the class in which it was declared becoming abstract.

Reality
Anything that can be represented using two dimensonal images.

Scene
World Modeling Scheme container graphic object possessing the ability to be composed of shapes and sub-scenes, and representing the larger and more complex parts, not the undivided whole, of the reality being simulated.

Shape
World Modeling Scheme graphic object representing the atomic parts of reality.

World
World Modeling Scheme container graphic object possessing the ability to be composed of shapes and sub-scenes and representing the undivided whole of the reality being simulated.

Tformer
World Modeling Scheme object that manages the manipulation of a transformation matrix which tracks all transformations of a graphic object.

World Model
A model constructed using the World Modeling Scheme that describes reality as a collection of graphic objects and showing the relationships those graphic objects have to one another.

World Modeling
World Modeling Scheme process by which a world model is constructed using the dissected and then classified parts of the reality being simulated.

World Modeling Scheme
A way of graphically simulating reality using objects that simulate the individual parts making a reality whole.

1. Simulating Reality
The WMS-Library delivers to a programmer an assortment of tools allowing the writing of code that, when executed, will build and graphically render two dimensional realities, i.e., two dimensional images. The term world signifies a C++ object representing a world model. A world model is a design developed using the World Modeling Scheme that represents the undivided whole of reality. The World Modeling Scheme is a way of graphically simulating reality using objects that simulate the individual parts making a reality whole. With the World Modeling Scheme only a small semantic gap between reality and the model will exist. An example of reality is my good friend George, Figure 1. Using the World Modeling Scheme a simulation is broken into the following processes:

(a) Dissecting reality into simplified parts.
(b) Classifying those parts dissected.
(c) Building a world model using the classified parts.
(d) Coding of the world model.
(e) Writing code that graphically renders different aspects of the world model.

1.1. Dissection and Classification The construction of a world model begins with the processes of Disssection and Classification. These processes are the very beginning stages of a simulation. Disssection is the process by which reality is simplified by the breaking of it up into smaller parts that can be built individually. These parts are then ready for Classification. Classification is the categorizing of those parts dissected to be one of three graphic object types. Once dissected and classified, the reality parts are used in the World Modeling process (1.2. World Modeling) to construct a world model, but before constructing a world model we must develop an understanding of graphic objects, examine the hierarchy of the World Modeling Scheme, and then look at one possible Disssection and Classification of George.

1.1.1. Graphic Objects
Graphic objects are abstractions that represent specific aspects of reality by defining a part of the reality's substance. Each graphic object is unique to the World Modeling Scheme, autonomous, and known as either a shape, scene, or world. The level of abstraction of a graphic object is arbitrary. That is, each graphic object type has the ability to represent equal definitions of a reality's substance. Their difference, however, is in how the graphic object is constructed. Each graphic object is provided by its corresponding class of the WMS-Library: shape, scene, and world. As is, the shape, scene, and world class define objects that have no drawable characteristic. Drawable substance of a graphic object is defined by a programmer using class inheritance. Shapes, scenes, a world and the use of derivation to give them substance is the underlying principle of a world model and is that which gives rise to the name a World Modeling Scheme.

1.1.2. World Modeling Scheme Hierarchy

The World Modeling Scheme hierarchy (Figure 2) shows the structuring logic of graphic objects within a world model. Shapes are the atomic parts of reality, scenes the larger and more complex ones, and a world is all of reality. A world model has at least one world graphic object plus any number of scene and shape graphic objects. Worlds contain shapes and scenes, a scene contains shapes and sub-scenes, and a shape is single. Because a scene is recursive in that it's able to contain other scenes within it, the hierarchy has an infinite number of levels, thus allowing the construction of world models representing realities of any complexity.

1.1.3. George's parts
Before dissecting and classifying George's parts lets expand on the idea of a graphic object being an arbitrary abstraction defining some aspect of the reality to be modeled. Using this definition, a graphic object's abstraction is dependent upon one's interpretation of reality. Nevertheless each graphic object type has a name that signifies something different about the particular type: shape, scene, and world. The key understanding of the World Modeling Scheme and its graphic objects is that a shape is atomic, a scene is something more than a shape, and a world is everything. Thus, upon one's Disssection of George we may find his reality consisting of two 'ears', a 'face', two 'eyes', two 'pupils', a 'nose', and a 'mouth'. Using the same notion we could then classify George's face as being the only scene and all remaining parts to be shapes. However because an 'eye' contains a 'pupil', one could have classified George's two eyes as being scenes. Now we see that the abstractions of the graphic objects used in the construction of a world model are dependent upon one's interpretation of reality and are truly arbitrary.

.2. World Modeling World Modeling is the construction of a world model from the dissected and then classified parts of reality. This is the third stage of a simulation and the last stage before writing of the code that will create and build a world model in computer memory. We begin exploring the idea of World Modeling by expanding our example of George the robot. As you will see, in the construction of a world model there are three technique's of World Modeling. After constructing a world model of George and discussing the three construction techniques we conclude our discussion of World Modeling with an examination to some variations to George's world model. From this examination will come a greater understanding of the power of a graphic object that results from our definition of a graphic object being an arbitrary abstraction defining some aspect of reality.

1.2.1. George's World Model
Building upon our Dissection and Classification of George's parts (1.1. Dissection and Classification),
Figure 3 is a world model of George. The model shows the whole of George’s reality as a collection of those parts dissected and the relationship those parts have to one another. Collectively these parts behave as one. Classified as either a world, scene, or shape graphic object, each part defines a distinct aspect of George's reality. From his world model we see all of George existing within one world graphic object that's instantiated from the world class, his world being composed of two graphic objects derived from the shape abstract plus one from the scene class, and the scene graphic object being composed of four shape graphic objects derived from the shape abstract.

One can extrapolate two reasons for choosing to derive George's face from the scene class; a 'face' has an outline and it consists of several additional parts. Both the scene and shape class are capable of defining the substance of a graphic object. However, unlike the scene class, the shape class has no means of containing other graphic objects. Thus, by deriving a new class from the scene class we are able to create a new graphic object type having all the qualities of the scene class and some drawable substance. In this case the drawable substance is a rectangular filled region.

This version of George's world model demonstrates three construction techniques. Special in definition when speaking of World Modeling, but general in nature to OOP, these construction techniques are called instantiation, composition, and derivation.

1.2.2. The Three Construction Techniques
Instantiation is the creation of graphic object type variables. The shape, scene, and world class in addition to any class derived from them, is a graphic object type. The scene and world class as supplied by the WMS-Library will instantiate objects, however the abstract class shape (the shape abstract) embodies at least one pure virtual function, thus all shape graphic objects are instantiated from a derived-shape class.

Composition is the containment of one or more graphic objects within another graphic object. There are two flavors of composition. The first flavor of composition is that of OOP container technology where an object is allocated from the heap and then added to an already existing object. The second flavor of composition is the aggregation of one or more objects within a class definition. When specifically speaking of World Modeling, composition refers to a modified version of OOP container technology. In general, the World Modeling idea of composition is somewhat simplistic in that a scene or world object or any object derived from them are the only graphic objects having the ability to contain graphic objects.

Derivation is the process of creating new class types using class inheritance. Inheritance is the property of all classes that allows a new class to be defined which inherits all data and method definitions that were previously defined by another class. This construction technique is by far the most complex, thus is the focus of 1.3. Model Coding. For now, simply remember that only classes directly or indirectly derived from the shape, scene, or world class are applicable to the construction of a world model.

1.2.3. Variations to George's World Model
Now that we have a better understanding of World Modeling and the three construction techniques, we are ready to look at some variations to George's world model (figure 3). Using instantiation we could have created eye and ear objects from preexisting circle and rectangle class's that where derived from the shape abstract. With composition we could have composed his world entirely of shape objects. Or using derivation we could have derived a new world class called George. The George class would be nothing more than a world class having its own drawable substance. In this case the drawable substance is a rectangular filled region representing an outline of George's face.

1.3. Model Coding After the construction of a world model (1.2. World Modeling) and before the rendering of images of the model (1.4. Model Rendering) comes the fourth stage of a simulation ¾ the process of Model Coding. Model coding entails the writing of C++ software instructions that at minimum derive new shape classes, instantiate world, scene, and shape objects, and attaches scene and shape objects to a world object, thus creating in computer memory a representation of reality. We begin exploring the Model Coding process with an in depth look at the shape abstract, followed by derivation, and then the attachment process. We conclude our discussion of Model Coding by looking at an example Psudocode of George's world model. Information presented during our discussing of Model Coding will greatly enhance our understanding of the World Modeling process.

1.3.1. The Shape Abstract
The abstract class shape is the most fundamental graphic object. A solid grasp of its workings will lead to better understandings of scene and world class usage, and from every aspect is the corner stone to understanding the World Modeling Scheme. Three highlights of the abstract class shape are:

(a) It has its own set of graphics primitives called PDMs (Primitive Drawing Methods) that are used exclusively within the class's scope.

(b) It has a pure virtual and public method called draw, who’s only purpose it to provide a means of giving a shape drawable substance.

(c) It's the primordial base class from which all other graphic object class's are derived.

Most unusual is in how a programmer uses PDMs to define the function body of the draw method within a derived shape class (a class derived from the shape abstract), thus giving a shape or other graphic object substance that renders an image upon the execution of draw. The use of PDMs in derivation to create a new graphic object class that has drawable substance is that which distinguishes primitive drawing methods from standard graphics primitives that per-say just draw lines.

1.3.2. Derivation
Derivation is the most powerful World Modeling technique. In fact, without derivation there's no way of giving a graphic object its own drawable substance. Previously, in 1.2.2. The Three Construction Techniques, we found that to give a shape graphic object drawable substance one derives a new class redefining a member function draw that executes PDM calls and that the shape abstract is the primordial base class of all graphic objects.

Because the shape abstract is a primordial base class within the heritage of the scene and world class, the scene and world class inherit all abilities of the shape abstract. However, unlike the shape abstract, the scene and world class (not derived scene and world classes) have redefined the function body of draw so that they have the may call the draw method of the graphic objects they contain.

Understanding how the scene and world class are derived and that they posses the abilities of the shape abstract, we can see that derived scene and world classes (classes derived from the scene and world class) are given drawable substance just as if deriving a shape class from the shape abstract. However, by deriving a new scene or world class and redefining draw just as would be done in deriving a new shape class, results in introducing a second draw within the same scope. Thus, if we want the derived graphic object class to have the same drawable substance as the base class from which it was derived as well as the drawable substance defined by the derived class, then we must explicitly call draw of the base class within draw method of the derived class.

Now we can formulate a clear definition for draw. The draw method of a graphic object is a public member of a derived graphic object class that when executed call's several PDMs and/or the draw method of the base graphic object class from which it was derived.

1.3.3. Attachments
An attachment is the vehicle through which the composition of shapes and scenes within a World Modeling Scheme container are accomplished. This form of composition is an extremely modified version of traditional OOP container technology where objects are simply added. With an attachment we not only have the containment of one graphic object by another, but we also have a translation of the object being attached out of its coordinate system into the coordinate system of the container. In section 1.4.1. Space, we will see that each graphic object has its own coordinate system. For now, understand that a container object does not perform the actual containment procedure. This is done by World Modeling Scheme linker object.

Each attachment is carried out relative to a point within the rectangular boundary that defines the area of drawable substance of the graphic object being attached in addition to a point within the boundary of the container graphic object. Once attached, a graphic object becomes owned by and part of the container graphic object that it was attached to. Without deriving new graphic object types, only the shape and scene class graphic objects are attachable, and only scene and world class object types (called graphic containers) are able to contain other graphic objects.

1.3.4. Linker Objects
A linker object facilitates the containment of graphic objects by a container object via attachment. This is done by first associating the linker object with a container object at which point the linker object is able to perform the attachment by carrying out a coordinate system translation of the object being attached and the memory management of the container object.

There are two types of linker objects, one for attaching graphic objects to a scene object and another for attaching graphic objects to a world object. The use of linker objects is needed to prevent multiple friendship across the graphic object classes: shape, scene, and world, that would otherwise be needed in order for the graphic objects involved with the attachment to access each other's data members to obtain information needed for the coordinate system translation and memory management. Thus we only need two linker object types' requiring attachment knowledge of three graphic object types' verses having three graphic object types requiring attachment knowledge of each other. The accessibility of the graphic object data members is achieved through class friendship. This can be viewed as a two to one cardinality verses a three to three cardinality.

1.3.5. George's Pseudocode
Having an understanding of the shape abstract, attachments, and the three construction techniques, we conclude our discussion of Model Coding with an example Pseudocode of George's world model. Written in the part's derivation section, but not shown in Figure 4, are the defining bodies of the draw methods of each graphic object class. Also not shown in this figure is the instantiation of linker objects. However, upon translation to and execution of a block of C++ code, this pseudocode will create and build in computer memory a model that represents George's reality. Once this model is created in computer memory, George can then be observed from many different aspects by the process of Model Rendering.

1.4. Model Rendering Immediately following Model Coding (1.3. Model Coding), and being the final stage in the World Modeling Scheme approach to simulating reality, model rendering is the process of writing C++ software containing instructions that display in the graphics terminal images that reflect different aspect of a world model. We begin our introduction to the Model Rendering process by introducing a concept called space, followed by the concept of a graphics port, and then end with an example Psudocode that renders different images of George's world model.

1.4.1. Space
Space is the name given to a module of traditional C code provided by the WMS-Library that supplies the tools of an advanced interface to the graphics terminal. The name space is used because this module is that which allows the drawable substance of a graphic object to be rendered. Without space a graphic object can not be seen, i.e., a graphic object must have space in which to exist before existing.

Space provides several coordinate systems in which to navigate. Of these coordinate systems and the focus at this point in time is ACS (Absolute Coordinate System). In an ACS, coordinates specify the absolute position of a graphic object within space, that is, the final position within the overall region of space occupied by a world model. Thus, the ACS is the coordinate system used by the graphic object PDMs. Nevertheless, until becoming attached to a world model this system is implied and another system called SCS (Setup Coordinate System) is actually in play.

Space does not recognize the SCS. The SCS is specific to a graphic object. In fact, the SCS is actually an ACS to a graphic object. It's not until a graphic object is directly or indirectly attached to a world graphic object that the SCS becomes an ASC and then recognized by space. Both the SCS and the ASC are based on world coordinate theory which is discussed in section 2. World Coordinate Theory.

Graphic object PDM specified SCS positions are never permanently changed to ACS positions. SCS positions are actually transformed through a series of transformations built by attachments to an absolute position by a transformation performed within the graphic object PDMs. Once a graphic object PDM has carried out the transformation it then invokes the corresponding space graphics primitive, passing to it, the ACS positions, thus producing on the fly ACS positions upon execution of a graphic object's method draw.

1.4.2. The Graphics Port
Space graphics primitives (not primitive drawing methods) are responsible for the mapping of continuous space into device space of the graphics terminal by means of a graphics port. Graphics ports are the eyes through which an observer can see images of a world model. Each graphics port has its own window and view. The window is that portion of absolute space being observed and the view is that portion of device space where the windows image is rendered. The window can be positioned and moved anywhere within a world, plus its size decreased or increased to give the effect of a camera lens zooming in and out and the view can be positioned and moved anywhere within the graphics terminal, plus its size decreased or increased making the image smaller or larger. Figure 5, shows the window to view mapping of a graphics port.

1.4.3. Rendering George
Writing code that renders images is rather simplistic compared with that of the Model Coding or World Modeling. A graphics driver, mode, and number of graphics ports is specified upon initialization of space. Each graphics port is then focused on a specific region of the world model to be rendered. Completing all necessary initializations, current or active space access is chosen. Active access results in simultaneous rendering of all active graphics ports while current access results in rendering of a pre-selected port only. The Pseudocode listing of Figure 6 sets up three graphics ports that simultaneously render George's entirety, look into his mouth, and inspect his right eye.

2. World Coordinate Theory World coordinate theory describes the multiple coordinate system behavior of the World Modeling Scheme and the means by which a graphic object's substance is described. Recall from section 1.4.1. Space that the PDMs of a graphic object use an ACS and that the context of this system depends upon whether or not a graphic object is attached to a world. A graphic object's substance defined by its method draw never changes, but goes through different stages of development. These stages produce enough difference to warrant regarding them as being of a different coordinate system: SCS, ICS (Intermediate Coordinate System), or ACS. Nevertheless, each system uses real number Cartesian coordinate geometry; having an origin at (0.0,0.0) with vertical and horizontal axis extending to infinity in both directions. This gives the ability to place graphic objects about the origin of a graphic container via attachment and to specify positions using fractional units.

2.1. Setup Coordinates
Setup coordinates refers to those positions passed as formal argument values in the call of a PDM. Recall from section 1.3.1. The Shape Abstract and 1.3.2. Derivation, that the substance of a graphic object is created by the derivation of a new graphic object class that uses the inherited PDMs of the base graphic object class to define the function body of draw. This actually begins by first sketching the lines or other characteristics that define the substance image on a piece of graph paper. PDM calls corresponding to that which is constructed on the graph paper and having formal argument values the same as that found on the graph paper are then placed in the function body of draw. Those coordinate values taken from the graph paper and passed to the PDMs are the setup coordinates of a graphic object.

2.2. Intermediate Coordinates
Intermediate coordinates specifically refer to the attachment and transformation of a graphic object. Within the inheritance of each graphic object is a transformation object called a tformer. The tformer provides the functionality needed for the manipulation of a transformation matrix that tracks all transformations of a graphic object. The representation of these transformations by the matrix is that of an intermediate coordinate system. The transformation matrix is thought of in this way because it establishes the passage through which a graphic object transforms its setup coordinates to absolute coordinates.

2.3. Absolute Coordinates
Absolute coordinate, as stated in section 1.4.1. Space, represent the absolute position of a graphic object in space. Scene and shape graphic objects become absolute upon directly or indirectly being attached to a world graphic object, until then their PDMs are not able to render. Thus, the coordinate system of world graphic object is always absolute.

2.4. Tformers
As discussed in section 2.2. Intermediate Coordinates, each graphic object inherits a transformation object that is called a tformer and the tformer manages a transformation matrix that tracks all transformation of a graphic object. The tformers function within the World Modeling Scheme is to provide an intermediate coordinate system through which a graphic object's substance is transformed from a setup coordinate system to an absolute coordinate system. Thus, the intermediate coordinate system represents the translation of graphic object out of its coordinate system into the coordinate system of the container object that it's attached to as well as non-attachment transformation that the graphic object may have gone through, but to this point we have yet to explore the transformation capabilities of a tformer, how a linker object uses a tformer in an attachment, or the possibilities of transforming a graphic object other than attachment .

2.4.1. Tformer Transformations
A tformer has member functions for translation, scaling, rotation, reflections, shears, and composition of any sequence of the previous operations. All the operations with the exception of translation are carried out relative to an arbitrary point. Rotations may be in the clockwise or counter clockwise direction. The shearing operations are y-shear and x-shear. And among the reflection operations is reflection in the y-axis, x-axis, origin, y = x, and y = -x. Compositions of these operations are achieved with the overloaded multiplication and negation operators. With the multiplication operator multiplying on the left is pre-multiplication and multiplying on the right is post-multiplication.

2.4.2. Attachments
During an attachment a linker object has access to both the tformer of the graphic object being attached and the tformer of the container object in which the graphic object is attached to. As discussed in section 1.3.4. Linker Objects, this accessibility is provided by class friendship. A linker object carries out an attachment as follows:

(a) Store the placement value within the container object of the graphic object in the tformer of the graphic object, preserving any previous transformation, preserving all previous transformations.

(b) Transform the graphic object tformer with that of the container graphic object's tformer, preserving all previous transformations. This is done using matrix multiplication.

(c) Transform the graphic object tformers of any graphic objects contained within the graphic object being attached with that of the container graphic object's tformer just as was done with the graphic object being attached.

2.4.3. Non-Attachment Transformations
Non-attachment transformations are carried out by calling the transformation member functions of a graphic object or the multiplication of one graphic object by another. In the case of a scene or world container graphic object the tformer functions have been overloaded with corresponding transformation functions that in addition to transforming the container graphic object, transform the graphic objects being contained. The tformer transformation functions are virtual. Thus, transformation functions of the tformer can not be called outside of the overloading class.

2.5. Graphics Port Geometry
Figure 7 is an example of the window to view mapping of a graphics port as applied to absolute space. This mapping is based on world coordinate theory and should serve as a better example of the theory than trying to show how the theory carries through the tranformations executed in the coding of George's world model.

3. Non-Object uses of the WMS-Library Space has three different plains of drawing. Each plain is accessible without having to derive a graphics object. That is images can be created using these plains via space. However, there is no means of associating those images to the World Modeling Scheme other than a graphics object. Thus, applications can be entirely developed without using the World Modeling Scheme, but by using the space graphics primitives only. Furthermore, calls within the draw method of graphic objects can be made to the space graphics primitives in addition to those of the graphic object PDMs.

3.1. The Three Plains of Drawing
Of the three drawing plains, each has two attributes determining the outcome of a space graphics primitive call: mapping (absolute, normalized, and device) and access (active, current, and terminal). Mapping refers to a specific plain, distinguished by the coordinate system used to specify the formal arguments of a graphics primitive. When using an absolute or normalized drawing plain the arguments of a graphics primitive are transformed into device coordinates. The device plain assumes that the formal arguments are already in device form. For each plain the access determines whether or not a graphics primitive maps into all active graphics ports, the current graphics port, or directly to the graphics terminal. Graphic object PDMs always operate in the absolute plain and conform to world coordinate theory. Each plain and its access has their own CP (current position) and CC (current draw color). Figure 8 sumarizes the three drawing plains and their modes of access.

3.2. Normalized Device Coordinates

NDC (Normalized Device Coordinates) represent positions within the graphics terminal or the graphics port in a device independent manner. Such a coordinate system describes the view surface of the graphics terminal as a set of real numbers ranging from 0 to 1 in both the X and Y direction, with the origin in the upper left hand corner. When using terminal access the these coordinates represent device positions within the terminal ranging from 0 to screen width in the X direction and 0 to screen height in the Y direction. When using active or current access these coordinates represent device positions within the graphics port view ranging from 0 to view-width in the X direction and 0 to view-height in the Y direction.

3.3. One Theoretical Application
I have conjured an example pertaining to George that demonstrates the power of space singelton PDMs combined with World Modeling Scheme PDMs. To get things rolling, lets add a shape graphic object to George's world model. This shape is that of a push button labeled open and is arbitrarily attached to George's world object. The shape is constructed with the space singelton PDMs using a normalized drawing plain and terminal access so that the button is always rendered in the upper left hand corner of the graphics terminal. Also associated to this shape is another world model separate that of George and a mouse interrupt type object. The new world model simulates George's a