This document contains the ActiveVRML reference.

Introduction

This document presents a technical summary of the Active Virtual Reality Modeling Language (ActiveVRML). It provides a brief but reasonably precise definition of ActiveVRML version 1.0. Those seeking an introduction to ActiveVRML should consult the companion paper, An Introduction To ActiveVRML.

ActiveVRML is intended to provide a framework for constructing models that manipulate media including sound, two dimensional (2D) images, and three dimensional (3D) geometry. There are two characteristics that make ActiveVRML unique and especially well suited for this task: all values in ActiveVRML potentially vary with time, and values may change in reaction to events.

Every value in ActiveVRML may change with time. For example, there is an image type in ActiveVRML. An image object is not like a static photograph, but more like a video, continuously changing with time. Similarly, geometry in ActiveVRML is not like a static geometry model, but is (potentially) animated, moving, and reacting to events. This is an important principle in ActiveVRML; every value may change with time. Even simple objects, like numbers, may change with time. Values that can vary with time are called behaviors in ActiveVRML. A reactive behavior is one that (potentially) varies in response to events.

One way that values change with time is in response to particular events. For example, a user input event, or a mouse event. Events may be caused by internal events (to the ActiveVRML model) as well. For example, a particular number value being zero may cause an event.

Finally, ActiveVRML is a language for describing media via reactive behaviors. The language part of ActiveVRML is actually very small. The bulk of this document is spent describing a large collection of useful library functions for manipulating media types.

Expressions and Declarations

ActiveVRML is fundamentally very simple; it has just a handful of expression forms and only two forms for declaring identifiers. This section describes these forms. The power of ActiveVRML comes from the underlying model which includes time varying values, reactivity, and a rich set of media types. These are described in subsequent sections.

Literal and Constructor Expressions

Associated with most types in ActiveVRML is a literal or constructor form for each. Examples include 17 which represents a literal number, and [1, 2, 3], which uses the constructor form for lists to build a list of numbers. Allowable constructor forms are defined below in the sections defining each type.

Variable Expressions

An identifier in ActiveVRML is an arbitrarily long string of alpha-numeric characters beginning with a letter. Identifiers are case sensitive and there are some keywords (listed in Appendix A) that may not be used as identifiers.

Variables in ActiveVRML are statically scoped.

Application Expressions

An expression of the form expression1 expression2 is an application expression and represents the value of applying the function value of expression1 to the value of expression2. ActiveVRML is strict; that is, it obeys call by value semantics for argument evaluation. The order of the evaluation of arguments is not specified. Argument application associates left; for example, f(x)(y) equates to (f(x))(y).

Parenthetical Expressions

Parentheses may be used to group expressions and override operator precedence. Parentheses are also useful for improving the readability and presentation of ActiveVRML models. Operator precedence in ActiveVRML is listed in Appendix A.

If Expressions

A construct of the form if expression1 then expression2 else expression3

is an IF expression. It represents the value returned by evaluating the boolean test of expression1 and selecting expression2 or expression3 depending upon the true value of expression1. The types of the two branches of the IF expression are required to match (or unify). There is no corresponding IF THEN statement; all IF statements have both branches. Since ActiveVRML is functional (operations do not have side effects), such one-armed IF statements would not be very useful.

Let Expressions

A construct of the form let declaration1; . . . declaration[n]; in expression

is a LET expression. It represents the value of an expression when evaluated in a context that includes the bindings for declaration1 through declarationn. The LET expression is used to introduce a local scope. The declarations are evaluated simultaneously. The names are in scope of the right hand sides of the declarations, allowing for forward declarations and mutual recursion automatically. All of the declared identifiers are required to be distinct. The scope of the declarations is limited to the LET expression itself. The semicolon following the last declaration is optional.

Declarations

The simplest declarations declare an identifier to have a value: identifier = expression

Or, they declare a function: identifier(identifier, ..., identifier) = expression

The following is an example function declaration: swizzle(seed) = if seed = 1 then 1 else if odd(seed) then swizzle(seed * 3 + 1) + seed else swizzle(seed / 2) + seed

This declares a function, swizzle, that takes one formal argument, seed. The function is recursive. All function declarations are assumed to be recursive. When you use the name of the function you are declaring within the expression, you are referring to the function itself, not a new or different function.

The following is an example variable declaration: swizit = swizzle(27)

This declares the variable swizit to be the evaluation of the expression swizzle(27). We can illustrate scoping in LET expressions by combining these declarations along with a declaration for the predicate odd used in the declaration of swizzle: let swizzle(seed) = if seed = 1 then 1 else if odd(seed) then swizzle(seed * 3 + 1) + seed else swizzle(seed / 2) + seed; odd(i) = (mod(i, 2) = 1); swizit = swizzle(27) in swizit

Notice that the declaration for odd comes after its use in the declaration of swizzle. Since all of the declarations within a LET expression are assumed to be mutually recursive, this is legal. However, for better readability and because the declarations are not truly mutually recursive, the definition of odd should probably appear first.

Within the scope of the LET expression, three identifiers are declared, swizzle, odd, and swizit. Beyond the scope of this expression, the three declarations are not available. The value of the LET expression is the value of swizit which evaluates to 101440.

In addition to these simple forms of variable and function declarations, it is also possible to use pattern matching to specify the destructuring of values within a declaration. This is described later in this document.

In addition to the local declarations in an ActiveVRML file, it's possible to augment the environment with declarations from a second file with the use keyword: use (pathname)

This keyword must be used as a top level declaration.

Basic Types

ActiveVRML includes a very powerful and useful typing system. Each expression and declaration in ActiveVRML is given a type either explicitly by the user, or implicitly by the system. Consider the following example declaration: successor(nat) = nat + 1

ActiveVRML assigns successor the type number -> number, meaning it will map any value of type number to another value of type number. This typing is strong in the sense that ActiveVRML will catch all type errors during authoring. It is also convenient; the user did not have to explicitly give the type as the system inferred it.

Finally, types are polymorphic, meaning that a given type may stand for many different type instances. Consider the following declaration: nada(val) = val

When applied, nada will return its actual argument unchanged. Thus nada(3) evaluates to the number 3 and nada("hello") evaluates to the string "hello". The type that ActiveVRML infers for nada is polymorphic: α -> α . Here α is a type identifier and may be replaced everywhere uniformly to create an instance of a polymorphic type. Thus, number -> number is an instance of α -> α, and so is string -> string.

Note that number -> string is not an instance of α -> α , since one α was replaced by a number and the other by a string (not uniformly). A polymorphic type can contain more than one type identifier, for example, α -> β. In this case, each identifier can be replaced separately. Thus, number -> β, α -> string, number -> string, number -> number, and γ -> string are all instances of the polymorphic type α -> β. Note

In deference to the typographically-challenged ASCII character set, the type assigned to nada is actually written 'a -> 'a. In typeset ActiveVRML documents, including this one, Greek letters are often used instead of the ASCII syntax for type identifiers in order to improve readability.

Every expression and declaration in ActiveVRML is assigned a type by the system using a standard Milner-Damas polymorphic type checker. Except to improve exposition and occasionally to resolve ambiguity with an overloaded operator, it is not necessary for the programmer to explicitly give types. An expression may be qualified with a type using the syntax like the following: expression: type-expression

For example, the following syntax can be used to restrict nada to a particular type (desirable for clarity): nada(val: string) = val

This will assign nada the monomorphic type string -> string.

The following sections define the basic types for ActiveVRML and list the constructor forms and functions for these types. Later sections define types for reactivity and for modeling (geometry, images, and associated types). Unit Type: unit Type unit The unit type is a trivial type containing only one member. This type is often used for functions that take or return uninteresting data, similar to the way that the void type is used in C++ programs. Constructors () The unique member of the unit type, pronounced "trivial." Function Type: type -> type Type type -> type The function type α -> β represents mappings from type α to type β. Functions in ActiveVRML may be higher-order, meaning that a function can take another function as an argument or return a function as its result. For example, function f might have type (number -> number) -> number. This means that f can be applied to a function with type number -> number to produce a result of type number. Another function g might have type number->(number -> number). This means that g will take a number as an argument, and produce a function with type number -> number as its result. Constructors function pattern . expression This constructor is used to create anonymous function values. The pattern part of this constructor (described in the "Pattern Matching" section) may be thought of as a list for formal arguments. For example, the declaration f (x, y) = x * y + 1 can be thought of as an abbreviation for: f = function (x, y). x * y + 1 Function declarations are value declarations where the value is a function value. Functions infix o : (α -> β) * (β -> γ) -> (α -> γ) The expression f o g is the composition of the functions f and g. The notation infix o means that o is an infix operator (like the familiar + as in 14 + 3). The value of (f o g)(x) is f(g(x)). Note that o has a type like a higher-order function. It takes two functions as arguments and returns a function as its result. Its type can be written as ((α -> β) * (β -> γ)) -> (α -> γ) since * has higher precedence than -> in ActiveVRML types. (See ActiveVRML Type Precedence later in this document.) Product Type: type * type Type type * type The product type α * β represents pairs of elements: for example, (e1, e2) where e1 has type α and e2 has type β. Constructors expression, expression The pairing constructor is a comma. The precedence of the comma is extremely low in ActiveVRML, so it is usually desirable (and visually clearer) to write pairing with parentheses: (3, "hello"). The pairing operator associates to the right. Thus, (3, "hello", 17) is the same as (3, ("hello", 17)). It is a pair, the second element of which is also a pair. Functions first: α * β -> α first(α, β) returns the first element of a pair, α. second: α * β -> β second(α , β) returns the second element of a pair, β. List Type: list Type α list The type α list is a list (or finite sequence). Each element is of type α . For example, number list is a list of numbers, and (string list) list is a list where each element is a list of strings. Constructors [expression-list] A list of expressions (zero or more) separated by commas. For example, [] is the null list (of type α list) and [1, 2, 3] is a number list. Functions head: α list -> α head(list) returns the first element of the list list. It is illegal to apply head to an empty list. tail: α list -> α list tail(list) returns a list comprising all but the first element of the original list. It is illegal to apply tail to an empty list. infix:: : α * α list -> α list The operator :: is read as "cons." The expression elt:: list returns a new list formed by prepending ("cons'ing") elt to list. empty: α list -> boolean empty(list) is true if and only if list is an empty list. length: α list -> number length(list) returns the length of list. map: (α -> β) * (α list) -> βlist map(fun, list) returns a list by applying fun to each element of list. reduce: (α * β -> β) * β * (α list) -> β reduce([e1,...,en], base, fun) returns: fun(e1, fun(e2, fun(...,fun(en-1, fun(en, base))...))) nth: α list * number -> α nth(list, n) returns the nth element of list, where the first element of the list is 1. Boolean Type: boolean Type boolean The boolean type represents true and false values in ActiveVRML. Constructors true Constructs a true value. false Constructs a false value. Functions infix and: boolean * boolean -> boolean The expression x and y is true if x and y are true. infix or: boolean * boolean -> boolean The expression x or y is true if either x or y are true. not: boolean -> boolean The expression not x is true if x is false. infix =: α * α -> boolean Equality may be used to compare any two values with the same type. Equality in ActiveVRML is structural: pairs are equal if each side of the pair is equal; lists are equal if their lengths are the same and corresponding elements of each list are equal. Equality applied to functions and modeling types is not defined (since it is not always possible to determine equality on these types). infix <>: α * α -> boolean The expression x <> y is true if x is not equal to y. Number Type: number Type number The number type in ActiveVRML does not distinguish between "fixed point" and "floating point" numbers; both are considered numbers. The implementation will choose an appropriate representation. Constructors number-literal The number-literal constructor is any sequence of characters satisfying the following regular expression: digit+ ('.' digit*)? (['e' 'E'] ['+' '-']?digit+)? time A time-varying number representing the local time of a behavior. This important constructor is the basis for many interesting time-varying behaviors. random A pseudo-random number in [0, 1] that is time-varying. All instances of random that start at the same global time have the same time-varying value. pi A constant number representing π. Functions infix +: number * number -> number The expression x + y returns the value of x added to y. infix *: number * number -> number The expression x * y returns the value of x multiplied to y. infix -: number * number -> number The expression x - y returns the value of y subtracted from x. infix /: number * number -> number The expression x /y returns the value of x divided by y. Division by zero is an error. prefix -: number -> number The expression -x returns the value of x multiplied by -1. prefix +: number -> number The prefix + operator does not change the value of a number. infix <: number * number -> boolean The expression x < y returns true if the value of x is less than the value of y. infix <=: number * number -> boolean The expression x <= y returns true if the value of x is less than or equal to the value of y. infix >: number * number -> boolean The expression x >y returns true if the value of x is greater than the value of y. infix >=: number * number -> boolean The expression x >=y returns true if the value of x is greater than or equal to the value of y. abs: number -> number abs(x) returns the absolute value of x. sqrt: number -> number sqrt(x) returns the square root of x. mod: number * number -> number mod(x, y) returns the modulus of x divided by y. ceiling: number -> number ceiling(x) returns the smallest integer that is greater than or equal to x. floor: number -> number floor(x) returns the largest integer that is less than or equal to x. round: number -> number round(x) returns the nearest integer to x. radiansToDegrees: number -> number radiansToDegrees(x) returns x expressed in degrees. degreesToRadians: number -> number degreesToRadians(x) returns x expressed in radians. asin: number -> number asin(x) returns the arcsine of x. acos: number -> number acos(x) returns the arccosine of x. atan: number * number -> number atan(h, w) returns the arctangent of h divided by w in radians. atan: number -> number atan(x) returns the arctangent of x. cos: number -> number cos(x) returns the cosine of x in radians. sin: number -> number sin(x) returns the sine of x in radians. tan: number -> number tan(x) returns the tangent of x in radians. infix ^: number * number -> number The expression x ^ y returns x raised to the power of y. exp: number -> number exp(x) returns the exponential value of x. ln: number -> number ln(x) returns the natural logarithm of x. log10: number -> number log10(x) returns the base 10 logarithm of x. seededRandom: number -> number Pseudorandom behavior is parameterized by a random seed. SeededRandom returns x in [0, 1], implicitly parameterized by time. cubicBSpline: number list * number list -> (number -> number) cubicBSpline(knots, control points) creates a polynomial, real-valued B-spline function of degree three. nurb: number list * number list * number list -> (number -> number) nurb(knots, control points, weights) creates a rational, real-valued B-spline function of degree three.

Reactive Behaviors

Recall that all values in ActiveVRML are potentially time-varying. Variation is achieved by specifying the time explicitly (for example, 3 + time), input such as mouse motion, or by reactivity. This section defines reactivity and the constructs used to build reactive behaviors.

Reactivity

A reactive behavior is one that can potentially react to an event. The following is a very simple reactive behavior: red until leftButtonPress => green

In this expression, UNTIL is the main operator. The expression is parsed into the following: red until (leftButtonPress => green)

The reactive behavior changes the color from red to green when the mouse button is pressed.

The subexpression leftButtonPress => green is called a handler. It pairs up an event, leftButtonPress, with a value, green. The value is the action taken when the event occurs.

The UNTIL construct can also be used to watch for more than one event, reacting to the first one that occurs. For example: red until leftButtonPress => green | rightButtonPress => yellow

This is parsed into the following: red until ( (leftButtonPress => green) | (rightButtonPress => yellow) )

The color remains red until either the left or right mouse buttons are pressed. If the left button is pressed, the color changes to green. If the right button is pressed, the color changes to yellow.

In general, the logic of the UNTIL operator follows this pattern: b0 until e1 => b1 | e2 => b2 . . . | en => bn

The reactive behavior is b0 until any one of the events occurs, e1 through en. The first event to occur, ei, results in the behavior, bi.

A more advanced form of events uses event data to produce the next behavior. For example: 0 until numEvent => function x. x + 1

In this case, numEvent is an event that produces a number. The value of this behavior is zero until the event occurs, and then becomes the value associated with the event plus 1.

The type checking of an UNTIL expression is as follows: If b is an α behavior and e is an α event, then b until e is a reactive behavior with type α behavior.

The next section describes events in more detail.

Events and Reactivity

An event can trigger a discrete change in a behavior. In addition to marking the occurrence of an event at a particular time, an event may also produce data. An α event produces some data of type α that can be used as new behavior after the event.

Events are constructed from one of the following types of events.

System Events

System events represent user input events. All of the following are system events:

• LeftButtonPress: unit event
• RightButtonPress: unit event
• KeyPress: character event

Boolean Events

Events corresponding to a boolean behavior occur as soon as the boolean value is evaluated as true. The predicate function is used to construct an event from a boolean behavior. For example, predicate(x = 0) is a unit event (one that returns a unit as its data) that occurs the first time that behavior x is zero.

Simple Handler Events

New events may be constructed from other events. If e is a unit event (an event that does not produce interesting data) and b is an α behavior, then e=>b is a simple handler. In this example, the behavior is to wait for event e and before becoming b.

Handler Events

More complex handler events use the data produced by an event to construct the resulting behavior. If e is an α event and f is a function with type α -> β, then e=>f is a β event. In this example, the behavior is to wait for event e, and then use the data produced by the event to run function f. The result is an event, e=>f, that occurs at the same time that e does and produces f(x) as the data, where x is the data produces by event e.

Alternative Events

If e and e' are α events, then e | e' is an α event. This means to choose whichever event, e or e', happens first, and then return the corresponding data for the event. If e and e' happen at the same time, it is undefined which will be chosen.

Filtered Events

If e is an α event, and p is a function that maps α values to boolean values, then suchThat(e, p) is an α event that allows through only occurrences of e whose data satisfies the predicate p.

Snapshot Events

The snapshot construct may be used to make a time varying behavior into a static one (i.e., no longer varying with time). This is useful when one wishes to record the instantaneous value of a time varying behavior for some other use. For example, 0 until snapshot(xComponent(mousePosition), leftButtonPress,)

is the static behavior 0 until the left mouse button is pressed, and then becomes the static value of the x coordinate of the mouse position at the time of the press. The behavior 0 until leftButtonPress => xComponent(mousePosition)

is different in that it produces a behavior that continues to vary with time and track the x coordinate of the mouse after the button event occurs. The following subsections define the types and operators used in constructing reactive values.

Behavior Termination

Behaviors can terminate. The end construct is of type α (any type) and means to immediately terminate the behavior. For example: b = time until leftButtonPress => end

This behavior varies with time until the leftButtonPress event occurs. Then it terminates.

A behavior will also terminate if one of its defining behaviors terminates. For example, consider b' = f(b, 3)

If behavior b terminates, then b' terminates at the same time.

The DONE construct is a unit event that can be used to detect when a behavior has terminated and react accordingly. Consider the following: repeat(b) = b until done => repeat(b)

This function takes behavior b and runs it until it terminates. Then it starts b again. (Note: This a built-in function in ActiveVRML.)

Does this last note mean that the example is a built-in function or that there is another function available that does the same thing?

Behaviors and Time

To improve modularity, all behaviors are defined to begin at local time zero. That is, when a behavior begins, no matter what the system time is, the time for the behavior starts at zero. Consider the following: b until e => b'

When b is performed, there are two times to consider, the system time and the local time for b. Let the system time be tg. Whatever the value of tg is, at the time b is performed, it represents time zero for b. Event e occurs some time after tg. Let this time be te. This behavior can then be broken down by time: Do behavior b from time tg to te. Then do behavior b'. The local time for b is zero to (te -tg). When b' starts, its local time is zero as well.

Here is a simple behavior that uses the local time as its events: green until time = 2 => (blue until time = 1 => red)

This behavior makes the color green for 2 seconds, blue for 1 second, and then red.

The TimeTransform Construct

The timeTransform construct can be used to change the interpretation of local time. The function has the type timeTransform: α * number -> α . It uses the number to redefine how local time is interpreted within a behavior. Consider the following: doubleSpeed = time * 2; b = playVideo(video); doubleVideo = timeTransform(b, doubleSpeed)

In this example, the video is played at twice its original speed. From the perspective of global time, each 1 second interval corresponds to 2 seconds of local time. The effects of time transformation are cumulative. For example: timeTransform(doubleVideo, doubleSpeed)

This line would play the video at four times its original speed.

To be consistent and predictable, the number argument (n) for the time transformation must satisfy two rules:

• Monotone&em;For all times t0 and t1 when t0 is less than t1, n at time t0 must be less than n at time t1.
• Nonnegative&em;For all times t, n at time t is nonnegative.

Monotonicity is required to make event reaction sensible (event transitions cannot be undone). Nonnegativity is required to prevent definition of a behavior before local time zero; that is, it may not make sense to sample a behavior like a video before local zero.

Certain behaviors, principally those defined by system or user input devices, may not be transformed in time in the same way that artificial behaviors can be. Such devices ignore user-defined time transformations when they are sampled.

Reactive Types

Following are definitions for the basic types for events, reactivity, and time. Event Type: event Type &alpha; event Constructors done: unit event The done constructor detects the termination of a behavior. Functions infix |: α event * α event -> α event The expression e1 | e2 is an alternative event. The first of the two events is chosen, and the data associated with that event becomes the data for the alternative event. predicate: boolean -> unit event predicate(b) turns a boolean value into an event with trivial (unit) data. The event occurs the first time after local time 0 that the predicate b is true. infix =>: α event * (α -> β) -> βevent The expression e => h is a handler event. It occurs the same time that e does, and returns as the function h applied to the data produced by e. infix =>: α event * β -> βevent This second form of e => b is a syntactic convenience, valid only when b is not a function. It is roughly equivalent to e => function x.b and is useful when the handler does not need the value of the data produced by the event. This is a special form and does not immediately evaluate the second argument. suchThat: α event * (α -> boolean) -> α event suchThat(e, p) is a filter event that occurs when e does, producing the data that e would, but only if the predicate p is true on that data. andEvent: α event * βevent -> (α * β)event andEvent(e1, e2) occurs when e1 and e2 occur simultaneously. The data returned is the pair of data from e1 and e2. snapshot: α * unit event -> α event snapshot(b, e) creates a new event that happens at the same time as the e event, and associates a static snapshot of the b behavior with the event. When the e event occurs, b is sampled. A new event with the static sampled value of b associated with the e event is created. Snapshot is a special form that does not immediately evaluate the second argument. Reactivity Type Constructors end : α The end constructor causes the behavior to finish immediately. infix until: α * α event -> α The expression b until e is equivalent to the behavior is b until the event e occurs, in which case, the behavior becomes b'. repeat: α -> α repeat(b) is equal to b until b ends. Then it restarts with b at that time. Time Type Functions timeTransform: α * number -> α timeTransform(b, timeTrans) adjusts the local time line for behavior b to follow the (time-varying) number timeTrans. For example, timeTransform(b,time * 2) makes a behavior that runs twice as fast as b normally would. See the "Behaviors and Time" section above for more information.

Modeling Types

This section defines a broad range of types and associated functions that are useful for creating and manipulating ActiveVRML media values.

ActiveVRML uses the following conventions:

• Time is specified in seconds.
• Angles are specified in radians.
• Distances are specified in meters.
• Coordinate system conventions:
• The three-dimensional coordinate system used is a right-handed system with positive X to the right, positive Y up, and negative Z into the screen.
• Canonical 3D objects that are used (lights, cameras, microphones) are all positioned at the origin, facing -Z with +Y up.
• The two-dimensional coordinate system used has positive X to the right and positive Y up.
• The window in which images are viewed has its center at (0,0).

2D Point Type: point2 Type point2 A two dimensional point. Constructors origin2: point2 Functions point2Xy: number * number -> point2 point2Xy(x, y) takes two Cartesian coordinates, x and y, and returns a point2 type. point2Polar: number * number -> point2 point2Polar(theta, rho) takes two polar coordinates, theta and rho, and returns a point2 type. infix +: point2 * vector2 -> point2 Adds vector to point and returns a point2 type. infix -: point2 * vector2 -> point2 Subtracts vector from point and returns a point2 type. infix -: point2 * point2 -> vector2 The expression p1 - p2 creates a vector from p1 to p2. distance: point2 * point2 -> number distance(p1, p2) returns the distance between p1 and p2 by finding the square root of the x and y coordinates of p1 and p2 squared. distanceSquared: point2 * point2 -> number distanceSquared(p1, p2) returns only the x and y coordinates of p1 and p2 squared. Use distanceSquared when you need to compare the distance between one set of points and another. xComponent: point2 -> number xComponent(point) returns the first coordinate of point (Cartesian). yComponent: point2 -> number yComponent(point) returns the second coordinate of point (Cartesian). transformPoint2: transform2 -> (point2 -> point2) transformPoint2(transformation) (point) returns a point2 type by applying transformation to point. thetaComponent: point2 -> number thetaComponent(point) returns the theta component of point (polar). phiComponent: point2 -> number phiComponent(point) returns the phi component of point (polar) 2D Vector Type: vector2 Type vector2 A two dimensional vector. Constructors xVector2: vector2 yVector2: vector2 zeroVector2: vector2 Functions vector2Xy: number * number -> vector2 vector2Xy(x, y) constructs a vector2 type from two Cartesian coordinates, x and y. vector2Polar: number * number -> vector2 vector2Polar(theta, rho) constructs a vector2 type from two polar coordinates, theta and rho. normal: vector2 -> vector2 normal(vector) normalizes vector such that its length is equal to 1. length: vector2 -> number length(vector) returns the length of vector by finding the square root of the x and y coordinates squared. lengthSquared: vector2 -> number lengthSquared(vector) returns from vector only the x and y coordinates squared. Use lengthSquared when you need to compare the length of one vector to another. infix +: vector2 * vector2 -> vector2 The expression v1 + v2 adds the two vectors to form a third vector. infix -: vector2 * vector2 -> vector2 The expression v1 - v2 subtracts v2 from v1 to form a third vector. infix *: vector2 * number -> vector2 Multiplies vector2 by scale and returns a vector2 type. infix /: vector2 * number -> vector2 Divides vector2 by scale and returns a vector2 type. infix *: number * vector2 -> vector2 Multiplies scale by vector2 and returns a vector2 type. dot: vector2 * vector2 -> number dot(v1, v2) returns the dot product of the two vectors, v1 and v2. xComponent: vector2 -> number xComponent(vector) returns the x component of vector (Cartesian). yComponent: vector2 -> number yComponent(vector) returns the y component of vector (Cartesian). transformVector2: transform2 -> (vector2 -> vector2) transformVector2(transformation) (vector) returns a point2 type by applying transformation to vector. thetaComponent: vector2 -> number thetaComponent(vector) returns the theta component of vector (polar). rhoComponent: vector2 -> number rhoComponent(vector) returns the rho component of vector (polar). 2D Transformation Type: transform2 Type transform2 2D transformations represent a mapping between 2D-space and 2D-space. They are used to transform various 2D objects, including point2, vector2, and image, all by the overloaded apply function listed in their relevant sections. Constructors identityTransform2: transform2 Creates a transformation that does not change the object it is applied to. Functions infix o: transform2 * transform2 -> transform2 The expression t1 o t2 composes two transformations into a single transform2 type. translate: number * number -> transform2 translate(tx, ty) moves an object by adding tx and ty to the object's x and y coordinates. translate: vector2 -> transform2 translate(vector) moves an object by adding the x and y coordinates from vector to the object's x and y coordinates. scale: number * number -> transform2 scale(sx, sy) scales an object by multiplying sx and sy by the object's x and y coordinates. scale: vector2 -> transform2 scale(vector) scales an object by multiplying the x and y coordinates from vector by the object's x and y coordinates. scale2: number -> transform2 scale(number) scales an object by multiplying number by the object's x and y coordinates. rotate2: number -> transform2 rotate2(number) rotates an object number radians ccw. shear2: number -> transform2 shear2(number) shears an object in the x direction such that the object's x coordinate is increased by the product of its y coordinate multiplied by number. transform3x2: number * number * number * number * number * number -> transform2 transform3x2 converts a matrix of two rows and three columns to two coordinates, x and y. The third column contains the translation components. inverse: transform2 -> transform2 inverse(trans) creates a transformation that is inverse to trans. isSingular: transform2 -> boolean isSingular(trans) returns true if trans does not contain an inverse transformation. Image Type: image Type image A value of type image is a spatially continuous image behavior of infinite spatial extent. Operations on it include the application of 2D transforms and opacity, and overlaying images. Continuous images are constructed by importing bitmap files, projecting 3D geometry, or by rendering text into an image. Constructors emptyImage: image import(pathname.[bmp | jpeg | gif]): image * vector2 * number Import a bitmap file. The return value image is the imported image, centered at (0,0). point2 is the upper-righthand coordinate of the resultant image, and number is the resolution of the image in pixels per meter. Functions renderedImage: geometry * camera -> image The renderedImage(geometry, viewingCamera) function is the primary 3D to 2D interface. The viewingCamera parameter determines the projection by which the geometry will be imaged. The resultant image is spatially infinite, and the camera performs no cropping. Cropping, if desired, can be achieved by using the crop function below. It may be useful to think of this function as returning the entire projection plane as an image. The description of the camera type discusses the projection plane and projection point. infix over: image * image -> image The expression top over bottom constructs a new image by placing the top image over the bottom image. Is this a continuation of renderedImage definiation above or a typo? opacity2: number * image -> image opacity2(value, img), given a value from 0.0 to 1.0, returns a new image identical to img, but with a certain percentage of opacity, determined by value: percentage opacity = value * 100. This function composes multiplicatively; thus opacity2(0.5, opacity2(0.2, myOpaqueImg)) results in an image with opacity of 0.1 (or 90% transparent). The default opacity is 1 (fully opaque). crop: point2 * point2 * image -> image crop(min, max, img) returns an image identical to img inside of the box defined by min and max, and a transparent layer outside of this box. tile: point2 * point2 * image -> image tile(min, max, img) returns an infinitely tiled image given by cropping img to the region (min,max), and then replicating that region infinitely in all directions. transformImage: transform2 -> (image -> image) transformImage(transformation) (image) returns an image type by applying transformation to image. Composite 2.5D Image Type: montage Type montage A montage is a set of images with associated depth values. Montages are useful for creating multilayered, image-based (cel) animation. Constructors emptyMontage Constructs a montage object with no images. Functions imageMontage: image * number -> montage imageMontage(image, depth) builds a 2.5D image set with a single image at depth. infix union: montage * montage -> montage The expression m1 union m2 combines the contents of two image sets, m1 and m2, into a single collection. renderedImage: montage -> image renderedMontage(montage) converts the set of images and depths encapsulated in montage into a flattened image. Image elements with larger depths will be layered underneath. 3D Point Type: point3 Type point3 A three dimensional point. Constructors origin3 Functions point3Xyz: number * number * number -> point3 point3Xyz(x, y, z) constructs a point3 type from three Cartesian coordinates. point3Spherical: number * number * number -> point3 point3Spherical(theta, phi, rho) constructs a point3 type from three polar coordinates. infix -: point3 * point3 -> vector3 The expression p1 - p2 creates a vector from p1 to p2. distance: point3 * point3 -> number distance(p1, p2) returns the distance between p1 and p2 by finding the square root of the x, y, and z coordinates of p1 and p2 squared. distanceSquared: point3 * point3 -> number distanceSquared(p1, p2) returns only the x, y, and z coordinates of p1 and p2 squared. Use distanceSquared when you need to compare the distance between one set of points and another. infix +: point3 * vector3 -> point3 Adds vector to point to return a new point3 type. infix -: point3 * vector3 -> point3 Subtracts vector to point to return a new point3 type. transformPoint3: transform3 -> (point3 -> point3) transformPoint3(transformation) (point) returns a point3 type by applying transformation to point. xComponent: point3 -> number xComponent(point) returns the x component of point (Cartesian). yComponent: point3 -> number yComponent(point) returns the y component of point (Cartesian). zComponent: point3 -> number zComponent(point) returns the z component of point (Cartesian). thetaComponent: point3 -> number thetaComponent(point) returns the theta component of point (Polar). phiComponent: point3 -> number phiComponent(point) returns the phi component of point (Polar). rhoComponent: point3 -> number rhoComponent(point) returns the rho component of point (Polar). 3D Vector Type: vector3 Type vector3 A three dimensional vector. Direction and magnitude. ??? Constructors xVector3: vector3 yVector3: vector3 zVector3: vector3 zeroVector3: vector3 Functions vector3Xyz: number * number * number -> vector3 vector3Xyz(x, y, z) constructs a vector3 type from three Cartesian coordinates, x, y, and z. vector3Spherical: number * number * number -> vector3 vector3Spherical(theta, phi, rho) constructs a vector3 type from three polar coordinates, theta, phi, and rho. normal: vector3 -> vector3 normal(vector) normalizes vector such that its length is equal to 1. length: vector3 -> number length(vector) returns the length of vector by finding the square root of the x, y, and z coordinates squared. lengthSquared: vector3 -> number lengthSquared(vector) returns from vector only the x, y, and z coordinates squared. Use lengthSquared when you need to compare the length of one vector to another. infix +: vector3 * vector3 -> vector3 The expression v1 + v2 adds the two vectors to form a third vector. infix -: vector3 * vector3 -> vector3 The expression v1 - v2 subtracts v2 from v1 to form a third vector. infix *: vector3 * number -> vector3 Multiplies vector by scale and returns a vector3 type. infix *: number * vector3 *-> vector3 Multiplies scale by vector and returns a vector3 type. infix /: vector3 * number-> vector3 Dividesvector by scale and returns a vector3 type. dot: vector3 * vector3 -> number dot(v1, v2) returns the dot product of the two vectors, v1 and v2. cross: vector3 * vector3 -> vector3 dot(v1, v2) returns the cross product of the two vectors, v1 and v2. transformVector3: transform3 -> (vector3 -> vector3) transformVector3(transformation) (vector) returns a vector3 type by applying transformation to vector. xComponent: vector3 -> number xComponent(vector) returns the x component of vector (Cartesian). yComponent: vector3 -> number yComponent(vector) returns the y component of vector (Cartesian). zComponent: vector3 -> number zComponent(vector) returns the z component of vector (Cartesian). thetaComponent: vector3 -> number thetaComponent(vector) returns the theta component of vector (polar). phiComponent: vector3 -> number phiComponent(vector) returns the phi component of vector (polar). rhoComponent: vector3 -> number rhoComponent(vector) returns the rho component of vector (polar). 3D Transformation Type: transform3 Type transform3 3D transformations represent a mapping between 3D-space and 3D-space and are used to transform various 3D entities, including point3, vector3, geometry, microphone, and camera, all by using the overloaded apply function listed in their relevant sections. Constructors identityTransform3:transform3 Creates a transformation that does not change the object it is applied to. Functions infix o: transform3 * transform3 -> transform3 The expression t1 o t2 composes two transformations into a single transform3 type. translate: number * number * number -> transform3 translate(tx, ty, tz) moves an object by adding tx, ty, and tz to the object's x, y, and z coordinates. translate: vector3 -> transform3 translate(vector) moves an object by adding the x, y, and z coordinates from vector to the object's x, y, and z coordinates. scale: number * number * number -> transform3 scale(sx, sy, sz) scales an object by multiplying sx, sy, and sz by the object's x, y, and z coordinates. scale: vector3 -> transform3 scale(vector) scales an object by multiplying the x, y, and z coordinates from vector by the object's x, y, and z coordinates. scale3: number -> transform3 scale(number) scales an object by multiplying number by the object's x, y, and z coordinates. rotate: vector3 * number -> transform3 rotate(vector, number) rotates an object number radians about the axis specified by vector. xyShear: number -> transform3 xyShear2(number) shears an object in the x direction such that the object's x coordinate is increased by the product of its y coordinate multiplied by number. yzShear: number -> transform3 yzShear2(number) shears an object in the y direction such that the object's y coordinate is increased by the product of its z coordinate multiplied by number. zxShear: number -> transform3 zxShear2(number) shears an object in the z direction such that the object's z coordinate is increased by the product of its x coordinate multiplied by number. transform4x4: number * number * number * number * number * number * number * number * number * number * number * number * number * number * number * number -> transform3. The fourth column contains the translation components. transform4x4 converts a matrix of four rows and four columns to four coordinates, x, y, z, and w. lookAtFrom: point3 * point3 * vector3 -> transform3 lookAtFrom(from, to, up) creates a transformation when applied to an object centered at the origin, with +Y up and directed toward -Z, moves the object to from, pointing towards to, with its up direction as close to up as possible. inverse: transform3 -> transform3 inverse(trans) creates a transformation that is the inverse of trans. isSingular: transform3 -> boolean isSingular(trans) returns true if trans does not have an inverse transformatin. 3D Geometry Type: geometry Type geometry A value of type geometry is a spatially continuous behavior of infinite spatial extent in three dimensions. A geometry value is constructed by importing other geometry formats (such as VRML), applying modeling transformations, material properties, aggregating multiple geometries, positioning sound in 3D, and specifying other attributes. Constructors emptyGeometry: geometry import(filename.[wrl]): geometry * point3 * point3 In the import constructor, geometry is the result of importing the specified file. The two points returned are the minimum and maximum extents of the tightest axis-aligned, rectangular bounding volume containing the geometry. import(beginLiteral wrl ascii body endLiteral): geometry * point3 * point3 An alternative import constructor, allows VRML 1.0 geometry to be expressed in line. body is a sequence of ascii characters in VRML 1.0 format. Functions infix union: geometry * geometry -> geometry g1 union g2 aggregates two geometries into their geometric union. soundSource3: sound -> geometry The soundSource3 function allows sounds to be embedded into a geometry. It creates a geometry with the specified sound positioned at the local coordinate system origin. The resultant geometry may be transformed in space, and has no visible presence when rendered. The function renderedSound, described in the Sound section below, takes a geometry and a microphone, and creates a sound by spatializing all of the sounds embedded into that geometry with respect to the microphone. transformGeometry: transform3 -> (geometry -> geometry) transformGeometry(transformation) (geometry) returns a geometry type by applying transformation to geometry. opacity3: number * geometry -> geometry opacity3(value, geo), given a value from 0.0 to 1.0, returns a new geometry identical to geo, but with a certain percentage of opacity, determined by value: percentage opacity = value * 100. This function composes multiplicatively, making opacity3(0.5, opacity3(0.2, myOpaqueGeo)) result in a geometry with opacity of 0.1 (or 90% transparent). The default opacity is 1 (fully opaque). texture: image * geometry -> geometry The texture function is the means by which texture mapping onto geometry is specified. The coordinates of the image are mapped onto the texture map coordinates associated with the vertices of the primitive geometries comprising the geometry being mapped, resulting in textured geometry. If the primitive geometries have no texture coordinates, texturing is ignored. Note that textures are applied in an outer-overriding fashion. That is, texture(im1, texture(im2, geo)) results in geo textured with im1. The default texture is none. Note

The following functions create light geometries, all of which have no visible appearance themselves; but, they do cast light onto other objects they are aggregated with. The point and spot lights are both located at the origin. The directional and spot lights both point along the -Z axis. ambientLight: geometry directionalLight: geometry pointLight: geometry spotLight: number * number * number -> geometry The spotLight function has arguments fullcone, cutoff, and exponent. lightColor: color * geometry -> geometry lightColor(col, geo) creates a geometry identical to geo where all the lights are col color. By default, the light colors are white. lightAttenuation: number * number * number * geometry -> geometry lightAttenuation(c, l, q, geo) creates a geometry identical to geo where Lambertian light attenuation equation is set to 1 / (c + ld + qdd) where d is the distance from the light to the object. By default, the attenuation coefficients are (1, 0, 0). Note

The following functions allow for attributing geometry with standard Lambertian shading characteristics. The outermost applied attribute overrides other attributes of the same kind; that is, diffuseColor(red, diffuseColor(blue, geo)) results in a red geometry. diffuseColor: color * geometry -> geometry The default diffuseColor is white. ambientColor: color * geometry -> geometry The default ambientColor is white. specularColor: color * geometry -> geometry The default specularColor is black. emissiveColor: color * geometry -> geometry The default emmisiveColor is black. specularExponent: number * geometry -> geometry The default specularExponent is 1. Camera Type: camera Type camera The camera type is used to project geometry into an image via the renderedImage function. Constructors defaultCamera : camera The canonical camera, defaultCamera, has its projection point at (0,0,1), looking along the -Z axis with +Y up, with the projection plane in the XY plane at z = 0. New cameras may be created by applying transformations to existing cameras in order to position and orient the camera, and also to affect projection properties. Translation and orientation position and orient the camera, respectively. Scaling in X or Y stretches the resulting projection accordingly. Scaling in Z changes the distance between the projection point and the projection plane. For instance, a Z scale of less than one yields a wide-angle camera, while a Z scale of &infinity; will yield a parallel projecting camera. (In practice, scaling in Z by a very large number is sufficient.) Functions transformCamera: transform3 -> (camera -> camera) transformCamera(transformation) (camera) returns a camera type by applying transformation to camera. Sound Type: sound Type sound A sound value is constructed by importing primitive sounds, mixing, rendering of geometry with embedded sounds, and by the application of audio attributes, such as gain. Note that sound is always considered to be single channel. Stereo is supported by constructing two separate sounds. Certain audio effects can be achieved by using the general time transformation mechanism. For example, both phase shift and rate control can be achieved by time transformations. Constructors silence import(pathname.[wav | au | aiff]): sound * sound * number Importing a .wav, .au, or .aiff file constructs a pair of sounds, one for the left and right channels of the sound. If the imported file is monophonic, the two returned sounds are identical. When a sound is finished, it terminates. Sounds can be looped by using the repeat facility. The third returned value, a number, is the length in seconds of the longer of the two returned channels. Functions infix mix: sound * sound -> sound The expression s1 mix s2 combines two sounds, s1 and s2, into a single sound, with each component sound contributing equally. renderedSound: geometry * microphone -> sound renderedSound(geo, mic) produces an audio rendering of the sounds embedded within geo (by using soundSource3), with respect to mic. gain: number * sound -> sound gain(value, sound) multiplicatively adjusts the gain of a sound. Thus, gain(0.3, gain(5.0, origSound)) results in a sound 1.5 times as loud as the original sound. Microphone Type: microphone Type microphone The microphone type represents an audio perceiver in 3D. ActiveVRML 1.0 supports a simple microphone that may be spatially transformed by using modeling transformations. Future versions will add other attributes to the microphone. Constructors defaultMicrophone: microphone The canonical camera, defaultMicrophone, lies at the origin, looking along the -Z axis with +Y up. Functions transformMicrophone: transform3 -> (microphone -> microphone) transformMicrophone(transformation) (mic) returns an microphone type by applying transformation to mic. Color Type: color Type color Constructors red: color green: color blue: color cyan: color magenta: color yellow: color white: color black: color Functions colorRgb: number * number * number -> color colorRgb(red, green, blue) constructs a color type using the color values, red, green, and blue. colorHsl: number * number * number -> color colorHsl(hue, saturation, lightness) constructs a color type using the values, hue, saturation, and lightness. redComponent: color -> number redComponent(color) returns the red color value of color. greenComponent: color -> number greenComponent(color) returns the green color value of color. blueComponent: color -> number blueComponent(color) returns the blue color value of color. hueComponent: color -> number hueComponent(color) returns the hue value of color. saturationComponent: color -> number saturationComponent(color) returns the saturation value of color. lightnessComponent: color -> number lightnessComponent(color) returns the lightness value of color. Character Type: char Type char Constructors 'c' In the 'c' constructor, c is an ASCII character or one of the following escape forms: Escape Code Result \n Newline \t Tab \' Apostrophe \" Quote \\ Backslash \integer The ASCII character with this value Functions ord: char -> number ord(c) returns the ASCII code for character c. chr: number -> char chr(n) returns the ASCII character corresponding to n. String Type: string Type string Constructors "string-literal" In the "string-literal" constructor, string-literal is a sequence of characters or escape characters. Functions infix &: string * string -> string The expression string1 & string2 concatenates the two strings and returns the result. implode: char list -> string implode(char list) returns a string built from char list. explode: string -> char list explode(string) returns a char list built from string. numberToString: number * number -> string numberToString(num, precision) function formats num to a string with precision digits after the decimal point. If precision is zero, then the decimal point is elided. It is illegal for precision to be negative. Font Family Type: fontFamily Type fontFamily Constructors serifProportional: fontFamily sansSerifProportional: fontFamily monospaced: fontFamily Text Type: text Type text ActiveVRML supports the construction of simply formatted text, which can then be rendered into an image. ActiveVRML 1.0 supports simple scaling, coloring, emboldening, italicizing, and choosing from a fixed set of typefaces. Functions simpleText: string -> text The text created by simpleText has a default color (black), family (serif proportional), and is neither bold nor italic. It has a nominal scale of 1 point. textColor: color * text -> text textColor creates an attributor such that you can assign the color of the text. textFamily: fontFamily * text -> text textFamily selects the text font. This could use some explanation. bold: text -> text bold(text) returns a new text type with emboldened text. italic: text -> text italic(text) returns a new text type with italicized text. renderedImage: text -> image * point2 The renderedImage function takes text and returns an image, with the text centered around (0,0) in the image. The returned point2 is the coordinate of the upper-right hand corner of the the nontransparent region of the resultant image. The resultant image may subsequently be scaled by applying image transformations. The resultant image is transparent in all places other than where the text is actually rendered. There are explicitly no alignment operators (align left, right, center, etc.), as these can be achieved by transformation of the resultant image.

Integration, Differentiation, and Interpolation

Derivatives, integrals, and linear interpolation apply to the following types: Type Derivative number number point2 vector2 vector2 vector2 point3 vector3 vector3 vector3 derivative: T -> DT integral: DT -> DT

Note that the derivative of a point is a vector, but the integral of a vector is a vector.

Single User Interactivity

Interaction with a user is one way in which a behavior value can change over time. For ActiveVRML 1.0 models, a very simple user input model for a single user is provided. It provides behaviors that represent the user's mouse and keyboard, and facilities for specifying reactions to picking (clicking on) geometry and images.

User Input

The keyboard and mouse of the user are modeled through a number of additional, pervasive library functions and values in ActiveVRML: leftButtonState: boolean Tracks the state of the left button of the mouse. leftButtonDown: unit event Occurs each time the user presses the left mouse button. leftButtonUp: unit event Occurs each time the user releases the left mouse button. rightButtonState: boolean rightButtonDown: unit event rightButtonUp: unit event Analogous to the corresponding left button behaviors. keyState: character -> boolean For keyDown(c), the value is true if key c is presently depressed. keyDown: character event Occurs when a key is pressed. It produces the pressed character in the event data. keyUp: character event Occurs when a key is released. It produces the released character in the event data. mousePosition: point2 Tracks the current position of the mouse in world image coordinates.

Picking Images and Geometry

ActiveVRML 1.0 provides a simple model of picking geometry and images within a model. Picking is based upon a continuously probing input device, the user's mouse. The following probe functions take images (for 2D) or geometry (for 3D) and return events that occur when any ray, cast from the assumed eye point of the viewer, "touches" the specified image or geometry.

Occlusion is taken into account. That is, probing rays do not pass through one nontransparent image and into another image, or through one nontransparent geometry into another.

The ActiveVRML functions for picking are: pickable: image * (string list) -> image * (point2 * vector2) event pickable: geometry * (string list) -> geometry * (point3 * vector3) event

As an example of how these functions are used, consider: origGeo = ...; pickableGeo, pickEv = pickable(origGeo, ["my geo"]); newGeo = pickableGeo until pickEv => function (pt,vec) . emptyGeometry;

(The discussions here are focused on geometry, but the identical properties hold for image picking.)

When the user's probe device (typically his mouse) is over pickableGeo, pickEv will fire causing the event transition, making the geometry invisible (by transitioning to the empty geometry). Somewhat more formally, in, geo', ev = pickable(geo, path)

geo' behaves identically to geo, however, when the probe device is over it, the event ev is fired. The event data that ev is invoked with is the static point of intersection between the probe and the geometry (in the local coordinates of geo), and a vector-valued behavior that tracks the probe as it moves relative to the pick point (also in local coordinates).

The resultant geometry and event are completely insulated from any subsequent modeling transformations applied to the geometry. Thus, the geometry can be arbitrarily transformed (and multiply referred to), but the event will still fire when any instance, in any coordinate frame, is picked.

The path argument to pickable is a string list that is intended to allow the author to disambiguate between multiple instances of the same geometry or image in situations where different reactions are desired. For instance: origGeo = ...; pickableGeo, pickEv = pickable(origGeo, ["my geo"]); geo1 = transformGeometry(xf1)(pickableGeo); geo2 = transformGeometry(xf2)(pickableGeo); reactive1 = geo1 until pickEv => function (pt,vec) . emptyGeometry; reactive2 = geo2 until pickEv => function (pt,vec) . emptyGeometry; model = reactive1 union reactive2;

In this scenario, if either reactive1 or reactive2 is picked, both change to emptyGeometry, since the same event is being used. If the author wants each instance to behave independently under picking, multiple invocations of pickable would be used, each with a different path argument: origGeo = ...; geo1, ev1 = pickable(origGeo, ["first"]); geo2, ev2 = pickable(origGeo, ["second"]); rg1 = transformGeometry(xf1)(geo1); rg2 = transformGeometry(xf2)(geo2); reactive1 = rg1 until ev1 => function (pt,vec) . emptyGeometry; reactive2 = rg2 until ev2 => function (pt,vec) . emptyGeometry; model = reactive1 union reactive2;

The path argument exists in order to allow different events to be returned from the same base geometry. Because the return values of all of the functions in ActiveVRML are dependent solely upon the value of the inputs, multiple calls to pickable with the same geometry and the same path will return the same geometry * event pair each time. Thus, the path argument exists to allow the author to explicitly disambiguate multiple instances, when desired.

Pattern Matching

This section describes a useful, somewhat more advanced feature of ActiveVRML: the ability to specify declarations using pattern matching. The general syntax for a value declaration is as follows: pattern = expression

The general syntax for a function declaration is as follows: identifier pattern = expression

Patterns may be used to destructure values and specify bindings for (potentially) more than one identifier. Patterns are denoted in one of the following forms: () Matches the trivial value, (), of the type unit. identifier Matches any value and effectively binds the value to the identifier in the right hand side of the declaration. All of the identifiers in a pattern must be distinct. pattern1, pattern2 Matches a pair value if pattern1 and pattern2 match the left and right components of the pair. The comma pattern associates right. (pattern) Groups pattern syntax for readability and precedence. The form matches if pattern matches the value. pattern: type Matches the value if pattern does, and constrains the pattern to have a particular type.

The following are a few example declarations: x = 3 This declares an identifier x as a variable with a value of 3. (x, y) = (4, 17) This pattern matches since the pattern (x, y) and value (4, 17) are both pairs. Effectively, x is bound to 4, and y is bound to 17. (x, y) = p This declaration matches x to the first component of p (which must be a pair) and y to the second component of p. (x, y) = 3 This declaration will cause an error to be reported since 3 is not a pair. f() = 5 This declares a constant function. The application expression f() will always produce a value of 5. g(x) = x + 1 This declares a function g that takes an argument x. h(x, y) = x + y This declares a function h that takes a single argument that must be a pair. The function may be applied either as h(3, 4) or as h(p) where p is a pair of numbers. k(x, y, z) = x + y / z This declares a function k that takes a single argument that is a pair, the second component of the pair being also a pair. Because comma patterns associate right, the argument takes the following form: (int, (int, int)) An application of k could look like k(1, 2, 3), k(1, (2, 3)), k(1, p) where p is a pair, or k(t) where t is a triple of type int*int*int. The application k((1,2),3) will report an error.

Pattern matching can also be used to specify the formals for a function expression. The following is an anonymous addition function: function (x, y). x + y

And, this function returns the first element of a pair: function (x, y). x

ActiveVRML Models and World Wide Web Browsing

This section describes the conventions that must be followed to connect an ActiveVRML model to a World Wide Web (web) browser for single-user interactive animations.

ActiveVRML 1.0 models should contain the following comment as their first source line: // ActiveVRML 1.0 ASCII

An ActiveVRML model consists of a list of top-level declarations.

An external entry point will be a declaration with type geometry or image*sound*sound. These are the only behaviors that may be indexed from a uniform resource locator (URL). The former will cause the browser to enable a 3D navigational user interface to allow the user to navigate the geometry, and have the embedded sounds rendered (played). The latter form allows the creator of the model to use their own camera, and to directly specify the sounds to play to the left and right speakers.

Embedding and Hyperlinking To ActiveVRML

To specify a URL from an HTML document to an ActiveVRML model, use the following syntax:

This will create an active link to the model myModel in the file model.av on the server www.microsoft.com. The entry point for the model, mainEntry, must be type geometry or image*sound*sound. The former is for geometries that will use the default camera from the view (including embedded sound). The latter is for images with stereo sound.

When the user clicks the URL, an ActiveVRML 1.0 capable viewer will begin executing the ActiveVRML (at local time 0).

In order to display an ActiveVRML model on an HTML page, it's necessary to use the OBJECT and PARAM tags. <OBJECT CLASSID="uuid:{389C2960-3640-11CF-9294-00AA00B8A733}" ID="AVView" WIDTH=300 HEIGHT=300> <PARAM NAME="DataPath" VALUE="earth.avr"> <PARAM NAME="Expression" VALUE="model"> </OBJECT>

The OBJECT Tag

The OBJECT tag has the following attributes:CLASSID, ID, WIDTH, and HEIGHT.

The CLASSID attribute specifies the GUID that's assigned to the OLE control which displays ActiveVRML models. This 128-bit value is: 389C2960-3640-11CF-9294-00AA00B8A733.

The ID attribute is a string used to identify the control if you need to set the control's properties programmatically. For example, the tutorial specifies an identifier of "AVView". Using this identifier and VB Scripting, it's possible to assign a new model 'on the fly' by setting the control's Frozen, Expression, and DataPath properties: AVView.Frozen = True AVView.Expression = "second_model" AVView.DataPath = "\avrml\avr\rotate.avr" AVView.Frozen = False

The Frozen property disables the control temporarily so that both the Expression and DataPath properties can be set. (If Frozen were not set, the control would attempt to set reset the Expression property immediately using the current DataPath.)

The WIDTH and HEIGHT attributes specify the dimensions of the rectangle used by the control to display the model.

The PARAM Tag

The PARAM tag has the following attributes: NAME and VALUE. As the name implies, the PARAM tags specify parameters that are used by the OLE control.

In the previous example and the first use of the PARAM tag, the corresponding NAME attribute specifies that a data path is being specified by the VALUE attribute.

In the previous example and the second use of the PARAM tag, the corresponding NAME attribute specifies that an string expression is being specified by the VALUE attribute.

Hyperlinking from ActiveVRML

The hyperlinking interfaces in ActiveVRML are very basic: hyperlink3: string * geometry -> geometry hyperlink2: string * image -> image

These act as attributers for geometries and images. For example: im2 = hyperlink2("http://www.microsoft.com")(im1)

The variable im2 is now an image that when selected will be noticed by the browser, causing a jump to the specified URL. Note that the URL can be any valid web content type, not just ActiveVRML.

Viewer Conventions and Information

The ActiveVRML viewer window will obey a number of conventions that define its interaction with the model being displayed:

The origin of displayed images will be at the center of the viewer window.

When the window is resized, its coordinate range changes. Thus, generally, more or less of the model comes into view, rather than the model being stretched or squeezed. The model only reacts to changes in the size of the window if it is making use of the viewerUpperRight behavior described below.

The following information is available to the model in ActiveVRML: viewerResolution: number The resolution of the view in pixels per meter. In general, this number will be an approximation, as, among other things, monitor size varies. viewerUpperRight: point2 The coordinate of the upper-right corner of the viewer window. This is useful for models which size or position objects explicitly with respect to the window the model is being viewed in.

ActiveVRML Grammar and Lexical Conventions

The following information describes ActiveVRML grammatical formations and lexical conventions.

Identifiers

An ActiveVRML identifier is an alphanumeric string beginning with a letter. Identifiers are case sensitive. No length limit is imposed.

Type identifiers

A type identifier is an apostrophe (') followed by an identifier. In typeset documents, including this one, Greek letters are often used instead of ASCII for type identifiers to improve readability.

Comments

Comments are of the form /* This is a comment. */ // This is a comment.

where the first form encloses any characters up to the first */ pattern and the latter form ignores everything until the end of line. Nested comments are handled correctly.

White Space

Spaces, tabs, carriage-returns, line-feeds, and comments are considered white space and are ignored other than as token separators.

ActiveVRML Keywords

Following are the keywords of ActiveVRML. These words are reserved and may not be used as identifiers: and else event function if import in let list mix not o or over then union until

ActiveVRML Precedence Table Operator Associativity -> Right * (product type) Left list event Non Associative , (comma) Right .(dot in function) else in Non Associative until Right | (or event) Left => (event handler) Left o union over mix Left :: (list cons) Right or Left and Left not Non Associative = < <= > >= <> Left + - (binary) & Left * / Left ^ Right + - (unary) Non Associative : (type qualification) Non Associative

ActiveVRML Syntax program: declarations | epsilon declarations: declaration | declaration ; | declaration ; declarations declaration: pattern = commaexpression | identifier pattern = commaexpression pattern: () | identifier | pattern , pattern | (pattern) | pattern: typeexp expressionlist: nonemptyexpressionlist | epsilon nonemptyexpressionlist: expression , nonemptyexpressionlist | expression commaexpression: expression , commaexpression | expression expression: if expression then expression else expression | let declarations in expression | function pattern . expression | expression binaryoperator expression | unaryoperator expression | applyterm binaryoperator: + | - | * | / | ^ | = | > | >= | < | <= |:: | & | and | or | until | | | union | over | mix | o unaryoperator: + | - | not applyterm: applyterm term | term term: numberliteral | characterliteral | stringliteral | identifier | () | (commaexpression ) | [ expressionlist] | term: typeexp typeexp: typeidentifier | identifier | typeexp * typeexp | typeexp -> typeexp | typeexp identifier | (typeexp) epsilon: /* empty */