808 Part VII Animation

Then click the Debug button. The Expression Debug window appears, in which you can see the variable values change as items in the scene change. With the expression complete, you can drag the Time Slider back and forth and watch the pupil follow the ball from side to side. If you’re happy with the motion, click the Evaluate button and then the Close button to exit the interface.

This is a simple example, but it demonstrates what is possible. Figure 32-7 shows the resulting face.

Figure 32-7: The Expression controller was used to animate the eye’s following the ball in this example.

Understanding Expression Elements

The key to making the Expression controller work is building expressions. Before you can build expressions, you need to understand the various elements that make up an expression. Expressions can consist of variables (including predefined variables), operators, and functions, so we start by explaining these. It is also important to understand what the expression will return and the various return types.

Predefined variables

Max includes several predefined variables that have constant values. These predefined variables are defined in Table 32-1. They are case sensitive and must be typed exactly as they appear in the Syntax column.

Chapter 32 Using the Expression Controller

809

Table 32-1: Predefined Variables

Variable	Syntax	Value
PI	pi	3.14159
Natural Logarithm	e	2.71828
Ticks Per Second	TPS	4800
Frame Number	F	Current frame number
Normalized Time	NT	The entire time of the active number of frames
Seconds	S	The number of seconds based on the frame rate
Ticks	T	The number of ticks based on the frame rate, where 4,800 ticks
		equal 1 second

In addition to these predefined variables, you can choose your own variables to use. These variables cannot contain spaces, and each variable must begin with a letter. After a variable is defined, you can assign it a constant value or have it pick up its value from a controller track.

Operators

An operator is the part of an expression that tells how to deal with the variables. One example of an operator is addition: It tells you to add a value to another value. These operators can be grouped as basic operators, which include the standard math functions such as addition and multiplication; logical operators, which compare two values and return a true or false result; and vector operators, which enable mathematical functions between vectors. Tables 32-2 through 32-4 identify the operators that are in each of these groups.

Tip Although it isn’t required, scalar variables are typically lowercase and vector variables are uppercase.

Table 32-2: Basic Operators

Operator	Syntax	Example
Addition	+	i+j
Subtraction	-	i-j
Negation	-	-i
Multiplication	*	i*j
Division	/	i/j
Raise to the power	^ or **	i^j or i**j

810 Part VII Animation

Table 32-3: Logical Operators

Operator	Syntax	Example
Equal to	=	i=j
Less than	<	i<j
Greater than	>	i>j
Less than or equal to	<=	i<=j
Greater than or equal to	>=	i>=j
Logical Or (Returns a 1 if either value is 1)	|	i|j
Logical And (Returns a 1 if both values are 1)	&	i&j

Table 32-4: Vector Operators

Operator	Syntax	Example
Component (Refers to the x component of vector V)	.	V.x
Vector Addition	+	V+W
Vector Subtraction	-	V-W
Scalar Multiplication	*	i*V
Scalar Division	/	V/i
Dot Product	*	V*W
Cross Product	x	VxW

The order in which the operators are applied is called operator precedence. The first equations to be calculated are the expressions contained inside of parentheses. If you’re in doubt about which expression gets evaluated first, place each expression in separate parentheses. For example, the expression (2 + 3) * 4 equals 20 and 2 + (3 * 4) equals 14. If the equation doesn’t contain any parentheses and only simple operators, then the precedence is from left to right with multiplication and division coming before addition and subtraction.

Functions

Functions are like mini-expressions that are given a parameter and return a value. For example, the trigonometric function for sine looks like this:

Sin()

It takes an angle, which is entered within the parentheses, and returns a sine value. For example, if you open the Expression Controller dialog box and enter the expression

Sin(45)

Chapter 32 Using the Expression Controller

811

	and then open the Debug window, the Expression Value will be 0.707, which is the sine value
	for 45 degrees.
	Table 32-5 lists the functions that are used to create expressions.
Tip	You can see a full list of all the possible functions with explanations by clicking the Function
	List button in the Expression Controller dialog box.

Table 32-5: Expression Functions

Function	Syntax	Description
Sine	sin(i)	Computes the sine function for an angle.
Cosine	cos(i)	Computes the cosine function for an angle.
Tangent	tan(i)	Computes the tangent function for an angle.
Arc Sine	asin(i)	Computes the arc sine function for an angle.
Arc Cosine	acos(i)	Computes the arc cosine function for an angle.
Arc Tangent	atan(i)	Computes the arc tangent function for an angle.
Hyperbolic Sine	hsin(i)	Computes the hyperbolic sine function for an angle.
Hyperbolic Cosine	hcos(i)	Computes the hyperbolic cosine function for an
		angle.
Hyperbolic Tangent	htan(i)	Computes the hyperbolic tangent function for an
		angle.
Convert Radians to Degrees	radToDeg(i)	Converts an angle value from radians to degrees.
Convert Degrees to Radians	degToRad(i)	Converts an angle value from degrees to radians.
Ceiling	ceil(i)	Rounds floating values up to the next integer.
Floor	floor(i)	Rounds floating values down to the next integer.
Natural Logarithm (base e)	ln(i)	Computes the natural logarithm for a value.
Common Logarithm (base 10)	log(i)	Computes the common logarithm for a value.
Exponential Function	exp(i)	Computes the exponential for a value.
Power	pow(i,j)	Raises i to the power of j.
Square Root	sqrt(i)	Computes the square root for a value.
Absolute Value	abs(i)	Changes negative numbers to positive.
Minimum Value	min(i,j)	Returns the smaller of the two numbers.
Maximum Value	max(i,j)	Returns the larger of the two numbers.
Modulus Value	mod(i,j)	Returns the remainder of i divided by j.

Continued

812 Part VII Animation

Table 32-5 (continued)

Function	Syntax	Description
Conditional If	if(i,j,k)	Tests the value of i, and if it’s not zero, then j is
		returned, or if it is zero then k is returned.
Vector If	vif(i,V,W)	Same as the if function, but works with vectors.
Vector Length	length(V)	Computes the vector length.
Vector Component	comp(V,i)	Returns the i component of vector V.
Unit Vector	unit(V)	Returns a vector of length 1 that points in the same
		direction as V.
Random Noise Position	noise(i,j,k)	Returns a random position.

Return types

	A return type is the type of value that is expected to be returned to the track that was assigned
	the Expression controller. For example, if you assign the Position Expression controller to the
	Position track of an object, then the return type is a vector, expecting coordinate values.
Tip	In the Expression Controller dialog box, the object and the track that is assigned to the con-
	troller are displayed in the title bar.

The return type can be a number (called a scalar), a collection of coordinates (called a vector), or a collection of color values (called a Point3). Scalars are used to control parameter values, vectors define actual coordinates in space, and the Point3 return type defines colors. Each of these types has its own format that you need to know before you can use it in an expression.

Caution	Variables used in an expression need to match the return type. For example, if you have a
	Vector return type describing an object’s position, it can use a scalar or vector variable, but
	the expression result needs to be a vector. A scalar return type (such as a sphere’s Radius)
	cannot be multiplied by a vector; otherwise, an error appears.

Scalar return type

A scalar value is a single value typically used for an object parameter, such as a sphere’s radius or the length of a Box object. This is used for tracks that have the Float Expression controller applied. Any resulting value from the expression is passed back to the assigned parameter. Scalars have no special format — only the number.

Vector return type

If a transform such as position is assigned, the Expression pane shows three values separated by commas and surrounded by brackets. These three values represent a vector, and each value is a different positional axis. You can refer to an individual axis value by placing a dot and the axis after the variable name. For example, if a vector variable named boxPosition exists, you can refer to the X-axis position component with the variable boxPosition.x. Component values are actually scalars.

Chapter 32 Using the Expression Controller

813

Point3 return type

Materials work with yet another return type called a Point3. This type includes three numbers separated by commas and surrounded by brackets. Each of these values, which can range between 0 and 255, represents the amount of red, green, or blue in a color.

Note Any value that is out of range is automatically set to its nearest acceptable value. For example, if your expression returns a value of 500 for the green component, the color is shown as if green were simply 255.

Sample expressions

If you scan through your old physics and math books, you can find plenty of equations that you can use to create expressions. Following are some sample motions and their respective expressions.

For example, if the sphere object has the Float Expression controller applied to its Radius track, then a simple expression for increasing the radius from an initial radius value as the frame number increases would look like this:

initialRadius + F

You can use the trigonometric functions to produce a smooth curve from 0 to 360. Using the Sine function, you can cause the sphere’s radius to increase to a maximum value of 50 and then decrease to its original radius. The expression would look like this:

50 * sin(360*NT)

To make our sphere example move in a zigzag path, we can use the modulus function. This causes the position to increase slowly to a value and then reset. The expression looks like this:

[0, 10*mod(F,20), 10*F]

You can use the square root function to simulate ease in and ease out curves, causing the object to accelerate into or from a point. The expression would look like this:

[100*sqrt(NT*200), 10, 10]

Building complex expressions takes a little bit of math to accomplish, but it isn’t difficult to do and the more experience you get with expressions, the easier it becomes. To help get you started, Table 32-6 includes several prebuilt expressions that can be entered into the Expression Controller Interface to get certain motions.

Table 32-6: Sample Expressions

Motion	Expression	Variable Description
Circular motion	[Radius1 * cos(360*S),	Where Radius1 is the radius of the orbiting
	Radius1 * sin(360*S), 0]	path.
Elliptical motion	[Radius1 * cos(360*S),	Where Radius1 and Radius2 are the radii of
	Radius2 * sin(360*S), 0]	the elliptical path.

Continued