Advanced Formula Usage in Cinema 4D

Cinema 4D offers a powerful way to perform complex calculations and transformations by allowing you to enter formulas in any field that accepts numerical values. Whether you are working with the Formula spline primitive, Formula deformer, or any numerical parameters in the Attribute Manager, understanding the range of units, operators, functions, and constants available can greatly enhance your creative workflow.

In this article, we will explore all the essential components available in Cinema 4D’s formulae, how to use them effectively, and several practical examples to illustrate their application.

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General Overview

Cinema 4D’s formula system permits you to type in expressions anywhere numerical values are used. The system automatically converts units for you, regardless of the basic units defined in your preferences. This flexibility means you can mix and match measurements (such as kilometers, meters, and feet) without having to perform manual conversions.

Key points:

• You can enter formulas in any field (e.g., the Attribute Manager).
• Formulas support values with different units (km, m, cm, mm, etc.).
• All computations are automatically scaled based on the basic unit settings.

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Formula Entry and Syntax

When writing formulas, you can:

• Combine numbers using standard mathematical operators.
• Use units directly in your formulas (e.g., 1km, 10cm).
• Enclose function arguments with parentheses, ensuring that the number of open brackets equals the number of closing brackets.

Tip: Always check that each function’s arguments are correctly bracketed, especially when using nested functions, such as:
sin(sqr(exp(pi)))

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Mathematical Operators

Cinema 4D implements a variety of arithmetic operators which adhere to standard mathematics:

• Addition (+):
  Example: 144 + 14 results in 158.
• Subtraction (-):
  Example: 144 - 14 results in 130.
• Multiplication (*):
  Example: 144 * 2 results in 288.
• Division (/):
  Example: 144 / 12 results in 12.
• Parentheses ( ):
  Use them to explicitly define order of operations.
  Example: 3 + 4 * 2 evaluates to 11 by following standard precedence rules.

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Units of Measurement

The formula system supports a number of units and performs conversions automatically:

• km: Kilometer
  Example: 1km = 1000m
• m: Meter
  Example: 144m - 14m = 130m
• cm: Centimeter
  Example: 1cm = 0.01m
• mm: Millimeter
  Example: 1mm = 0.001m
• um: Micrometer
  Example: 1um = 0.000001m
• nm: Nanometer
  Example: 1nm = 0.000000001m
• mi: Mile
  Example: 1mi = 1609.344m
• yd: Yard
  Example: 1yd = 0.914m
• ft: Foot
  Example: 1ft = 0.305m
• in: Inch
  Example: 1in = 0.025m
• B: Frame Number

Using these units, you can write formulas that automatically adjust the value depending on the basic unit settings in the preferences.

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Logical Operators

You can also incorporate logical comparisons and conditions into your formulas:

• Equal 
  Example: 1km = 1000m or 144 - 14 == 130
• Greater than 
  Example: 1cm > 0.01m
• Less than 
  Example: 1mm < 0.001m
• Greater than or equal 
  Example: 1um >= 0.000001m
• Less than or equal 
  Example: 1nm <= 0.000000001m
• Not equal 
  Example: 1mi != 1609.344m
• Not 
  Logical negation operator.
• Or 
  Logical OR operator.
• And &&, and. , and
  &&,
  Logical AND operator.

Additional bitwise operators are also provided:

• Bitwise and (&)
• Bitwise or (|)
• Bitwise xor (^)
• Bitwise not (~)

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Constants

Cinema 4D provides several built-in constants that are useful in many calculations:

• e: Euler’s Number (≈ 2.71828)
• pi: Pi (≈ 3.14159)
• pi05: Half Pi (≈ 1.5708)
• pi2: Double Pi (≈ 6.28318)
• piinv: Inverse Pi
• pi05inv: Half Inverse Pi
• pi2inv: Double Inverse Pi

These constants allow you to write more readable expressions without having to re-calculate or re-enter these values.

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Functions

A rich set of mathematical functions is available in Cinema 4D’s formula language:

Trigonometric Functions

• sin(a): Returns the sine of a (note: function arguments for trigonometric functions are interpreted as degrees when entered in parameter text boxes and as radians when used in the Formula time curve).
• cos(a): Cosine.
• acos(a): Arc cosine.
• asin(a): Arc sine.
• tan(a): Tangent.
• atan(a): Arc tangent.
• cosh(a): Hyperbolic cosine.
• sinh(a): Hyperbolic sine.
• tanh(a): Hyperbolic tangent.

Rounding and Truncation Functions

• floor(a): Rounds downwards.
  Example: floor(11.8) returns 11.
• ceil(a): Rounds upwards.
  Example: ceil(11.2) returns 12.
• round(a): Rounds to the nearest integer.
  Example: round(11.8) returns 12.
• trunc(a): Truncates the decimal portion.
  Example: trunc(-11.8987) returns -11.

Working with Numbers

• abs(a): Absolute value.
  Example: abs(-11) returns 11.
• sqr(a): Squares the number.
  Example: sqr(5) returns 25.
• sqrt(a): Square root.
  Example: sqrt(49) returns 7.
• exp(a): Exponential function.
  Example: exp(5) yields approximately 148.41.
• log10(a): Logarithm base 10.
  Example: log10(100) returns 2.
• log(a): Natural logarithm (base e).
  Example: log(e) returns 1.
• rnd(a{;b}): Generates a random number.
  • rnd(100) is equivalent to rnd[100].
  • rnd(100;234) is equivalent to rnd([100][234]).
• pow(a;b): Exponentiation.
  Example: pow(2;3) returns 8.
• mod(a;b): Modulo operation.
  Example: mod(10;4) returns 2.
• clamp(a;b;c): Clamps a value between two limits.
  Example: clamp(2;6;10) returns 6.
• min(a;b): Returns the minimum of two values.
  Example: min(4;7) returns 4.
• max(a;b): Returns the maximum of two values.
  Example: max(4;7) returns 7.
• len(a;b{;…}): Computes vector length.
  Example: len(1;1) returns approximately 1.414.

Bitwise Shift Functions

• (a)<<(b) or (a)shl(b): Bitwise shift to the left.
  Example: 1<<4 yields 16.
• (a)>>(b) or (a)shr(b): Bitwise shift to the right.
  Example: 1000>>4 yields 16.

Note: Trigonometric functions in parameter text boxes always expect degrees, whereas in the Formula time curve they are interpreted in radians. Therefore, writing sin(2*pi) in a parameter box does not compute the sine of 360° but of approximately 6.283°.

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Expanded Formula Entry for Multiple Selections

One of Cinema 4D’s most attractive features is the ability to manipulate multiple objects simultaneously using formulas. When you have multiple selections (for example, several objects, light sources, or cubes), you can use special variables to make relative adjustments easily:

• x: Represents the original value of the parameter.
• num: Represents the index (or order) of the selected element.
• tot: Represents the total number of selected elements.

Practical Examples:

1. Adjusting Object Positions:Imagine you have three spheres positioned at 100, 200, and 300 meters on the X-axis. By selecting all spheres and entering x+200 in the P.X field, the new positions will be adjusted to 300, 400, and 500 meters, respectively.
2. Incrementally Increasing Light Intensity:For ten light sources that need their intensity increased in steps of 20, enter num*20 in the Intensity field. The light sources will then be assigned intensities based on their order in the Object Manager (0, 20, 40, etc.).
3. Randomizing Light Intensity:If you want to randomly disperse the intensity of ten light sources over a total value of 200, use the formula:
   x + rnd(200)
   This applies a random adjustment to each light source’s original intensity.
4. Equal Interval Values for Cube Fillet Radii:Suppose you have 28 cubes and wish to evenly distribute values from 0 to 100 for the Fillet Radius. By selecting all cubes and entering:
   100 * (num/(tot-1))
   each cube will receive a proportionate fillet radius value along the specified range.

Tip: You can also define color values using this method. Right-click the Color parameter and select “Show Subchannels” from the context menu to work with individual color subchannels.

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Basic Elements

In many 3D systems, you can drive object parameters using formulas. Some of the most common basic parameters include:

• Position: px, py, pz
• Rotation: rx, ry, rz
• Scale: sx, sy, sz
• UVW Coordinates: u, v, w
• Object Index and Count: id (object index), count (total object count)
• Other Attributes:
  • falloff – Falloff weight for deformers or effects.
  • t – Project time, which can vary from -∞ to +∞.
  • f – Frequency, usable for periodic effects, also ranging over -∞ to +∞.

These values form the backbone of how you control transformations, texturing, and animation in the scene.

Conditional Expressions

Conditional operators allow you to execute a formula based on a condition. The basic syntax uses the ?(a; b) format, where if the condition is met, value a is returned; otherwise, value b is used. For example:plaintext

(3 > 4) ? (10; 20)  // returns 20 because 3 is not greater than 4

This can also be written in an alternative style such as if(3 > 4; 10; 20).

Arguments and Function Syntax

Some functions, like mod(a;b), require that arguments are separated by a semicolon ; or enclosed in square brackets. For instance:

• rnd(100) is equivalent to rnd[100].
• rnd(100;234) is the same as rnd([100][234]).

This syntax ensures that each function correctly receives its intended parameters.

Mathematical and Logical Operators

Mathematical Operators

The standard arithmetic operators are used to build your expressions:plaintext

144 + 14  // Addition: returns 158
144 - 14  // Subtraction: returns 130
144 * 2   // Multiplication: returns 288
144 / 12  // Division: returns 12
3 + 4 * 2 // Parentheses can override precedence.

Logical Operators

Logical operators let you perform comparisons and boolean evaluations:plaintext

=    // Equal comparison (e.g., 1km = 1000m)
==   // Alternative equal compare (144 - 14 == 130)
>    // Greater than, < for less than, >=, <=, != for not equal
!    // Logical NOT
||   // Logical OR (can also be 'or.' or 'or')
&&   // Logical AND (also written as 'and.' or 'and')

Bitwise operators are also available (e.g., &, |, ^, ~).

Units of Measurement

Units play a significant role in 3D calculations. When writing formulas, you can use a variety of units:plaintext

1km = 1000m
1m  = 1m
1cm = 0.01m
1mm = 0.001m
1um = 0.000001m
1nm = 0.000000001m
1mi = 1609.344m
1yd = 0.914m
1ft = 0.305m
1in = 0.025m
B   // Frame number

These units allow you to express dimensions directly, with automatic conversion based on your system’s settings.

Constants

Some mathematical constants are predefined for convenience:plaintext

e      // Euler's number (≈ 2.71828)
pi     // Pi (≈ 3.14159)
pi05   // Half of Pi
pi2    // Double Pi
piinv  // Inverse Pi
pi05inv// Half Inverse Pi
pi2inv // Double Inverse Pi

Using constants like pi makes your formulas more readable and easier to maintain.

Built-in Functions

A wide range of built-in functions lets you perform complex mathematical operations:

Trigonometric Functions

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sin(a)   // Sine (input in degrees in parameter text boxes, radians in time curves)
cos(a)   // Cosine
tan(a)   // Tangent
atan(a)  // Arc Tangent
acos(a)  // Arc Cosine
asin(a)  // Arc Sine

Hyperbolic and Exponential Functions

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cosh(a)   // Hyperbolic cosine
sinh(a)   // Hyperbolic sine
tanh(a)   // Hyperbolic tangent
exp(a)    // Exponential function
log10(a)  // Base-10 logarithm
log(a)    // Natural logarithm

Rounding and Absolute Functions

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floor(a)   // Round down (e.g., floor(11.8) = 11)
ceil(a)    // Round up (e.g., ceil(11.2) = 12)
round(a)   // Normal rounding (e.g., round(11.8) = 12)
trunc(a)   // Truncate (e.g., trunc(-11.8987) = -11)
abs(a)     // Absolute value (e.g., abs(-11) = 11)

Advanced Numerical Functions

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sqr(a)          // Square – for example: sqr(5) = 25
sqrt(a)         // Square root – for example: sqrt(49) = 7
pow(a; b)       // Exponentiation – for example: pow(2; 3) = 8
mod(a; b)       // Modulo – for example: mod(10; 4) = 2
clamp(a; b; c)  // Clamps a value between two limits – e.g., clamp(2; 6; 10) = 6
min(a; b)       // Minimum of a or b
max(a; b)       // Maximum of a or b

Bitwise Shift Functions

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(a) << (b) or (a) shl(b)  // Bitwise shift to left (e.g., 1 << 4 = 16)
(a) >> (b) or (a) shr(b)  // Bitwise shift to right (e.g., 1000 >> 4 = 16)

Getting Started with Formula Expressions

In practice, you can start experimenting with different formulas to drive visual effects or parameter animations. Here are some quick-start examples based on common patterns:

Creating a White-to-Dark Vertical Gradient Based on Object Index

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1 - abs(mod(id/count; 1) - 0.5)*2

This formula creates a gradient where objects transition from white to dark vertically through the list.

Horizontal Gradient Based on UVW Coordinates

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1 - abs(mod(u; 1) - 0.5)*2

This drives a horizontal transition by manipulating the u coordinate.

Calculating a Radial Gradient

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sqrt(px*px + py*py + pz*pz)/1000

This calculates the distance from the origin (px, py, pz) to create a radial gradient effect.

Formula Library Examples

• Default/Wave Animation:plaintext/0.5 + sin((subfields + (id / count) + d) * f * 360)*0.5
• Alternating Lines Effect:plaintextmod((id/80);2) ? (t*f; -t*f)

These examples can be used directly or serve as inspiration when constructing your own formula-based animations and effects.

Usage Guide

1. Identify Parameters: Determine which parameters you wish to drive—whether it’s for position, rotation, scale, UVW mapping, intensity, or other attributes.
2. Compose Your Formula: Use a combination of basic values, units, operators, and functions. Experiment with conditional expressions to create adaptive effects.
3. Test in Context: Enter your formulas in the corresponding parameter fields within Cinema 4D or your preferred 3D software. Remember that the input context (time curves vs. parameter boxes) may affect how functions like trigonometric operations are interpreted.
4. Debug and Iterate: Use simple test cases (such as the examples provided) to validate the behavior of your formulas before integrating them into more complex setups.