user: Provide the full official documentation entry for Types of Modifiers, preserving all code examples.

assistant: ### Generator Modifier

Creates a polynomial function.
These are basic mathematical formulas that represent lines, parabolas,
and other more complex curves, depending on the values used.

Mode
:   Method used to represent the equation.

    Expanded Polynomial
    :   Equation in the form \(y = A + Bx^1 + Cx^2 + ... + Dx^n\).

    Factorized Polynomial
    :   Equation in the form \(y = (Ax + B)(Cx + D)\).

Additive
:   Add the polynomial to the curve rather than replacing it.

Order
:   The highest power of `x` for this polynomial.

Coefficient
:   The constants A, B, C… in the equation.

### Built-in Function Modifier

These are additional formulas, each with the same options to control their shape.
Consult mathematics reference for more detailed information on each function:

Type
:   The built-in function to use:

    - [Sine](https://en.wikipedia.org/wiki/Sine)
    - [Cosine](https://en.wikipedia.org/wiki/Trigonometric_functions)
    - [Tangent](https://en.wikipedia.org/wiki/Trigonometric_functions)
    - [Square Root](https://en.wikipedia.org/wiki/Square_root)
    - [Natural Logarithm](https://en.wikipedia.org/wiki/Natural_logarithm)
    - Normalized Sine: \(sin(x)/x\)

Additive
:   Add the function to the curve rather than replacing it.

Amplitude
:   Adjusts the Y scaling.

Phase Multiplier
:   Adjusts the X scaling.

Phase Offset
:   Adjusts the X offset.

Value Offset
:   Adjusts the Y offset.

### Envelope Modifier

Lets you reshape the curve. First, you define an envelope, which consists of two horizontal
lines that more or less match the curve’s lower and upper bounds. Then, you add control points,
where each point can push, squeeze, and stretch the envelope (and the curve along with it)
at a certain frame.

[image reference removed]

The Envelope modifier.

Reference
:   The value which the envelope is centered around.

Min/Max
:   The offset from the reference value to the envelope’s initial lower/upper bound.

Add Control Point
:   Adds a control point at the current frame.

Point
:   Frame
    :   The frame of the control point.

    Min/Max
    :   The offset from the reference value to the envelope’s adjusted lower/upper bound
        at this frame.

### Cycles Modifier

Makes the curve repeat itself.

Before/After Mode
:   No Cycles
    :   Do not repeat the curve before/after the original.

    Repeat Motion
    :   Repeats the curve, keeping the values of each copy the same.

    Repeat with Offset
    :   Repeats the curve, offsetting each copy vertically so that its first keyframe matches the
        previous last keyframe.

    Repeat Mirrored
    :   Repeats the curve, flipping every other copy horizontally.

Count
:   The number of copies to create. A value of 0 means infinite.

#### Trivially Cyclic Curves

When the *Cycle Mode* for both ends is set to either *Repeat Motion* or
*Repeat with Offset*, and no other options of the modifier are
changed from their defaults, it defines a simple infinite cycle.

This special case receives some additional support from other areas of Blender:

- Automatic Bézier handle placement is aware of the cycle and adjusts to achieve a smooth transition.
- The [Cycle-Aware Keying](../../../animation/animation_editors.xhtml#bpy-types-toolsettings-use-keyframe-cycle-aware)
  option can be enabled to take the cycle into account when inserting new keyframes.

### Noise Modifier

Modifies the curve with a noise formula.
This is useful for adding subtle or extreme randomness to animated movements,
like camera shake.

Blend Type
:   Replace:
    :   Adds noise in the range [-0.5, 0.5].

    Add:
    :   Adds noise in the range [0, 1].

    Subtract:
    :   Subtracts noise in the range [0, 1].

    Multiply:
    :   Multiplies by noise in the range [0, 1].

Scale
:   Changes the horizontal scale of the noise. Higher values make for less dense oscillation.

Strength
:   Changes the vertical scale of the noise.

Offset
:   Offsets the noise in time.

Phase
:   Adjusts the random seed of the noise.

Depth
:   Adjusts how detailed the noise function is.

Lacunarity
:   Gap between successive frequencies. Depth has to be greater than 0 for this to have an effect.

Roughness
:   The amount of high frequency detail. Depth has to be greater than 0 for this to have an effect.

### Limits Modifier

Limits the curve to specific time and value ranges.

Minimum, Maximum X
:   Removes the original curve data to the left of the minimum frame and to the right of the maximum,
    replacing it by [Constant extrapolation](../channels/editing.xhtml#bpy-ops-graph-extrapolation-type).

Minimum, Maximum Y
:   Clamps the curve values, never letting them go below the minimum or above the maximum.

### Stepped Interpolation Modifier

Gives the curve a stepped appearance by sampling it every N frames and making it hold its value
after each sample. In a sense, this lowers the curve’s frame rate by letting it change its value
less frequently, producing choppy movement as a result.

Step Size
:   The number of frames to hold each step.

Offset
:   The number of frames to offset the sample points.

Start Frame
:   The frame where to start applying the effect.

End Frame
:   The frame where to stop applying the effect.