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ArtMatic 1 in 2 out components
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12 components act as either "dual filters" or "motion functions". Dual filters (such as Modulo & Int) implement two related functions (such as floating point modulo and integer quantize) one of which is passed out the left output and the other of which is passed out the right output. These are useful in cases where you need to perform two calculations on the same value. The Dual Filters versions are computationally more efficient than connecting two different 11 Filters to the same input. A few of the 12 components (such as Circular Motion) are motion functions and translate a value (usually time) into a position. The motion function allow you to set up trees where motion is automatically created by time.
An essential in-depth discussion of ArtMatic structures Trees and components is found in ArtMatic Designer References and in Building trees.

• Vector:
  Returns a 2D vector constant (x,y)
• Unity vector :
  Returns a 2D vector in the range -1 +1
• Normalised vector :
  Return a normalized 2D vector (magnitude 1)
• Cx + time derivative
  Allows an creative arbitrary animation path where the orientation can follow the trajectory. It returns in output B the derivative of the constant value changing over time when parameter A is keyframed. If no curve animates the parameter B the derivative will be zero. The keyframes value itself is returned in output A; The parameter B 'Dv (derivative) gain' allows to scale the derivative magnitude to adjust for amount of rotation. Typically the B output will be connected to a Z rotation tile to orient the object along a path.
• Polar Coordinates (8.08):
  Sets the vector using polar coordinates with radius (parameter A) ranging 0 to 2 and Angle (parameter B) in degree. The result will have a norm of radius.
  This mode is often used to constrain the constant vector to be of a particular length and handy to set seeds for complex polynomials fractals.

discussion :
Scale the input value using parameters A and B. Left output is A*x. Right output is B*x.
In ArtMatic Engine 8.5 the A B range is set to +-12 and the standard scaling options have been added for consistency with the 13 Set Vector (Ax,Bx,Cx) .

The component options picker provides the Scale options set.

discussion :
2D derivative dx,dy returns the partial derivatives df/dX, df/dY. When Normalize is at 1 the vector is normalized and will have an unit length.
The derivatives are taken by evaluating the tree above several times using finite differences. The Derivative component has to "see" the function above which means it cannot be used within a CT to compute the derivative of a function depending on values computed outside. Practically if you create a CT for a function and needs its derivative, put the derivative component outside the CT and connect the derivative to the CT output.
When used with loopers the derivative component shall be placed outside the loop, that is, after a memory or accumulating component like 11 Memory Add or 21 Orbit_analyzer # .

discussion :
This component provides a few different flavors of Modulo and Integer functions. The left output is a continuous floating point version of the modulo/remainder function (see 11 Modulo). The right output is the discontinuous integer modulo function (which is equivalent to the 11 Quantize component).

• Modulo & int Mod C:
  The left-output is a continuous floating point modulo (remainder) function equivalent to 11 Modulo. The right output is a discontinuous integer-quantized modulo (equivalent to 11 Quantize ) where parameter C determines the number of steps.
  
  
  Modulo & int Mod C with the int constant part (in red) looping over 4 integer values with modulo 4
  
  
• Modulo & integer part:
  Left-output is the floating point (continuous) modulo. The right-output returns the integer part of the input. The parameter option provides tiling options that limit the number of steps. This output can be very useful for differentiating copies and making them behave differently.


• Modulo & Random Value:
  Right-output: return a unique random value B for each step of the modulo function. The parameter option provides tiling options that limit the number of steps.
  
  
  With Modulo & Random Value a constant random value (in red) is associated with each ramp cycle
  
  
• Modulo & smooth step:
  Similar to "Modulo & integer part" but the step function is made continuous with smoothing of the steps. The modulo returned by the x-output is the modulo of the input's absolute value and so is symmetrical. This mode was used to animate the legs in Legs Mechanics ArtMatic Voyager example. The parameter option provides tiling options that limit the number of steps.

discussion :
Indexed Saw provides a saw wave where each saw is indexed and the index value is returned in output 2. Saw wave is an useful primitive to repeat symetrical shapes. Period sets the space between each repeats. When the index is further passed into random tiles it can be used to modify each saw instance separately or simply the index can give a particular color when passed into a color generating tile like 13 Main Gradient .


Example in Voyager Examples/DF Modeling/Geometry/CubeStar isaw.vy

The parameter options are "Infinite Tile" and "Repeats N". Use "Repeats N" when a fixed number of repetitions are needed otherwise the tiling is unbounded.

discussion :
Index to Array is an utility that maps the input index to an array cell position with 'Rows' number of item per row. The array cells are indexed from left to right and bottom to top. The input index is computed as int(input x * "input gain"). The array cell center position is given by (x=(index % Rows)*Period,y=(index / Rows)*Period). Negatives inputs are supported.
The "Period" parameter sets the dimension of the grid and the "Randomize" parameter allows to jitter the grid positions.
In general you will need to substract the cell center to the main (x,y) coordinate to get the full cell coordinates and use an iteration object to display all cells.
Index to Array can be used in motion equations to generate position on a grid for components like 2D Motion Pict Atlas or Motion Path render. When used directly it allows cool animation if you connect the input to time as each array cell will be displayed sequencially over time.
Examples in ArtMatic Designer CTX/Libraries/Component 8.5/Animation/

• log(Ax + B) & exp(Ax + B):
  The left output is the log and the right output is the exponential.
• exp(x s) & exp(-x s) B :slope:
  Parameter B acts as a slope value. The left output returns the positive exponent and the right output the negative exponent. The two curves are symmetrical. When creating DFRM objects, the second output could be used, for example, to thin the field while the first output is being used to scale the y-axis.
• tan(x) & atan(x):
  returns tan(x) in output 1 and atan(x) in output 2.
• Offset & Abs:
  Provides 2 abs function with independent offsets and height. This operator does not scale space and is DF friendly.
• Offset & N-Abs
  Provides 2 mirror function with independent offsets and height. This operator does not scale space and is DF friendly. N-Abs is basically the DF 2D line equation so you can think of Offset & N-Abs as a way to get 2 DF lines at once that you can position with the offsets A & B. A 2D xy DF line equation is actually a 3D DF plane, since it is infinite in z. Offset & N-Abs plot, Red curve is second output below:
  
  

discussion :
This component creates motion along a straight line when input is connected to Global Input Matrix Time (w) or z input in 2D systems. The direction is given by the "Direction" angle parameter and will be (cos(angle), sin(angle)). The 'Speed' parameter scales the input time. When negative motion will proceed backward. The motion is at constant speed if the input is linear.
The tile has the option to set the angle in degree or radian.
For a linear motion in 3D use the Linear motion tile, available starting ArtMatic Engine 8.0.

discussion :
When Speed X and Speed Y are the same and when the input is connected to Global Input Matrix Time (w), the result is a circular motion. The motion will be elliptical when the Speed X and Speed Y parameters are different. The 'Radius' sets the radius of the circle.The 'Speed X or Y' parameters scales the input time. When negative motion will proceed backward. The motion position has to be added to the target coordinates usually with 4 in 2 out add tile.
The tile has the options to interpret the radius in various units.
For a 3D circular motion you can use the 13 Circular motion tile, available starting ArtMatic Engine 8.0.

discussion :
This component creates continuous random 2D motion when the input is connected to Global Input Matrix Time (w).The 'Amplitude' sets the radius of the motion. The 'Speed' parameter scales the input time. When negative motion will proceed backward. The random positions are generated using a 1D vector valued (2D) perlin noise function. The motion position produced by Random motion has to be added to the target coordinates usually with a 4 in 2 out add tile when the target is a 3D object.
The tile has the options to interpret the radius in various units.
For a 3D random motion you can use the 13 Random motion tile, available starting ArtMatic Engine 8.0.

discussion :
Compiled Trees are groups of tiles that can be used in place of single tiles as a kind of macro or subroutine. 12 CT can hold any 1 input / 2 outputs tree and can be seen generally as a 2D vector valued 1D function.
12 CT are often used to define 2D coordinates derived from a single input (typically time). When input is time one can see the CT as the motion path equation or a 2D parametric curve of 't'. For example x=cos(t),y=sin(t) will define a circular motion. To efficiently render parametric curve ArtMatic Engine 8 offer a new component, 2D Motion Path render, that will work in tandem with 12 Compiled trees.

usage :
Select a 12 tile and use "New compiled tree" to create a new CT from the selection (Tree Edit menu or type 'n' key).
To save a CT on disk to use the function elsewhere use "Save compiled tree" from the Tree Edit menu.
You may also copy and paste the entire CT by using Copy Tile and Paste Tile from the Edit menu.
You may iterate the tree by "Iterations" number when the tree contains an Iterations tile to modify and accumulates various values. "Scale" provide an optional scaling of outputs which is set at 1 by default (no scaling).