Alternate Mapping: Transformed Value Minimum
The Fractal Science Kit fractal generator Alternate Mapping Transformed Value Minimum applies the Transformation to each orbit point and data for the point with the minimum magnitude is retained.
See also:
The Transformed Value Minimum properties pages are:
- Shape Transformation
- Transformation
- Properties
General
Before the orbit point P is passed to Transformed Value Minimum for processing, it is transformed by Transformation. When a point P is transformed, it is understood that P is passed through the transformation and the resulting transformed point placed back in P.
Next, the magnitude of P is computed using the properties for Distance Metric and Power. If Apply Shape is checked, we apply the Shape Transformation to the magnitude. This injects a simple orbit trap like quality into the value based on the shape defined by the Shape Transformation. Finally, the magnitude is used to determine which point P is kept as the minimum across all the points in the orbit.
The alternate point's Value is set to the minimum magnitude.
The alternate point's Angle is set to the angle of P based on the Angle Vertex property. The angle is defined by the vector from the Angle Vertex to P. Angle Vertex is one of:
- Origin
- Z Prior to Transformation
Origin is the point 0+0i. Z Prior to Transformation is the original orbit point value prior to applying the transformation.
The alternate point's Index is set to the dwell value associated with P.
Orbit Generation
The Orbit Generation section defines 3 properties: Min Dwell, Max Dwell, and Mod Dwell. These values control the set of orbit points that are considered when looking for the minimum value. Min Dwell is the 1st dwell to check. Max Dwell is the last dwell to check. Mod Dwell - 1 is the number of dwells to skip between checks. That is, a dwell is processed if the dwell is between MinDwell and MaxDwell inclusive, and (dwell - MinDwell) % ModDwell = 0.
Transformation Shift / Transformation Angle
The Transformation Shift and Transformation Angle sections define a transformation based on 1 or more orbit points. Transformation Shift defines a translation and Transformation Angle defines a rotation. These are combined into a single transformation (called S hereafter) composed of the translation followed by the rotation. S is used in conjunction with the transformation defined by the Transformation (called T). Instead of simply applying T, we apply inverse(S), then apply T, and finally apply S, where inverse(S) is the inverse transformation of S. The composite transformation is called the conjugate of T with respect to the conguating map S. These properties have no effect unless T is set to a transformation other than the identity transformation.
Transformation Shift
The Transformation Shift section includes a property Shift which is <None>, P, or P+Q. If Shift is <None>, no transformation is applied.
If Shift is P, a transformation is defined as:
f(z) = z + P
If Shift is P+Q, a transformation is defined as:
f(z) = z + (P+Q)
P and Q are one of the following values:
- C ( Pixel / Julia Constant )
- Z0 ( Initial Z / Pixel )
- Z ( Current Orbit Point )
- ZPrev1 ( Previous Orbit Point )
- ZPrev2 ( Second Previous Orbit Point )
- Orbit Transformation Point
- Triangle Metric Point
The 1st 2 settings differ based on the type of fractal (Mandelbrot or Julia). For Mandelbrot fractals, C is the pixel value associated with the orbit and Z0 is the orbit's initial z value. For Julia fractals, C is the Julia Constant and Z0 is the pixel value. The next 3 settings are the last 3 orbit points, respectively. The Orbit Transformation Point and the Triangle Metric Point are the special points defined by the Orbit Transformation and Triangle Metric Pages.
Factor and F(z) are applied to the given point before it is used in the transformation expression; i.e., the point is multiplied by Factor and then the function F(z) is invoked to transform the point. If F(z) is Ident, the function is not applied.
For example, if Shift is P, P is ZPrev1, Factor is -1, and F(z) is Pow2, the transformation is defined as:
f(z) = z + Pow2(-zprev1)
Transformation Angle
The Transformation Angle section includes a property Angle which is <None>, Arg(P), or Arg(P+Q). The Arg function returns the radian angle of a vector from the origin to the given point. P and Q are defined as above. If Angle is <None>, no transformation is applied.
If Shift is Arg(P), a transformation is defined as:
f(z) = z * Cis(Arg(P))
Multiplying z by Cis(A), where A is an angle given in radians, has the effect of rotating z by A radians counterclockwise.
If Angle is Arg(P+Q), a transformation is defined as:
f(z) = z * Cis(Arg(P+Q))
Factor and F(z) are applied to the given point before it is used in the transformation expression; i.e., the point is multiplied by Factor and then the function F(z) is invoked to transform the point. If F(z) is Ident, the function is not applied.
For example, if Angle is Arg(P), P is Z, Factor is 1, and F(z) is Ident, the transformation is defined as:
f(z) = z * Cis(Arg(z))