Parametric Curves

Parametric Curves is an orbit trap coloring method for Ultra Fractal. It exists in two forms: a plugin for use in UF5, and a stand-alone coloring method compatible with older versions of the program. They are included in the Ultra Fractal Formula Database or you can download them here. For the most part, they both work exactly the same; the differences will be addressed at the end of the page. This help file assumes that the reader already has a reasonable working knowledge of UF; if something seems confusing, try reading the main help files, or contact me if you have questions.

This coloring method draws Lissajous (familiar to anyone who has played with an oscilloscope) and roulette curves (the patterns made by Spirograph), which can then be applied to the base fractal the same as any other orbit trap shape.

Quick Links:
Epicycloid & Hypocycloid
General Parameters
Differences between .ucl and .ulb
Example images


Lissajous curves are produced by harmonic motion. Wikipedia and Wolfram both have articles describing the mathematics behind them. From a design point of view, it's useful to remember that they tend toward square shapes.

X Period and Y Period
These are the frequency of the oscillations. If they are equal, the result will be a circle.

X = 2, Y = 3

X = 4, Y = 3

X = 17, Y = 5

When the X and Y periods are both whole numbers, one cycle is enough to complete the curve. If the X or Y periods have decimals, multiple cycles will generally be needed, unless you want the effect of a partial pattern.

Lissajous with X = 3.98 and Y = 4.09

Cycles = 1

Cycles = 2.7

Cycles = 6.85

Number of Points
The placement of dots generally looks more orderly if the Number of Points is evenly divisible by both the X Period and the Y Period.

Lissajous with X = 4 and Y = 3

No. of Points = 230

No. of Points = 240

Epicycloid & Hypocycloid

These are the classic Spirograph patterns. The epicycloid is drawn by rolling a gear around the outside of a circle, while the hypocycloid has the gear rolling around the inside of a ring. The coloring method was written with whole-number values in mind for the X and Y, but it's interesting to use the Explore feature to go through the in-between decimal values and see what happens. Here are links to Wikipedia: Epicycloid & Hypocycloid, and Wolfram: Epicycloid & Hypocycloid.

X Period
This corresponds to the size of the stationary circle or ring around which the moving gear rolls. It determines the number of cusps around the circumference. So, for example, an X Period of 5 will make star-shaped curves, while an X Period of 3 will make triangular forms.

Several epicycloids
with X = 4

Several hypocycloids
with X = 7

Y Period
This is the size of the moving gear. If you think of the X period as a number of evenly-spaced points around the circumference of a circle, and the curve as a line drawn connecting those points, the Y period determines which dot gets connected next. 1 goes to the next closest point, 2 skips the closest and goes to the one beyond, and so on.

Hypocycloid with X = 9

Y = 1

Y = 2

Y = 4

Trochoid Parameter
This is the distance from the edge of the gear which is being used to draw the curve. If it's equal to the Y period, the cusps will be pointed. Less than the Y period makes the points more blunt. Greater than the Y period turns them into loops. In an epicycloid, if the trochoid parameter = X + Y, the loops will pass through the center point. In a hypocycloid, this happens if the trochoid parameter = X - Y.

Epicycloid with X = 5 and Y = 2

Trochoid = 1

Trochoid = 2

Trochoid = 3.5

Trochoid = 7
Hypocycloid with X = 5 and Y = 2

Trochoid = 1

Trochoid = 2

Trochoid = 2.3

Trochoid = 3

If X and Y are whole numbers, setting the cycles equal to Y will draw a complete curve.

Number of Points
For symmetrically-distributed dots, make sure this is evenly divisible by X.


A cycloid is a roulette curve drawn by rolling a circle along a straight line. Wikipedia and Wolfram have the mathematical specifics. Design-wise, it makes a series of arches or coils.

X Period
The width of each arch. There's actually not a lot of reason to change this from 1, because you can get the same effect by adjusting the Trochoid Parameter and the scale.

Trochoid Parameter
Less than the X Period makes a wavy line, equal to the X Period makes points, greater than the X Period makes loops.

Cycloid with X = 1

Tr = 0.27

Tr = 1

Tr = 2.4

Number of complete arches in the curve. These go along the x-axis in the positive direction (to the right), so you will need to move the trap center if you want the whole curve to be more generally centered. Using non-whole numbers often works well here.

Cycloid with X = 1

Tr = 1, Cycles = 1

Tr = 13, Cycles = 8.6

Number of Points
If this is evenly divisible by the number of cycles, the dots will line up neatly with the arches.

Cycloid with Cycles = 4

No. of Points = 35

No. of Points = 40

General Parameters

Dots plots a series of points tracing out the curve. Lines connects the dots, and can either resemble a smooth curve (using many points) or be used for angular designs (using few points). The Lines method will render more slowly than the Dots method.

Adjusts how the gradient is applied to the dots or lines. It's mostly useful for fine-tuning the colors.

Differences Between Versions

The .ucl includes Center, Rotation, Size, and Aspect Ratio, which should be self-explanatory. Repeat Spacing gives a sort of Gaussian Integer effect. In the plug-in version, these happen in the Trap Position section instead.

In the Color By parameter, the closest, iteration, and angle to trap work like their equivalents in other orbit trap colorings. The only thing that's available in the .ucl that I haven't so far been able to duplicate using the plug-in is the "angle to points" option. Here's what it looks like:

Angle to Trap

Angle to Points
(If anyone out there knows how to send the individual point values from the plug-in to one of the existing angle colorings, please tell me.)


Pretzel Logic

click for parameters
Pretty Beads

click for parameters
Star-Forming Region

click for parameters

Images © 1998-2009 Morgen Bell. This gallery is a division of