Using Slope/Lighting Formulas

This tutorial assumes you are already familiar with the material in  The Basics of UltraFractal 5 and Its Use of Classes,  Changing and Exploring Plug-ins, Transforms as Plug-ins, Application of Texture   and Advanced Topics  in Applying Textures. This tutorial will be using most of the plug-ins covered in the previous tutorials. For this tutorial we will use a formula class called Slope Enhanced, which is in reb5.ufm. It can also be accessed from Object formulas in reb5.ufm and from Generic Formula in standard.ufm through the fractal formula plug-in.. It has a fractal formula plug-in which can use any of the fractal formulas in the libraries. I  advise against "Slope" in standard.ulb as it has no plugins for height value and so much of the flexibility of of "slope" is lost. Similarly, I advise against using the coloring formula "Lighting" in standard.ulb as it has no options for managing colors and textures. Instead use "Lighting with Textures" in reb5.ucl. Lighting with Textures can also be accessed from Generic Coloring (Direct) in standard.ucl through the coloring algorithm plug-in.

So what is a slope formula, and how does it work? Essentially a slope formula is a poor man's raytracer.  You can think of a fractal as a 3D surface where the height of the fractal is equal to its iteration value at bailout. A raytracer colors an object based upon the angle between an incident light ray and a normal to the surface. What a slope formula does is calculate normals to the fractal surface, which are then passed to the coloring formula to determine the actual coloring. As slope formulas have evolved it was a simple extension to modify the fractal surface using a smoothing algorithm such as exponential smoothing or to base the normal calculations on some other aspect of the iterated value. These are called "height values", and slope formulas usually have multiple choices for height values.

Before UltraFractal 5 special code had to be written for each slope formula, so they were usually given names such as Slope Julia to differentiate them from the regular formulas. With UltraFractal 5 all the regular formula objects can be used, as the special code to calculate the normals is now in a hidden, linked piece of code.

Since a slope shape is a handy plug-in that can be used to modify the iterated value, they can also be used as height values, much as they are used for trap calculations in orbit trap type formulas. As with any trap shape plug-in the options of TrapShape Block and TrapShape Merge are also useable, along with the everything else in those wrappers. Slope with Shapes is a plug-in for Generic Formula which can be found in Standard.ufm. The snapshot below shows the basic parameter list:

Alternatively, the slope formulas can be accessed in two additional ways.

1. Directly using the Slope Enhanced ucl in reb5.ucl  The upr examples below use thr Slope Enhanced ucl.

2. Using the Object Formula Switch Lite in reb5.ucl. This method uses the Switch Slope plug-in in reb.ulb. With this plug-in the user can switch between the Mandelbrot and Julia forms in virtually the same manner as for formulas in UF4 using the "switch" utility of UF5.

The first image below shows the use of the Slope Enhanced ucl, and the second set of images is for use of Object Formula Switch Lite.

 The two images illustrate Switch Slope in its Mandelbrot and Julia Forms

On the coloring side the lighting formula is a direct coloring formula so that the coloring can take advantage of Color Trap plugins and other coloring special effects. The lighting formula can be accessed directly using the Lighting with Textures ucl in reb5.ucl, or by using the Lighting with Textures Direct plug-in in reb.ulb with Generic Coloring (Direct) in standard.ucl. The two images below illustrate parameter list for the two methods:

The first example is a simple case using smoothed iteration for the height values and a monochrome gradient with the default settings. This is not a particularly interesting image, but it illustrates some of the basic principles for using Slope/Lighting. The user should experiment with the effects of Light Rotation, Light Elevation and Ambient Light.

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More interesting images can be created using different trap shapes. In the following upr and image a Flower trap shape was used for the height value. A color pattern using a color trap has also been added. Experience has shown that adding some additional coloring in the Lighting formula works best if the "Height Transfer" function is set to log(). Click the arrow for height transfer (on the formula tab) so you can see the parameters. On the outside tab notice that the default "Colors" plug-in has been replaced with a color trap plugin. The user should play with the color merge parameters to see how they affect the image. Internally color trap coloring and  slope/lighting coloring behave as separate layers and can be combined in all the ways that layers can be combined on the outside tab.

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The image for the Tutorial_13 upr should look familiar. The image for the Transform_2 upr in the tutorial on transforms uses the point trap shape and the radial waves transform. This combination is also used for the height value for Tutorial_13. Different coloring and the application of texture has changed the appearance somewhat, but it is still the same basic shape as was seen in the Transform_2 upr. To do it proper justice It needs to be rendered with custom anti-aliasing: Threshold = 0, Depth = 1, Subdivisions 4 (2x2) and at a resolution of at least 1024 x 768. The transfer function, as previously, has been changed to log(). Notice that the upr has two layers, and the bottom layer is a simple black layer. If the user turns the black layer off large area of transparency can be seen. This is created by the "soften edges" parameter in Slope Lighting. What this parameter does is increase the transparency as the height value approaches zero. Effectively, this allows the height value passed from the Slope formula to act as a trap, with coloring outside the trap being determined by the color of the layer below. Unclick the soften edges parameter and notice the effect on the image.

Texture has been applied using the TrapShape Block technique that was covered in previous tutorials. In this case trap shape block was used so that Trap Transfer Bias, which will be covered in more detail in the next tutorial, can also be used. "Cut at Zero" is checked for Trap Transfer Bias. This has the effect of setting all negative texture values to zero. Uncheck Cut at Zero and observe the effect. Notice that the distance bias value in Trap Transfer Bias has been set to -0.4. What this does is reset the cut value to -0.4. Try changing the value of the distance bias to see the effects. When Cut at Zero is unchecked, an new parameter is visible, called "No Negative Values". This is also in Trap Transfer, but has a very different effect on texture. Instead of setting negative values to zero, it takes the absolute value of the negative values, producing a very different effect on the texture. Try checking No Negative Values to see the effect on the texture.

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The upr Tutorial_14 and the corresponding image have some elements in common with the Tutorial_8 image in Advanced Topics in Applying Textures. The texture mapping is a little different, but the repeating star texture should be visible. It was created using the same TrapShape Block combination as was used in the Tutorial_8 upr, which the user should be able to see by looking at the bottom layer. The upper layer demonstrates that, although Slope formulas are intended for use with Lighting formulas, they will still work with other standard coloring formulas. In this case Exponential Smoothing was used to color the upper layer. This helps facilitate using the fractal formula with any coloring method as could be done with UltraFractal 4.

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}

The last upr and image is an exercise in relatively subtle coloring and texturing. To do it proper justice It needs to be rendered with custom anti-aliasing: Threshold = 0, Depth = 1, Subdivisions 4 (2x2) and at a resolution of at least 1024 x 768.

The bottom layer is fairly simple. In the Slope formula the height function is sqr() and the height pre-scale is 2.0. In the Lighting formula  the texture plug-in is Modified Perlin Textures with convolution turned on. The texture amount is high, being set to 1.5. One major point of difference from the previous uprs in this tutorial is that "Use Fixed Iteration" is checked and "Iter number" is set to 5. With just the bottom layer turned on, try checking and unchecking Use Fixed Iteration to see the effect on the texture. Also try varying the Iter number.

For the upper layer, in the Slope formula, the height function is cuberoot(). Texture and coloring is managed in the Lighting formula with the use of the Soften edges parameter and the application of the Worley Colors plug-in. With the plug-in in place new parameters for managing color are visible under "color merge" near the bottom of the parameters list. The Worley Colors and the colors from Slope/Lighting are managed internally as two separate layer which are merged using the "color merge" parameters.

Some transparency has been set on the internal Slope/Lighting layer using Soften edges. Further color management is done under the color merge section with the merge mode, merge order and merge opacity. Also note "Make image transparent by"  settings. These apply to the colors layer created by the Worley Colors plug-in. For this upr "Luminance value" is checked. Try unchecking it and you will see the other options. Notice the change in the image. The user should experiment with the Soften edges settings and the color merge settings to see the effects on the image.

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}

The next two uprs involve effects that do not use texture shape (a.k.a. trap shape). Instead they use the "colors" plug-in under Images and Coloring. The first one uses Worley Colors. There is an option in Lighting with Textures called "Tile method" which has options of "none", "fixed iteration" and "cabs(z)" Each of them affects how the coloring (and also texture) is mapped onto the image. The image below uses the default value of "none". This option performs very poorly for some height values (a.k.a. trap shapes) used in the slope formula. The other tiling options can be of use. As an exercise the user should locate "pattern size" within Worley Colors. Decrease the value from 0.1 to 0.001. Now go back to the Tile Method and change it to "fixed iteration". The value for "Iter number" should be 1. Try increasing the "Iter number" in steps up to a value of 10 and observe the effects. Now change the pattern size to 0.01 and the Tile method to "cabs(z). and the "cabs limit" to 1.0. Observe the effect.  Now go back to the original upr and vary the "Fuzz power" to see the effects.

ColorTrapShapeBlock2 {
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}

The next upr and image use the ColorTrapShape Block wrapper. This wrapper includes slots for Trap Position, Trap Shape and Null Transfer. In addition it has a special coloring method that allows the user to use either the external gradient for coloring or an internal gradient. There are plug-in options for the internal coloring to select different sets of colors to use. The user can also change any of the colors to design their own internal gradient. For this example we use Monnier's S.F.B.M. II Texture. This is a trap shape texture, and is different from the ColorTrap plug-in that has a similar name. We could have used the Monnier ColorTrap plug-in instead of our current choice, and except for the actual coloring the results would be the same. ColorTrapShape Block has a much more flexible coloring method, and so it gives superior results. It also works with any trap shape. This dramatically expands the coloring options.

ColorTrapShapeBlock {
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