From Context Free Art
Contents |
Shape Adjustments:
Geometry, Color, and Time
There are three categories of shape adjustments: geometric, color, and temporal. Anywhere that you see num there can either be a numeric constant or an expression.
Geometry Adjustments
The geometry of shapes is defined by a two dimensional affine transform which is represented internally as a matrix. z is represented by a separate one dimensional affine transform.
| Adjustment | Meaning |
|---|---|
| x x1 | translation along the x-axis by x1 |
| x x1 y1 | translation along the x-axis by x1 and y-axis by y1 |
| x x1 y1 z1 | translation along the x-axis by x1, y-axis by y1, and z-axis by z1 † |
| y y1 | translation along the y-axis by y1 |
| z z1 | translation along the z-axis by z1 † |
| size num or s num | scale in x and y by num |
| size x1 y1 or s x1 y1 | scale in x by x1 and in y by y1 |
| size x1 y1 z1 or s x1 y1 z1 | scale in x by x1, in y by y1, and in z by z1 † |
| rotate num or r num | rotate in the x/y plane num degrees |
| flip num or f num | reflect in x/y plane across a line through the origin at num degrees |
| skew num1 num2 | shear transform in x/y plane num1 degrees from the y axis and num2 degrees from the x axis |
| transform x1 or trans x1 | translate along x-axis by x1 |
| transform x1 y1 or trans x1 y1 | translate along x-axis by x1 and y-axis by y1 |
| transform x1 y1 x2 y2 or trans x1 y1 x2 y2 | transform from unit square at origin to a square with one corner at (x1,y1) and the adjacent corner at (x2,y2) |
| transform x1 y1 x2 y2 x3 y3 or trans x1 y1 x2 y2 x3 y3 | transform from unit square at origin to a parallelogram with one corner at (x1,y1), the adjacent corner at (x2,y2), and the opposite corner at (x3,y3) |
† The z-axis position determines which shape is on top when they overlap. The z-axis position and scale are not stored in the affine transform matrix and do not affect the shape of objects.
This diagram demonstrates the 1 and 2 parameter transform adjustments, which have translation, but no rotation, scaling, or skew.
This diagram demonstrates the 4 parameter transform adjustments, which have translation, rotation, and scaling, but no skew.
This diagram demonstrates the 6 parameter transform adjustments, which have translation, rotation, scaling, and skew.
Color Adjustments
The color of a shape is represented by a point in the HSBA color space (HSB color plus alpha, or opacity). The HSBA coordinates are [0,360) for hue, [0,1] for saturation, [0,1] for brightness, and [0,1] for alpha (opacity).
- hue num
- add num to the drawing hue value, modulo 360
- saturation num
- range [-1,1]. If num<0 then change the drawing saturation num% toward 0. If num>0 then change the drawing saturation num% toward 1.
- brightness num
- range [-1,1]. If num<0 then change the drawing brightness num% toward 0. If num>0 then change the drawing brightness num% toward 1.
- alpha num
- range [-1,1]. If num<0 then change the drawing alpha num% toward 0. If num>0 then change the drawing alpha num% toward 1.
| Abbreviations: 'hue' can be abbreviated as 'h', 'saturation' can be abbreviated as 'sat', 'brightness' can be abbreviated as 'b', and 'alpha' can be abbreviated as 'a'. |
Targeting a Color
There are four more drawing color adjustments for changing the drawing color components with respect to the target color. This is useful if you have a recursive shape rule that adjusts a color component, and you don't want to overshoot a value. For example, the drawing hue is red and you want the drawing hue to trend toward yellow, but never go past yellow to green.
- hue num target
- range [-1,1]. If num<0 then change the drawing hue num% toward the target hue moving clockwise around the color wheel. If num>0 then change the drawing hue num% toward the target hue counter-clockwise around the color wheel.
- saturation num target
- range [-1,1]. If num>0 then change the drawing saturation num% toward the target saturation. If num<0 then change the drawing saturation num% toward 0 or 1, whichever is closer.
- brightness num target
- range [-1,1]. If num>0 then change the drawing brightness num% toward target brightness. If num<0 then change the drawing brightness num% toward 0 or 1, whichever is closer.
- alpha num target
- range [-1,1]. If num>0 then change the drawing alpha num% toward the target alpha. If num<0 then change the drawing alpha num% toward 0 or 1, whichever is closer.
Example: Targeting yellow
For example, the following code applied consistently when drawing shapes, will change the initial hue 0 (red) quickly to yellow (hue 60) but not let it go past yellow towards green:
MyShape [ h 0.99 60 ] // yellow hues
More on targeting colors can be found in the tutorial Targeting a color.
Some common hues
Remember that the initial hue is always 0, i.e red. Some common hues follow:
| Hue number | Primary | Secondary | Tertiary |
|---|---|---|---|
| 0 | Red | ||
| 30 | Orange ("Red-Yellow") | ||
| 60 | Yellow | ||
| 90 | Yellow-Green | ||
| 120 | Green | ||
| 150 | Aqua ("Green-Cyan") | ||
| 180 | Cyan | ||
| 210 | Turquoise ("Cyan-Blue") | ||
| 240 | Blue | ||
| 270 | Violet ("Blue-Magenta") | ||
| 300 | Magenta | ||
| 330 | Reddish Purple ("Magenta-Red") | ||
| 360 (=0) | Red |
Time Adjustment
When rendering an animation the timeline for the animation is divided into frames and each frame is rendered into the movie (or into discrete png files). Each shape has a birth time, where it starts to be drawn; and a death time, where it stops being drawn. This is defined as a pair of one dimensional affine transforms that share the same scale element. The birth time can be -∞, which means that the shape is drawn from the start of the animation until it dies. The death time can be ∞, which means that the shape is drawn from birth until the end of the animation.
| Adjustment | Meaning |
|---|---|
| time num1 num1 | translate birth time by num1 and death time by num2, in current time scale |
| timescale num | multiply current time scale by num |
Basic Shape Adjustments vs. Ordered Shape Adjustments
or what is the difference between [ … ] and [[ … ]]
The simplified manner for handling geometric shape adjustments is to take each adjustment and apply it to create the final shape geometry in a fixed and easy to understand order. In this example:
shape foo {
bar [ x 1 y 2 rot 30 s 2 0.5 skew 10 0 flip 45 ]
}
The bar shape is first translated (1, 2), then it is rotated 30 degrees, then it is scaled (2, 0.5), then it is skewed 10 degrees from the y-axis, and finally it is reflected across a 45 degree line. It doesn't matter if you shuffle the adjustments, they are still applied in this fixed order:
shape foo {
bar [ flip 45 x 1 rot 30 y 2 skew 10 0 s 2 0.5 ] // same as above
}
If you have multiple instances of an adjustment (multiple scales, multiple x-axis translates, etc.) then only the last one is used. The other are dropped with a warning message. This translate/rotate/scale/skew/flip (TRSSF) adjustment order simplifies CFDG design by allowing you to not worry about what order you place the adjustments.
But being able to control the order that the adjustments are applied to create the final shape geometry is also very useful and can lead to some powerful idioms. Context Free/CFDG also has a syntax for specifying a list of shape adjustments where the adjustment order is significant:
shape spike {
SQUARE []
spike [[ x 0.5 s 0.95 x 0.5 ]]
}
Each time the spike rule is drawn it draws a square then draws another spike shifted over by 0.5, shrunk by 0.95, and shifted over again by 0.5, but the second shift is done in the scaled geometry. The end result is a line of shrinking squares that are perfectly abutted edge-to-edge. You can change the scaling from 0.95 to some other value and they will still be perfectly abutted. If you tried to do this in the simple adjustment order:
shape spike {
SQUARE []
spike [ x 0.975 s 0.95 ]
}
you would have to carefully compute the translation whenever you change the scale. Another useful idiom is to put a rotation before a translation and/or scale to work in polar geometry:
startshape flower
shape flower {
// petals
CIRCLE [[ r 30 x 0.5 s 1 0.25 ]]
CIRCLE [[ r 90 x 0.5 s 1 0.25 ]]
CIRCLE [[ r 150 x 0.5 s 1 0.25 ]]
CIRCLE [[ r 210 x 0.5 s 1 0.25 ]]
CIRCLE [[ r 270 x 0.5 s 1 0.25 ]]
CIRCLE [[ r 330 x 0.5 s 1 0.25 ]]
//center
CIRCLE [ s 0.25 b 1 ]
}
These are just two of many possible ways to use ordered shape adjustments.
There is no order dependence on color coordinate changes (hue, brightness, etc.) so color changes are treated the same in both basic and ordered shape adjustment lists.
