L-systems II
From GenerativeArt
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(New page: __FORCETOC__ == A reminder about "The Algorithmic Beauty of Plants" == Essentially all of the information presented here, including illustrations and diagrams, are derived from the classi...) |
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Models intended for 3D modeling add turtle graphic control symbols to extend heading operations to 3D space. All other control symbols remain the same. | Models intended for 3D modeling add turtle graphic control symbols to extend heading operations to 3D space. All other control symbols remain the same. | ||
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| + | Other, and satisfy the equation H x L = U. | ||
| + | Rotations of the turtle are then expressed by the equation | ||
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| + | [H' L' U']=[H L U] R | ||
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| + | where R is a 3 x 3 rotation matrix | ||
| + | Specifically, rotations by angle α about | ||
| + | vectors U, L and H are represented by the matrices: | ||
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| + | | cos α sin α 0 | | ||
| + | R<sub>U</sub>(α) = |-sin α cos α 0 | | ||
| + | | 0 0 1 | | ||
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| + | | cos α -sin α 0 | | ||
| + | R<sub>L</sub>(α) = | 0 0 1 | | ||
| + | | sin α 0 cos α | | ||
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| + | | 1 0 0 | | ||
| + | R<sub>H</sub>(α) = | 0 cos α -sin α | | ||
| + | | 0 sin α cos α | | ||
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| + | The following symbols control turtle orientation in space | ||
| + | + Turn left by angle δ, using rotation matrix R<sub>U</sub> (δ). | ||
| + | - Turn right by angle δ, using rotation matrix R<sub>U</sub>(-δ). | ||
| + | & Pitch down by angle δ, using rotation matrix R<sub>L</sub>(δ). | ||
| + | ^ Pitch up by angle δ, using rotation matrix R<sub>L</sub>(-δ). | ||
| + | \ Roll left by angle δ, using rotation matrix R<sub>H</sub>(δ). | ||
| + | / Roll right by angle δ, using rotation matrix R<sub>H</sub>(-δ). | ||
| + | | Turn around, using rotation matrix RU(180°). | ||
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| + | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www-viz.tamu.edu/courses/viza658/wiki/lsys2/01.jpg|caption=}} | ||
