Forming the Gyroid unit cell from 8 of the 12 patch boundary cells:
Table 2 of the Gyroid paper of Gandy & Klinowski can be understood in terms of 4X4 homogeneous transformation matrices, and also in terms of MoI mirror and move commands acting upon boundary patch surface O1 which consists of 12 basic patches. Surface O1 is represented by all of the points (xprime, yprime, zprime) on its surface. Dropping the prime from the name, one generic point (x, y, z, 1) written in homogeneous coordinates as a column vector, {(x, y, z, 1)T, transposed}, is sufficient to create all of the matrices and corresponding Mirrors, (reflections), and Moves (translations), to build the complete unit cell of a Gyroid.
The move distance unit 'eta' is 1, and the distances moved are 0, 1, -1, or 2 units, along the various axes. The final cube is 2 units on a side.
O1 is located in quadrant 1, with the cube enclosing it aligned with the 3 positive xyz-axes, with lower left corner at the origin.
O2 corresponds to [(-x+2), (-y+1), z, 1]
The corresponding composite matrix, cM01, is: (remember zero indexing.)
row 0: -1 0 0 2
row 1: 0 -1 0 1
row 2: 0 0 1 0
row 3: 0 0 0 1
In Top View:
In row0, the "-1" (and position) denotes mirror in yz plane. For this mirror, select the origin for the 1st point, and select another point, on the y axis, for the 2nd point. The "2" (and position) denotes the Move in the +x direction, of 2 units.
Still in Top View:
In row1, the "-1" (and position) denotes mirror in the xz plane. For this mirror, select the origin for the 1st point, and select another point, on the x axis, for the 2nd point. The "1" (and position) denotes the Move in the +y direction, of 1 unit.
The composite matrix cM01 can be broken down into 2 matrices.
O2 = cM01 * O1 = moveM * mirror_yzM * mirror_xzM * O1.
The correspodiing MoI commands applied to surface O1 are:
mirror_xz, then to result: mirror_yz, then to result: move (2, 1, 0) units.
In this case, the three operations can be applied in any order (???)
(Note that for the Schwarz D surface, (some of) the matrices are rotation, reflection, then move, where order DOES make a difference.)
With practice, certain key row column locations in the 4X4 matrices are recognized for the mirror, and move results. Schwarz D rotations are a bit harder...
Next up is to see if the nodeeditor Mirror will work properly, or not.
Then build a nodeeditor program with Frames, macros for various mirrors, and moves, etc., for G, D, and P surfaces.
Also to do, decompose the 12 steps of Table 1 for the Gyroid fundamental patch, into matrices and nodeeditor program.
For rotations, with a D boundary patch aligned with xyz-axes, the rotations are PI/2, so the cosine matrix entries are zero, and the sine entries are +1. And reflections are xy, yz, and/or xz planes.
- Brian
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