// Your friendly neighbour https://en.wikipedia.org/wiki/Dihedral_group
//
// This file implements the dihedral group of order 16, also called
// of degree 8. That's why its called groupD8.
import { Matrix } from './Matrix';
/*
* Transform matrix for operation n is:
* | ux | vx |
* | uy | vy |
*/
const ux = [1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1];
const uy = [0, 1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1];
const vx = [0, -1, -1, -1, 0, 1, 1, 1, 0, 1, 1, 1, 0, -1, -1, -1];
const vy = [1, 1, 0, -1, -1, -1, 0, 1, -1, -1, 0, 1, 1, 1, 0, -1];
/**
* Cayley Table
* for the composition of each rotation in the dihederal group D8.
* @type {number[][]}
* @private
*/
const rotationCayley: number[][] = [];
/**
* Matrices for each `GD8Symmetry` rotation.
* @type {PIXI.Matrix[]}
* @private
*/
const rotationMatrices: Matrix[] = [];
/*
* Alias for Math.sign.
*/
const signum = Math.sign;
/*
* Initializes `rotationCayley` and `rotationMatrices`. It is called
* only once below.
*/
function init(): void
{
for (let i = 0; i < 16; i++)
{
const row: number[] = [];
rotationCayley.push(row);
for (let j = 0; j < 16; j++)
{
/* Multiplies rotation matrices i and j. */
const _ux = signum((ux[i] * ux[j]) + (vx[i] * uy[j]));
const _uy = signum((uy[i] * ux[j]) + (vy[i] * uy[j]));
const _vx = signum((ux[i] * vx[j]) + (vx[i] * vy[j]));
const _vy = signum((uy[i] * vx[j]) + (vy[i] * vy[j]));
/* Finds rotation matrix matching the product and pushes it. */
for (let k = 0; k < 16; k++)
{
if (ux[k] === _ux && uy[k] === _uy
&& vx[k] === _vx && vy[k] === _vy)
{
row.push(k);
break;
}
}
}
}
for (let i = 0; i < 16; i++)
{
const mat = new Matrix();
mat.set(ux[i], uy[i], vx[i], vy[i], 0, 0);
rotationMatrices.push(mat);
}
}
init();
type GD8Symmetry = number;
/**
* @memberof PIXI
* @typedef {number} GD8Symmetry
* @see PIXI.groupD8
*/
/**
* Implements the dihedral group D8, which is similar to
* group D4;
* D8 is the same but with diagonals, and it is used for texture
* rotations.
*
* The directions the U- and V- axes after rotation
* of an angle of `a: GD8Constant` are the vectors `(uX(a), uY(a))`
* and `(vX(a), vY(a))`. These aren't necessarily unit vectors.
*
* **Origin:**<br>
* This is the small part of gameofbombs.com portal system. It works.
* @see PIXI.groupD8.E
* @see PIXI.groupD8.SE
* @see PIXI.groupD8.S
* @see PIXI.groupD8.SW
* @see PIXI.groupD8.W
* @see PIXI.groupD8.NW
* @see PIXI.groupD8.N
* @see PIXI.groupD8.NE
* @author Ivan @ivanpopelyshev
* @namespace PIXI.groupD8
* @memberof PIXI
*/
export const groupD8 = {
/**
* | Rotation | Direction |
* |----------|-----------|
* | 0° | East |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
E: 0,
/**
* | Rotation | Direction |
* |----------|-----------|
* | 45°↻ | Southeast |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
SE: 1,
/**
* | Rotation | Direction |
* |----------|-----------|
* | 90°↻ | South |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
S: 2,
/**
* | Rotation | Direction |
* |----------|-----------|
* | 135°↻ | Southwest |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
SW: 3,
/**
* | Rotation | Direction |
* |----------|-----------|
* | 180° | West |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
W: 4,
/**
* | Rotation | Direction |
* |-------------|--------------|
* | -135°/225°↻ | Northwest |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
NW: 5,
/**
* | Rotation | Direction |
* |-------------|--------------|
* | -90°/270°↻ | North |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
N: 6,
/**
* | Rotation | Direction |
* |-------------|--------------|
* | -45°/315°↻ | Northeast |
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
NE: 7,
/**
* Reflection about Y-axis.
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
MIRROR_VERTICAL: 8,
/**
* Reflection about the main diagonal.
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
MAIN_DIAGONAL: 10,
/**
* Reflection about X-axis.
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
MIRROR_HORIZONTAL: 12,
/**
* Reflection about reverse diagonal.
* @memberof PIXI.groupD8
* @constant {PIXI.GD8Symmetry}
*/
REVERSE_DIAGONAL: 14,
/**
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} ind - sprite rotation angle.
* @returns {PIXI.GD8Symmetry} The X-component of the U-axis
* after rotating the axes.
*/
uX: (ind: GD8Symmetry): GD8Symmetry => ux[ind],
/**
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} ind - sprite rotation angle.
* @returns {PIXI.GD8Symmetry} The Y-component of the U-axis
* after rotating the axes.
*/
uY: (ind: GD8Symmetry): GD8Symmetry => uy[ind],
/**
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} ind - sprite rotation angle.
* @returns {PIXI.GD8Symmetry} The X-component of the V-axis
* after rotating the axes.
*/
vX: (ind: GD8Symmetry): GD8Symmetry => vx[ind],
/**
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} ind - sprite rotation angle.
* @returns {PIXI.GD8Symmetry} The Y-component of the V-axis
* after rotating the axes.
*/
vY: (ind: GD8Symmetry): GD8Symmetry => vy[ind],
/**
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} rotation - symmetry whose opposite
* is needed. Only rotations have opposite symmetries while
* reflections don't.
* @returns {PIXI.GD8Symmetry} The opposite symmetry of `rotation`
*/
inv: (rotation: GD8Symmetry): GD8Symmetry =>
{
if (rotation & 8)// true only if between 8 & 15 (reflections)
{
return rotation & 15;// or rotation % 16
}
return (-rotation) & 7;// or (8 - rotation) % 8
},
/**
* Composes the two D8 operations.
*
* Taking `^` as reflection:
*
* | | E=0 | S=2 | W=4 | N=6 | E^=8 | S^=10 | W^=12 | N^=14 |
* |-------|-----|-----|-----|-----|------|-------|-------|-------|
* | E=0 | E | S | W | N | E^ | S^ | W^ | N^ |
* | S=2 | S | W | N | E | S^ | W^ | N^ | E^ |
* | W=4 | W | N | E | S | W^ | N^ | E^ | S^ |
* | N=6 | N | E | S | W | N^ | E^ | S^ | W^ |
* | E^=8 | E^ | N^ | W^ | S^ | E | N | W | S |
* | S^=10 | S^ | E^ | N^ | W^ | S | E | N | W |
* | W^=12 | W^ | S^ | E^ | N^ | W | S | E | N |
* | N^=14 | N^ | W^ | S^ | E^ | N | W | S | E |
*
* This is a Cayley table
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} rotationSecond - Second operation, which
* is the row in the above cayley table.
* @param {PIXI.GD8Symmetry} rotationFirst - First operation, which
* is the column in the above cayley table.
* @returns {PIXI.GD8Symmetry} Composed operation
*/
add: (rotationSecond: GD8Symmetry, rotationFirst: GD8Symmetry): GD8Symmetry => (
rotationCayley[rotationSecond][rotationFirst]
),
/**
* Reverse of `add`.
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} rotationSecond - Second operation
* @param {PIXI.GD8Symmetry} rotationFirst - First operation
* @returns {PIXI.GD8Symmetry} Result
*/
sub: (rotationSecond: GD8Symmetry, rotationFirst: GD8Symmetry): GD8Symmetry => (
rotationCayley[rotationSecond][groupD8.inv(rotationFirst)]
),
/**
* Adds 180 degrees to rotation, which is a commutative
* operation.
* @memberof PIXI.groupD8
* @param {number} rotation - The number to rotate.
* @returns {number} Rotated number
*/
rotate180: (rotation: number): number => rotation ^ 4,
/**
* Checks if the rotation angle is vertical, i.e. south
* or north. It doesn't work for reflections.
* @memberof PIXI.groupD8
* @param {PIXI.GD8Symmetry} rotation - The number to check.
* @returns {boolean} Whether or not the direction is vertical
*/
isVertical: (rotation: GD8Symmetry): boolean => (rotation & 3) === 2, // rotation % 4 === 2
/**
* Approximates the vector `V(dx,dy)` into one of the
* eight directions provided by `groupD8`.
* @memberof PIXI.groupD8
* @param {number} dx - X-component of the vector
* @param {number} dy - Y-component of the vector
* @returns {PIXI.GD8Symmetry} Approximation of the vector into
* one of the eight symmetries.
*/
byDirection: (dx: number, dy: number): GD8Symmetry =>
{
if (Math.abs(dx) * 2 <= Math.abs(dy))
{
if (dy >= 0)
{
return groupD8.S;
}
return groupD8.N;
}
else if (Math.abs(dy) * 2 <= Math.abs(dx))
{
if (dx > 0)
{
return groupD8.E;
}
return groupD8.W;
}
else if (dy > 0)
{
if (dx > 0)
{
return groupD8.SE;
}
return groupD8.SW;
}
else if (dx > 0)
{
return groupD8.NE;
}
return groupD8.NW;
},
/**
* Helps sprite to compensate texture packer rotation.
* @memberof PIXI.groupD8
* @param {PIXI.Matrix} matrix - sprite world matrix
* @param {PIXI.GD8Symmetry} rotation - The rotation factor to use.
* @param {number} tx - sprite anchoring
* @param {number} ty - sprite anchoring
*/
matrixAppendRotationInv: (matrix: Matrix, rotation: GD8Symmetry, tx = 0, ty = 0): void =>
{
// Packer used "rotation", we use "inv(rotation)"
const mat: Matrix = rotationMatrices[groupD8.inv(rotation)];
mat.tx = tx;
mat.ty = ty;
matrix.append(mat);
},
};