/** This contains routines to derive equations for isometric projection.

    See: https://en.wikipedia.org/wiki/Isometric_projection
*/

use Matrix.frink

symbolicMode[true]
t1 = new Matrix[[[sqrt[3], 0, -sqrt[3]],
                 [1,       2,     1   ],
                 [sqrt[2], -sqrt[2], sqrt[2]]]]

c = new Matrix[([ax, ay, az]).transpose[]]

d = t1.multiply[c].multiplyByScalar[1/sqrt[6]]

println[d.formatMatrix[]]

bt = new Matrix[[[1,0,0], [0,1,0], [0,0,0]]]

f = bt.multiply[d]
println[f.formatMatrix[]]

// All of this comes out to:

// bx = sqrt[2] (ax - az)
// by = sqrt[1/6] (ax + 2 ay + az)
// bz = sqrt[3] (ax - ay + az)   
//     (There really isnt' a z but if you want to color code it)

// The minima and maxima are:
// min bx = sqrt[2] (axmin - azmax)
// max bx = sqrt[2] (axmax - azmin)
// min by = sqrt[1/6] (axmin + 2 aymin + azmin)
// max by = sqrt[1/6] (axmax + 2 aymax + azmax)
// min bz = sqrt[3] (axmin - aymax + azmin)
// max bz = sqrt[3] (axmax - aymin + azmax)

// Or, inverted,
// ay = -ax + bx/(2*Sqrt[2]) + Sqrt[3/2]*by
// az = ax - bx/Sqrt[2]