four.M4

.ortho `(l, r, b, t, n, f)`

local lib = { type = 'four.M4' } lib.__index = lib four.M4 = lib setmetatable(lib, { __call = function(lib, ...) return lib.M4(...) end })

local V3 = four.V3

-- h2. Constructor and accessors

e21, e22, e23, e24, -- row 2 e31, e32, e33, e34, -- row 3 e41, e42, e43, e44) -- row 4 local m if e12 then m = { e11, e21, e31, e41, -- col 1 e12, e22, e32, e42, -- col 2 e13, e23, e33, e43, -- col 3 e14, e24, e34, e44 } -- col 4 else if e11.type == "bt.Transform" then m = e11:toM4() else assert(false, string.format("Cannot convert %s to %s", e11.type, lib.type)) end end setmetatable(m, lib) return m end

local M4 = lib.M4

local b = (i - 1) * 4 return four.V4(m[b + 1], m[b + 2], m[b + 3], m[b + 4]) end

-- h2. Converters

return M4(r1[1], r1[2], r1[3], r1[4], r2[1], r2[2], r2[3], r2[4], r3[1], r3[2], r3[3], r3[4], r4[1], r4[2], r4[3], r4[4]) end

return M4(c1[1], c2[1], c3[1], c4[1], c1[2], c2[2], c3[2], c4[2], c1[3], c2[3], c3[3], c4[3], c1[4], c2[4], c3[4], c4[4]) end

return string.format("(% g % g % g % g )\n(% g % g % g % g )\n(% g % g % g % g )\n(% g % g % g % g )", m[1], m[5], m[ 9], m[13], m[2], m[6], m[10], m[14], m[3], m[7], m[11], m[15], m[4], m[8], m[12], m[16]) end

-- h2. Constants

return M4(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) end

return M4(1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1) end

-- h2. Functions

local r = {} for i in 1, 16 do r[i] = -a[i] end return r end

local r = {} for i in 1, 16 do r[i] = a[i] + b[i] end return r end

local r = {} for i in 1, 16 do r[i] = a[i] - b[i] end return r end

return M4 (a[1] b[1] + a[5] b[2] + a[9] b[3] + a[13] b[4], -- row 1 a[1] b[5] + a[5] b[6] + a[9] b[7] + a[13] b[8], a[1] b[9] + a[5] b[10] + a[9] b[11] + a[13] b[12], a[1] b[13] + a[5] b[14] + a[9] b[15] + a[13] b[16], a[2] b[1] + a[6] b[2] + a[10] b[3] + a[14] b[4], -- row 2 a[2] b[5] + a[6] b[6] + a[10] b[7] + a[14] b[8], a[2] b[9] + a[6] b[10] + a[10] b[11] + a[14] b[12], a[2] b[13] + a[6] b[14] + a[10] b[15] + a[14] b[16], a[3] b[1] + a[7] b[2] + a[11] b[3] + a[15] b[4], -- row 3 a[3] b[5] + a[7] b[6] + a[11] b[7] + a[15] b[8], a[3] b[9] + a[7] b[10] + a[11] b[11] + a[15] b[12], a[3] b[13] + a[7] b[14] + a[11] b[15] + a[15] b[16], a[4] b[1] + a[8] b[2] + a[12] b[3] + a[16] b[4], -- row 4 a[4] b[5] + a[8] b[6] + a[12] b[7] + a[16] b[8], a[4] b[9] + a[8] b[10] + a[12] b[11] + a[16] b[12], a[4] b[13] + a[8] b[14] + a[12] b[15] + a[16] b[16]) end

local r = {} for i in 1, 16 do r[i] = a[i] * b[i] end return r end

local r = {} for i in 1, 16 do r[i] = a[i] / b[i] end return r end

local r = {} for i in 1, 16 do r[i] = s * a[i] end return r end

return M4(a[ 1], a[ 2], a[ 3], a[ 4], a[ 5], a[ 6], a[ 7], a[ 8], a[ 9], a[10], a[11], a[12], a[13], a[14], a[15], a[16]) end

-- N.B. mij symbols for minors are written as zero-based row/column local d1 = (a[11] a[16]) - (a[12] a[15]) -- second minor local d2 = (a[7] a[16]) - (a[8] a[15]) local d3 = (a[7] a[12]) - (a[8] a[11]) local m00 = (a[6] d1) - (a[10] d2) + (a[14] d3) -- minor local m10 = (a[5] d1) - (a[9] d2) + (a[13] d3) local d4 = (a[9] a[14]) - (a[10] a[13]) local d5 = (a[5] a[14]) - (a[6] a[13]) local d6 = (a[5] a[10]) - (a[6] a[9]) local m20 = (a[8] d4) - (a[12] d5) + (a[16] d6) local m30 = (a[7] d4) - (a[11] d5) + (a[15] d6) local d7 = (a[3] a[16]) - (a[4] a[15]) local d8 = (a[3] a[12]) - (a[4] a[11]) local m01 = (a[2] d1) - (a[10] d7) + (a[14] d8) local m11 = (a[1] d1) - (a[9] d7) + (a[13] d8) local d9 = (a[1] a[14]) - (a[2] a[13]) local d10 = (a[1] a[10]) - (a[2] a[9]) local m21 = (a[4] d4) - (a[12] d9) + (a[16] d10) local m31 = (a[3] d4) - (a[11] d9) + (a[15] d10) local d11 = (a[3] a[8]) - (a[4] a[7]) local m02 = (a[2] d2) - (a[6] d7) + (a[14] d11) local m12 = (a[1] d2) - (a[5] d7) + (a[13] d11) local d12 = (a[1] a[6]) - (a[2] a[5]) local m22 = (a[4] d5) - (a[8] d9) + (a[16] d12) local m32 =(a[3] d5) - (a[7] d9) + (a[15] d12) local m03 = (a[2] d3) - (a[6] d8) + (a[10] d11) local m13 = (a[1] d3) - (a[5] d8) + (a[9] d11) local m23 = (a[4] d6) - (a[8] d10) + (a[12] d12) local m33 = (a[3] d6) - (a[7] d10) + (a[11] d12) local det = (a[1] m00) - (a[2] m10) + (a[3] m20) - (a[4] m30) return M4 ( m00 / det, -m10 / det, m20 / det, -m30 / det, -m01 / det, m11 / det, -m21 / det, m31 / det, m02 / det, -m12 / det, m22 / det, -m32 / det, -m03 / det, m13 / det, -m23 / det, m33 / det) end

-- h2. 3D space transforms

-- @move(d)@ is a matrix that translates 3D space by the vector @d@. return M4(1, 0, 0, d[1], 0, 1, 0, d[2], 0, 0, 1, d[3], 0, 0, 0, 1) end

-- @getMove(m)@ is a vector with the translation component of @m@.

-- @rotMap(u, v)@ is a matrix that rotates 3D space to map the unit -- vector @u@ on the unit vector @v@. local n = V3.cross(u, v) local e = V3.dot(u, v) local h = 1 / (1 + e) local x = n[1] local y = n[2] local z = n[3] local xy = x y local xz = x z local yz = y z return M4(e + h x x, h xy - z, h xz + y, 0, h xy + z, e + h y y, h yz - x, 0, h xz - y, h yz + x, e + h z * z, 0, 0, 0, 0, 1) end

-- @rotAxis(v, theta)@ is a matrix that rotates 3D space by @theta@ around -- the unit vector @v@. local xy = u[1] u[2] local xz = u[1] u[3] local yz = u[2] u[3] local c = math.cos(theta) local one_c = 1. - c local s = math.sin(theta) return M4(u[1] u[1] one_c + c, -- row 1 xy one_c - u[3] s, xz one_c + u[2] s, 0, xy one_c + u[3] s, -- row 2 u[2] u[2] one_c + c, yz one_c - u[1] s, 0, xz one_c - u[2] s, -- row 3 yz one_c + u[1] s, u[3] u[3] * one_c + c, 0, 0, 0, 0, 1) -- row 4 end

-- rotZYX(r) is a matrix that rotates 3D space first by @V3.x(r)@ around -- the x-axis, then by @V3.y(r)@ around the y-axis and finally by @V3.y(r)@ -- around the z-axis local cz = math.cos(r[3]) local sz = math.sin(r[3]) local cy = math.cos(r[2]) local sy = math.sin(r[2]) local cx = math.cos(r[1]) local sx = math.sin(r[1]) return M4(cy cz, sy sx cz - cx sz, sy cx cz + sx sz, 0, cy sz, sy sx sz + cx cz, sy cx sz - sx cz, 0, -sy, cy sx, cy cx, 0, 0, 0, 0, 1) end

-- @scale(s)@ is a matrix that scales 3D space in the x, y, and z dimensions -- according to @s@. return M4(s[1], 0 , 0 , 0, 0 , s[2], 0 , 0, 0 , 0 , s[3], 0, 0 , 0 , 0 , 1) end

-- @getScale(s)@ is a vector with the scale factors preformed by @m@ in -- each dimension. return V3(math.sqrt(m[1] m[1] + m[2] m[2] + m[3] m[3]), math.sqrt(m[5] m[5] + m[6] m[6] + m[7] m[7]), math.sqrt(m[9] m[9] + m[10] m[10] + m[11] * m[11])) end

-- @rigid(d, axis, theta)@ is the rigid body transform of 3D space -- that rotates by @axis@, @angle@ and then translate by @d@. local r = lib.rotAxis(axis, theta) r[13] = d[1]; -- set translation in col 4 r[14] = d[2]; r[15] = d[3]; return r end

-- rigidq(d, q) is the rigid body transform of 3D space -- that rotates by the quaternion q and then translates by d. local r =§§44(q) r[13] = d[1]; -- set translation in col 4 r[14] = d[2]; r[15] = d[3]; return r end

-- @rigidScale(d, axis, theta, s)@ is like @rigid(d, axis, theta)@ but -- it starts by scaling according to @s@. local r = lib.rigid(d, axis, theta) r[1] = r[1] s[1] -- scale col 1 r[2] = r[2] s[1] r[3] = r[3] s[1] r[5] = r[5] s[2] -- scale col 2 r[6] = r[6] s[2] r[7] = r[7] s[2] r[9] = r[9] s[3] -- scale col 3 r[10] = r[10] s[3] r[11] = r[11] * s[3] return r end

-- @rigidqScale(d, q, s)@ is like @rigid(d, axis, theta)@ but -- it starts by scaling according to @scale@. local r = lib.rigidq(d, q) r[1] = r[1] s[1] -- scale col 1 r[2] = r[2] s[1] r[3] = r[3] s[1] r[5] = r[5] s[2] -- scale col 2 r[6] = r[6] s[2] r[7] = r[7] s[2] r[9] = r[9] s[3] -- scale col 3 r[10] = r[10] s[3] r[11] = r[11] * s[3] return r end

-- h2. Projection

--[[-- @ortho(l, r, b, t, n, f)@ maps the axis aligned box with corners @(l, b, -n)@ and @(r, t, -f)@ to the axis aligned cube with corners (-1, -1, -1) and @(1, 1, 1)@.

.persp `(l, r, b, t, n, f)`

@persp(l, r, b, t, n, f)@ maps the frustum with top of the underlying pyramid at the origin, near plane rectangle corners @(l, b, -n)@, @(r, t, -n)@ and far plane at @-far@ to the axis aligned cube with corners @(-1, -1, -1)@ and @(1, 1, 1)@.

.scale4D `(s)`

scales 4D space in the x, y, z and w dimensions according to @s@

.map `(f, m)`

TODO MISSING DOCUMENTATION

.mapi `(f, m)`

TODO MISSING DOCUMENTATION

.fold `(f, acc, m)`

TODO MISSING DOCUMENTATION

.foldi `(f, acc, m)`

TODO MISSING DOCUMENTATION

.iter `(f, m)`

TODO MISSING DOCUMENTATION

.iter `(f, m)`

TODO MISSING DOCUMENTATION

.forAll `(p, m)`

TODO MISSING DOCUMENTATION

.exists `(p, v)`

TODO MISSING DOCUMENTATION

.equal `(a, b)`

TODO MISSING DOCUMENTATION

.equalF `(f, u, v)`

TODO MISSING DOCUMENTATION

.compare `(u, v)`

TODO MISSING DOCUMENTATION

.compareF `(f, u, v)`

TODO MISSING DOCUMENTATION

.__unm `= lib.neg`

h2. Operators

.__add `= lib.add`

.__sub `= lib.sub`

.__mul `= lib.mul`

.__tostring `= lib.tostring`

TODO

M4TODO MISSING DOCUMENTATION

Lubyk documentation

four.M4

.ortho (l, r, b, t, n, f)

.persp (l, r, b, t, n, f)

.scale4D (s)

.map (f, m)

.mapi (f, m)

.fold (f, acc, m)

.foldi (f, acc, m)

.iter (f, m)

.iter (f, m)

.forAll (p, m)

.exists (p, v)

.equal (a, b)

.equalF (f, u, v)

.compare (u, v)

.compareF (f, u, v)

.__unm = lib.neg

.__add = lib.add

.__sub = lib.sub

.__mul = lib.mul

.__tostring = lib.tostring

TODO

.ortho `(l, r, b, t, n, f)`

.persp `(l, r, b, t, n, f)`

.scale4D `(s)`

.map `(f, m)`

.mapi `(f, m)`

.fold `(f, acc, m)`

.foldi `(f, acc, m)`

.iter `(f, m)`

.iter `(f, m)`

.forAll `(p, m)`

.exists `(p, v)`

.equal `(a, b)`

.equalF `(f, u, v)`

.compare `(u, v)`

.compareF `(f, u, v)`

.__unm `= lib.neg`

.__add `= lib.add`

.__sub `= lib.sub`

.__mul `= lib.mul`

.__tostring `= lib.tostring`