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@loopspace loopspace/Codea Tube
Created Feb 26, 2018

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-- BezierTubes
function setup()
assert(craft, "Please include Craft as a dependency")
assert(OrbitViewer, "Please include Cameras (not Camera) as a dependency")
scene = craft.scene()
scene.camera:add(OrbitViewer)
local track = scene:entity()
local p = f(0)
local np,nt,nq,ax,ang
local t = (f(.001) - f(-.001))/0.002
local it = t
local nml = vec3(1,0,0)
local pts = {{p,it,nml}}
for k=1,100 do
np = f(k/100*2*math.pi)
nt = (f(k/100*2*math.pi+.001) - f(k/100*2*math.pi-.001))/0.002
ax = t:cross(nt)
ang = math.asin(ax:len()/(t:len()*nt:len()))
nml = nml:rotate(ang,ax)
table.insert(pts,{np,nt,nml})
t = nt
p = np
end
local __track = PseudoMesh()
local ends = 1
for k=1,100 do
if k == 100 then
ends = 2
end
__track:addCylinder({
startCentre = pts[k][1],
startWidth = pts[k][3],
startHeight = pts[k][2]:cross(pts[k][3]):normalize(),
endCentre = pts[k+1][1],
endWidth = pts[k+1][3],
endHeight = pts[k+1][2]:cross(pts[k+1][3]):normalize(),
ends = ends,
faceted = false,
size=50
})
ends = 0
end
track.model = __track:toModel()
track.material = craft.material("Materials:Standard")
end
function draw()
background(40,40,50)
update(DeltaTime)
scene:draw()
end
function update(dt)
scene:update(dt)
end
local mt = getmetatable(vec3())
mt.rotate = function(v,a,u)
u = u or vec3(1,0,0)
u = u:normalize()
local x = v - v:dot(u)*u
local y = u:cross(x)
return v - x + math.cos(a)*x + math.sin(a)*y
end
function f(t)
local r,h=5,5
return vec3(r*math.cos(t),r*math.sin(t),h*t)
end
PseudoMesh = class()
function PseudoMesh:init()
self.vertices = {}
self.texCoords = {}
self.normals = {}
self.colors = {}
self.size = 0
end
function PseudoMesh:vertex(k,v)
if v then
self.vertices[k] = v
else
return self.vertices[k]
end
end
function PseudoMesh:normal(k,v)
if v then
self.normals[k] = v
else
return self.normals[k]
end
end
function PseudoMesh:texCoord(k,v)
if v then
self.texCoords[k] = v
else
return self.texCoords[k]
end
end
function PseudoMesh:color(k,v)
if v then
self.colors[k] = v
else
return self.colors[k]
end
end
function PseudoMesh:resize(k)
self.size = k
end
function PseudoMesh:addCylinder(t)
t = t or {}
local nt = {}
for k,v in pairs(t) do
nt[k] = v
end
nt.mesh = self
return addCylinder(nt)
end
function PseudoMesh:invertNormals()
for k,v in ipairs(self.normals) do
self.normals[k] = -v
end
end
function PseudoMesh:toModel()
local m = craft.model()
local i = {}
local n = #self.vertices
for k=1,n,3 do
if (self.vertices[k+1] - self.vertices[k]):cross(self.vertices[k+2] - self.vertices[k]):dot(self.normals[k+1] + self.normals[k+2] + self.normals[k]) < 0 then
table.insert(i,k)
table.insert(i,k+1)
table.insert(i,k+2)
else
table.insert(i,k)
table.insert(i,k+2)
table.insert(i,k+1)
end
end
m.positions = self.vertices
m.normals = self.normals
m.uvs = self.texCoords
m.colors = self.colors
m.indices = i
return m
end
-- MeshExt
local __doJewel, __doSuspension, __doPyramid, __doBlock, __addTriangle, __doSphere, __threeFromTwo, __orthogonalTo, __doCylinder, __discreteNormal, __doCone, __doPoly, __doFacetedClosedCone, __doFacetedOpenCone, __doSmoothClosedCone, __doSmoothOpenCone, __doFacetedClosedCylinder, __doFacetedOpenCylinder, __doSmoothClosedCylinder, __doSmoothOpenCylinder, __initmesh
--[[
A "cone" is formed by taking a curve in space and joining each of its points to an apex.
If the original curve is made from line segments, the resulting cone has a natural triangulation which can be used to construct it as a mesh.
m mesh
p position in mesh
n number of points
o "internal" point (to ensure that normals point outwards)
a apex of cone
v table of base points
t table of texture points
col colour
to texture offset
ts not used
f faceted or not
cl closed curve or not
l light vector
--]]
function __doCone(m,p,n,o,a,v,t,col,to,ts,f,cl,l,am)
if f then
if cl then
return __doFacetedClosedCone(m,p,n,o,a,v,t,col,to,ts,l,am)
else
return __doFacetedOpenCone(m,p,n,o,a,v,t,col,to,ts,l,am)
end
else
if cl then
return __doSmoothClosedCone(m,p,n,o,a,v,t,col,to,ts,l,am)
else
return __doSmoothOpenCone(m,p,n,o,a,v,t,col,to,ts,l,am)
end
end
end
function __doFacetedClosedCone(m,p,n,o,a,v,t,col,to,ts,l,am)
local j,nml,c
for k=1,n do
j = k%n + 1
nml = (v[k] - a):cross(v[j] - a)
if nml:dot(a - o) < 0 then
nml = -nml
end
nml = nml:normalize()
c = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nml)))
__addTriangle(m,p,v[j],v[k],a,c,c,c,nml,nml,nml,to+t[j],to+t[k],to)
p = p + 3
end
return p
end
function __doFacetedOpenCone(m,p,n,o,a,v,t,col,to,ts,l,am)
local j,nml,c
for k=1,n-1 do
j = k + 1
nml = (v[k] - a):cross(v[j] - a)
if nml:dot(a - o) < 0 then
nml = -nml
end
nml = nml:normalize()
c = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nml)))
__addTriangle(m,p,v[j],v[k],a,c,c,c,nml,nml,nml,to+t[j],to+t[k],to)
p = p + 3
end
return p
end
function __doSmoothClosedCone(m,p,n,o,a,v,t,col,to,ts,l,am)
local j,nmla,nmlb,nmlc,cc,ca,nb,cb
nmlb = vec3(0,0,0)
nmlc = __discreteNormal(v[1],o,v[n],a,v[2])
cc = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nmlc)))
nb = vec3(0,0,0)
for k=1,n do
j = k%n + 1
nb = nb + __discreteNormal(v[j],o,v[k],a,v[j%n+1])
end
nb = nb:normalize()
cb = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nb)))
for k=1,n do
j = k%n + 1
nmla = nmlc
ca = cc
nmlc = __discreteNormal(v[j],o,v[k],a,v[j%n+1])
cc = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nmlc)))
__addTriangle(m,p,v[j],v[k],a,cc,ca,cb,nmlc,nmla,nmlb,to+t[j],to+t[k],to)
p = p + 3
end
return p
end
function __doSmoothOpenCone(m,p,n,o,a,v,t,col,to,ts,l,am)
local j,nmla,nmlb,nmlc,cc,ca,ll,nb,cb
ll = l:len()
nmlb = vec3(0,0,0)
nmlc = __discreteNormal(v[1],o,a,v[2])
cc = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nmlc)))
nb = vec3(0,0,0)
for k=1,n-2 do
j = k + 1
nb = nb + __discreteNormal(v[j],o,v[k],a,v[j%n+1])
end
nb = nb + __discreteNormal(v[n],o,v[n-1],a)
nb = nb:normalize()
cb = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nb)))
for k=1,n-2 do
j = k + 1
nmla = nmlc
ca = cc
nmlc = __discreteNormal(v[j],o,v[k],a,v[j%n+1])
cc = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nmlc)))
__addTriangle(m,p,v[j],v[k],a,cc,ca,cb,nmlc,nmla,nmlb,to+t[j],to+t[k],to)
p = p + 3
end
nmla = nmlc
ca = cc
nmlc = __discreteNormal(v[n],o,v[n-1],a)
cc = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nmlc)))
__addTriangle(m,p,v[n],v[n-1],a,cc,ca,cb,nmlc,nmla,nmlb,to+t[n],to+t[n-1],to)
return p + 3
end
--[[
This forms a surface which has boundary a given curve by forming a cone with the barycentre of the curve as its apex.
m mesh
p position in mesh
n number of points
o "internal" point (for normals)
v table of base points
t table of texture points
col colour
to texture offset
ts not used
f faceted or not
cl closed curve or not
l light vector
--]]
function __doPoly(m,p,n,o,v,t,col,to,ts,f,cl,l,am)
local a,b,r = vec3(0,0,0),vec2(0,0),0
for k,u in ipairs(v) do
a = a + u
r = r + 1
end
a = a / r
for k,u in ipairs(t) do
b = b + u
end
b = b / r
for k=1,r do
t[k] = t[k] - b
end
return __doCone(m,p,n,o,a,v,t,col,to+b,ts,f,cl,l,am)
end
--[[
| Option | Default | Description |
|:-------|:--------|:------------|
| `mesh` | new mesh | The mesh to add the shape to. |
| `position` | end of mesh | The place in the mesh at which to add the shape. |
| `colour`/`color` | white | The colour of the shape. |
| `faceted` | true | Whether to make it faceted or smooth. |
| `ends` | `0` | Which ends to fill in (`0` for none, `1` for start, `2` for end, `3` for both) |
| `texOrigin` | `vec2(0,0)` | If using a sprite sheet, this is the lower left corner of the rectangle associated with this shape. |
| `texSize` | `vec2(1,1)` | This is the width and height of the rectangle of the texture associated to this shape. |
There are various ways to specify the dimensions of the cylinder.
If given together, the more specific overrides the more general.
`radius` and `height` (`number`s) can be combined with `axes` (table of three `vec3`s) to specify the dimensions, where the first axis vector lies along the cylinder. The vector `origin` then locates the cylinder in space.
`startCentre`/`startCenter` (a `vec3`), `startWidth` (`number` or `vec3`), `startHeight` (`number` or `vec3`), `startRadius` (`number`) specify the dimensions at the start of the cylinder (if numbers, they are taken with respect to certain axes).
Similarly named options control the other end.
If axes are needed, these can be supplied via the `axes` option.
If just the `axis` option is given (a single `vec3`), this is the direction along the cylinder.
Other directions (if needed) are found by taking orthogonal vectors to this axis.
--]]
function addCylinder(t)
t = t or {}
local m = t.mesh
local p = t.position or (m.size + 1)
p = p + (1-p)%3
local ip = p
local col = t.colour or t.color or color(255, 255, 255, 255)
local f = true
local ends
local solid = t.solid
if solid then
ends = t.ends or 3
else
ends = t.ends or 0
end
if t.faceted ~= nil then
f = t.faceted
end
local l,am
if rl then
l = vec3(0,0,0)
am = 1
else
l = t.light or vec3(0,0,0)
if t.intensity then
l = l:normalize()*t.intensity
elseif l:lenSqr() > 1 then
l = l:normalize()
end
am = t.ambience or (1 - l:len())
end
local r = t.radius or 1
local h = t.height or 1
local to = t.texOrigin or vec2(0,0)
local ts = t.texSize or vec2(1,1)
local sc,si,sj,ec,ei,ej,a,o
if t.axis or t.axes or t.origin or t.centre or t.center then
if t.axis then
a = t.axis
elseif t.axes then
a = t.axes[1]
else
a = vec3(0,1,0)
end
if t.height then
a = h*a:normalize()
end
if t.origin or t.centre or t.center then
local o = t.origin or t.centre or t.center
sc,ec = o - a/2,o + a/2
end
end
sc = t.startCentre or t.startCenter or sc
ec = t.endCentre or t.endCenter or ec
sc,ec,a = __threeFromTwo(sc,ec,a,vec3(0,-h/2,0),vec3(0,h/2,0),vec3(0,h,0))
si = t.startWidth or t.startRadius or t.radius or 1
sj = t.startHeight or t.startRadius or t.radius or 1
ei = t.endWidth or t.endRadius or t.radius or 1
ej = t.endHeight or t.endRadius or t.radius or 1
local c,d
if t.axes then
a,c,d = unpack(t.axes)
end
if type(si) == "number" then
if type(sj) == "number" then
if not c then
c = __orthogonalTo(a)
end
si = si*c:normalize()
if not d then
sj = sj*a:cross(c):normalize()
else
sj = sj*d:normalize()
end
else
si = si*sj:cross(a):normalize()
end
elseif type(sj) == "number" then
sj = sj*a:cross(si):normalize()
end
if type(ei) == "number" then
if type(ej) == "number" then
if not c then
c = __orthogonalTo(a)
end
ei = ei*c:normalize()
if not d then
ej = ej*a:cross(c):normalize()
else
ej = ej*d:normalize()
end
else
ei = ei*ej:cross(a):normalize()
end
elseif type(ej) == "number" then
ej = ej*a:cross(ei):normalize()
end
local n = t.size or 12
local sa,ea,da = __threeFromTwo(t.startAngle,t.endAngle,t.deltaAngle,0,360,360)
local closed
if da == 360 then
closed = true
solid = false
else
closed = false
end
sa = math.rad(sa)
ea = math.rad(ea)
da = math.rad(da)/n
o = (sc + math.cos((sa+ea)/2)*si/2 + math.sin((sa+ea)/2)*sj/2 + ec + math.cos((sa+ea)/2)*ei/2 + math.sin((sa+ea)/2)*ej/2)/2
local ss = 1 + math.floor((ends+1)/2)
if solid then
ss = ss + 2
end
ts.x = ts.x / ss
local cs,sn,ti,tj
ti,tj = vec2(ts.x/2,0),vec2(0,ts.y/2)
cs = math.cos(sa)
sn = math.sin(sa)
si,sj = cs*si + sn*sj, -sn*si + cs*sj
ei,ej = cs*ei + sn*ej, -sn*ei + cs*ej
ti,tj = cs*ti + sn*tj, -sn*ti + cs*tj
local u,v,tu,tv,tw,cnrs = {},{},{},{},{},{}
cnrs[1] = {sc,sc+si,ec+ei,ec}
cs = math.cos(da)
sn = math.sin(da)
u[0] = sc+cs*si - sn*sj
v[0] = ec+cs*ei - sn*ej
for k=0,n do
table.insert(u,sc+si)
table.insert(v,ec+ei)
table.insert(tu,to + vec2(ts.x*k/n,0))
table.insert(tv,to + vec2(ts.x*k/n,ts.y))
table.insert(tw,ti)
si,sj = cs*si + sn*sj, -sn*si + cs*sj
ei,ej = cs*ei + sn*ej, -sn*ei + cs*ej
ti,tj = cs*ti + sn*tj, -sn*ti + cs*tj
end
u[n+2] = sc+cs*si + sn*sj
v[n+2] = ec+cs*ei + sn*ej
cnrs[2] = {sc, sc+cs*si - sn*sj, ec+cs*ei - sn*ej, ec}
local size = 6*n + math.floor((ends+1)/2)*3*n
if closed then
size = size + math.floor((ends+1)/2)*3
elseif solid then
size = size + 24
end
if p - 1 + size > m.size then
m:resize(p-1+size)
end
n = n + 1
p = __doCylinder(m,p,n,o,u,v,tu,tv,col,f,closed,l,am)
to = to + ts/2
if solid and not closed then
local tex = {-ts/2,vec2(ts.x/2,-ts.y/2),ts/2,vec2(-ts.x/2,ts.y/2)}
for i=1,2 do
to.x = to.x + ts.x
p = __doPoly(m,p,4,o,cnrs[i],tex,col,to,ts,f,true,l,am)
end
end
if ends%2 == 1 then
to.x = to.x + ts.x
p = __doCone(m,p,n,o,sc,u,tw,col,to,ts,f,closed,l,am)
end
if ends >= 2 then
to.x = to.x + ts.x
p = __doCone(m,p,n,o,ec,v,tw,col,to,ts,f,closed,l,am)
end
if ret then
return m,ip, p
else
return m
end
end
--[[
This adds a cylinder to the mesh.
m mesh to add shape to
p position of first vertex of shape
n number of points
o centre of shape (vec3)
a apexes (table of 2 vec3s)
v vertices (table of vec3s in cyclic order)
t texture coordinates corresponding to vertices (relative to centre)
col colour
to offset of texture region (vec2)
ts size of texture region (vec2)
f faceted
cl closed
l light vector
--]]
function __doCylinder(m,p,n,o,u,v,ut,vt,col,f,cl,l,am)
if f then
if cl then
return __doFacetedClosedCylinder(m,p,n-1,o,u,v,ut,vt,col,l,am)
else
return __doFacetedOpenCylinder(m,p,n,o,u,v,ut,vt,col,l,am)
end
else
if cl then
return __doSmoothClosedCylinder(m,p,n-1,o,u,v,ut,vt,col,l,am)
else
return __doSmoothOpenCylinder(m,p,n,o,u,v,ut,vt,col,l,am)
end
end
end
function __doFacetedClosedCylinder(m,p,n,o,u,v,ut,vt,col,l,am)
local i,j,nv,nu,cu,cv,ll
ll = l:len()
for k=1,n do
j = k%n + 1
nu = (u[j] - u[k]):cross(v[k] - u[k]):normalize()
nv = (v[j] - v[k]):cross(u[k] - v[k]):normalize()
if nu:dot(u[k]-o) < 0 then
nu = -nu
end
if nv:dot(v[k]-o) < 0 then
nv = -nv
end
cv = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv)))
cu = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu)))
__addTriangle(m,p,v[j],v[k],u[j],cv,cv,cu,nv,nv,nu,vt[j],vt[k],ut[j])
p = p + 3
__addTriangle(m,p,v[k],u[j],u[k],cv,cu,cu,nv,nu,nu,vt[k],ut[j],ut[k])
p = p + 3
end
return p
end
function __doFacetedOpenCylinder(m,p,n,o,u,v,ut,vt,col,l,am)
local i,j,nv,nu,cu,cv,ll
ll = l:len()
for k=1,n-1 do
j = k + 1
nu = (u[j] - u[k]):cross(v[k] - u[k]):normalize()
nv = (v[j] - v[k]):cross(u[k] - v[k]):normalize()
if nu:dot(u[k]-o) < 0 then
nu = -nu
end
if nv:dot(v[k]-o) < 0 then
nv = -nv
end
cv = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv)))
cu = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu)))
__addTriangle(m,p,v[j],v[k],u[j],cv,cv,cu,nv,nv,nu,vt[j],vt[k],ut[j])
p = p + 3
__addTriangle(m,p,v[k],u[j],u[k],cv,cu,cu,nv,nu,nu,vt[k],ut[j],ut[k])
p = p + 3
end
return p
end
function __doSmoothClosedCylinder(m,p,n,o,u,v,ut,vt,col,l,am)
local i,j,nv,nu,cu,cv
nv,nu,cv,cu = {},{},{},{}
nv[1] = __discreteNormal(v[1],o,v[n],u[1],v[2])
nu[1] = __discreteNormal(u[1],o,u[n],v[1],u[2])
cv[1] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv[1])))
cu[1] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu[1])))
for k=1,n do
j = k%n + 1
i = j%n + 1
nv[j] = __discreteNormal(v[j],o,v[k],u[j],v[i])
nu[j] = __discreteNormal(u[j],o,u[k],v[j],u[i])
cv[j] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv[j])))
cu[j] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu[j])))
__addTriangle(m,p,v[j],v[k],u[j],cv[j],cv[k],cu[j],nv[j],nv[k],nu[j],vt[j],vt[k],ut[j])
p = p + 3
__addTriangle(m,p,v[k],u[j],u[k],cv[k],cu[j],cu[k],nv[k],nu[j],nu[k],vt[k],ut[j],ut[k])
p = p + 3
end
return p
end
function __doSmoothOpenCylinder(m,p,n,o,u,v,ut,vt,col,l,am)
local i,j,nv,nu,cu,cv
nv,nu,cv,cu = {},{},{},{}
nv[1] = __discreteNormal(v[1],o,v[0],u[1],v[2])
nu[1] = __discreteNormal(u[1],o,v[0],v[1],u[2])
cv[1] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv[1])))
cu[1] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu[1])))
for k=1,n-2 do
j = k + 1
i = j + 1
nv[j] = __discreteNormal(v[j],o,v[k],u[j],v[i])
nu[j] = __discreteNormal(u[j],o,u[k],v[j],u[i])
cv[j] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv[j])))
cu[j] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu[j])))
__addTriangle(m,p,v[j],v[k],u[j],cv[j],cv[k],cu[j],nv[j],nv[k],nu[j],vt[j],vt[k],ut[j])
p = p + 3
__addTriangle(m,p,v[k],u[j],u[k],cv[k],cu[j],cu[k],nv[k],nu[j],nu[k],vt[k],ut[j],ut[k])
p = p + 3
end
nv[n] = __discreteNormal(v[n],o,v[n-1],u[n],v[n+1])
nu[n] = __discreteNormal(u[n],o,u[n-1],v[n],u[n+1])
cv[n] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nv[n])))
cu[n] = col:mix(color(0,0,0,col.a),am+(1-am)*math.max(0,l:dot(nu[n])))
__addTriangle(m,p,v[n],v[n-1],u[n],cv[n],cv[n-1],cu[n],nv[n],nv[n-1],nu[n],vt[n],vt[n-1],ut[n])
p = p + 3
__addTriangle(m,p,v[n-1],u[n],u[n-1],cv[n-1],cu[n],cu[n-1],nv[n-1],nu[n],nu[n-1],vt[n-1],ut[n],ut[n-1])
return p+3
end
--[[
This works out a normal vector for a vertex in a triangulated surface by taking an average of the triangles in which it appears.
The normals are weighted by the reciprocal of the size of the corresponding triangle.
a vertex under consideration
o a point to determine which side the normals lie
... a cyclic list of vertices, successive pairs of which make up the triangles
--]]
function __discreteNormal(a,o,...)
local arg = {}
local n = 0
for k,v in ipairs({...}) do
if v then
table.insert(arg,v)
n = n + 1
end
end
local na,nb
na = vec3(0,0,0)
for k=2,n do
nb = (arg[k] - a):cross(arg[k-1] - a)
na = na + nb/nb:lenSqr()
end
na = na:normalize()
if na:dot(a-o) < 0 then
na = -na
end
return na
end
--[[
Adds a triangle to a mesh, with specific vertices, colours, normals, and texture coordinates.
--]]
function __addTriangle(m,p,a,b,c,d,e,f,g,h,i,j,k,l)
m:vertex(p,a)
m:color(p,d)
m:normal(p,g)
m:texCoord(p,j)
p = p + 1
m:vertex(p,b)
m:color(p,e)
m:normal(p,h)
m:texCoord(p,k)
p = p + 1
m:vertex(p,c)
m:color(p,f)
m:normal(p,i)
m:texCoord(p,l)
end
--[[
Returns three things, u,v,w, with the property that u + w = v. The input can be any number of the three together with three defaults to be used if not enough information is given (so if any two of the first three are given then that is enough information to determine the third).
--]]
function __threeFromTwo(a,b,c,d,e,f)
local u,v,w = a or d or 0, b or e or 1, c or f or 1
if not a then
u = v - w
end
if not b then
v = u + w
end
if not c then
w = v - u
end
v = u + w
return u,v,w
end
--[[
Returns a vector orthogonal to the given `vec3`.
--]]
function __orthogonalTo(v)
local a,b,c = math.abs(v.x), math.abs(v.y), math.abs(v.z)
if a < b and a < c then
return vec3(0,v.z,-v.y)
end
if b < c then
return vec3(-v.z,0,v.x)
end
return vec3(v.y,-v.x,0)
end
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