Flying Release v1.7 -Using quaternions to fly a plane.

1aTabOrder

Flying Tab Order Version: 1.7
------------------------------
This file should not be included in the Codea project.
#Main
#JoyStick
#Quaternion
#Flight
#Plane
#Sky

Flight.lua

--Flight class
 
Flight=class()

local yaw = vec3(0,1,0)
local pitchroll = vec3(1,0,1)

function Flight:init(m) --m is mode 1-3: 1=Absolute, 2=Increment, 3=Delta
    self:Reset(m)
end
 
function Flight:Reset(m) --m is mode 1-3: 1=Absolute, 2=Increment, 3=Delta
    self.mode=m
    self.y=vec4(1,0,0,0)
    self.pr=vec4(1,0,0,0)
end
 
function Flight:Turn(v,m)
    m = m or self.mode
    self.y = self.y * qTangent(v * yaw)
    self.pr = self.pr * qTangent(v * pitchroll)
end
 
function Flight:Position(s) --d is vec3 distance in pixels
    return s^(self.y*self.pr)
end
 
function Flight:Rotate()
    rotate(self.y*self.pr)
end

function Flight:rotation()
    return self.y*self.pr
end

function Flight:Interpolate()
    local l = self.pr:slen()
    local sl = self.pr:make_slerp()
    return function(t)
        if t > l then
            self.pr = vec4(1,0,0,0)
            return false
        end
        self.pr = sl(1-t/l)
        return true
    end
end
 
 

JoyStick.lua

JoyStick = class()

function JoyStick:init(t)
    t = t or {}
    self.radius = t.radius or 100
    self.stick = t.stick or 30
    self.centre = t.centre or self.radius * vec2(1,1) + vec2(5,5)
    self.position = vec2(0,0)
    self.value = vec2(0,0)
    self.oldvalue = vec2(0,0)
    self.delta = vec2(0,0)
end

function JoyStick:draw()
    -- Codea does not automatically call this method
    ortho()
    viewMatrix(matrix())
    pushStyle()
    fill(160, 182, 191, 50)
    stroke(118, 154, 195, 100)
    strokeWidth(1) 
    ellipse(self.centre.x,self.centre.y,2*self.radius)
    fill(78, 131, 153, 50)
    ellipse(self.centre.x+self.position.x, self.centre.y+self.position.y, self.stick*2)
    popStyle()
end

function JoyStick:touched(t)
    if t.state == BEGAN then
        local v = vec2(t.x,t.y)
        if v:dist(self.centre)<self.radius-self.stick then
            self.mytouch = t.id
        end
    end
    if t.id == self.mytouch then
        if t.state~=ENDED then
            local v = vec2(t.x,t.y)
            if v:dist(self.centre)>self.radius-self.stick then
                v = (v - self.centre):normalize()*(self.radius - self.stick) + self.centre
            end  --set x,y values for joy based on touch
            self.position=v - self.centre
            self.value=self.position/(self.radius-self.stick)
        else --reset joystick to centre when touch ends
            self.position=vec2(0,0) 
            self.value=vec2(0,0)
            self.mytouch = false
        end
        return true
    end
    return false
end

function JoyStick:update()
    self.delta = self.value - self.oldvalue
    self.oldvalue = self.value
end

Main.lua

--v1.6
 
--the RotateAndDraw function needs code
VERSION = 1.7
function setup()
    -- [[
    if AutoGist then
        autogist = AutoGist("Flying","Using quaternions to fly a plane.",VERSION)
        autogist:backup(true)
    end
    -- ]]
 
    --**Flight controls **
    -- yaw = 0
    -- mode not currently used
    mode=vec3(0,1,0)
    speed=vec3(0,0,-60)  --pixels/sec  --XXXXX
    
    --** Plane settings **
    plane=Plane() --plane mesh
    planeQ=Flight(mode)--plane quaternion
    planePos=vec3(0,0,0) --starting position
    camOffset={vec3(0,3,-9),vec3(0,0,50),vec3(0,40,100)} --camera position relative to plane
    camDir = {vec3(0,3,-10),vec3(0,0,-1),vec3(0,0,0)}
    parameter.integer("state",1,3,1)
    parameter.integer("View",1,#camOffset,1)
    --** Joystick settings **
    rotjoy = JoyStick({centre = vec2(105,105)})
    mvjoy = JoyStick({centre = vec2(WIDTH - 105,105)})
    --** rotation settings and limits **
    --x, y, z map to pitch, yaw, roll
    --limits for slave rotation
    limit=vec3(90,90,90)*math.pi/180
    --sensitivity of rotation for incremental rotation
    sens=vec3(1,1,1)
    -- planeQ:Position(speed*DeltaTime)
    SetupSky()
    cam = {
        at = vec3(0,0,1),
        frame = vec4(1,0,0,0),
        vel = vec3(0,0,0),
        avel = vec3(0,0,0)
    }
    -- parameter.watch("cam.avel")
    parameter.watch("planeQ:rotation()")
    parameter.watch("cam.frame")
    --[[
    local n = 2
    local q = v:exp(1/n)
    print(q^n)
      ]]
end
 
function draw()
    rotjoy:update()
    mvjoy:update()
    background(179, 205, 220, 255)
    fill(255)
    perspective()
    camera(0,0,100,0,0,0)   
    RotateAndDraw()    
    rotjoy:draw()
    mvjoy:draw()

end
 
--this function adjusts the rotation based on joystick position, and draws the plane
function RotateAndDraw()
    if rotjoy.mytouch then
        local dv = vec3(-rotjoy.delta.y,-rotjoy.delta.x,-rotjoy.delta.x)
        local vv = vec3(-rotjoy.value.y,-rotjoy.value.x,-rotjoy.value.x)
        local v = (1 - mode) * dv * limit + mode * vv * sens * DeltaTime
        
        planeQ:Turn(v,mode)
        turning = true
    elseif turning then
        interpolating = true
        turning = false
        stime = ElapsedTime
        slerpQ = planeQ:Interpolate()
    elseif interpolating then
        interpolating = slerpQ(ElapsedTime - stime)
    end
    if mvjoy.mytouch then
        planePos=planePos+planeQ:Position(mvjoy.value.y*speed*DeltaTime)
    end
    --  --XXXX plane position, going in wrong vertical direction!

    if state == 1 then
        -- stationary camera
        cam.at = camOffset[View] --planePos+camOffset  --XXXX camera position - incorrect
        cam.frame = vec4(1,0,0,0)
    elseif state == 2 then
        -- slavishly follows plane
        cam.at = planePos + planeQ:Position(camOffset[View])
        cam.frame = planeQ:rotation()

    elseif state == 3 then
        -- follows, but with a delay
        local ca,cq = cam.at,cam.frame
        cam.at = cam.at + cam.vel * DeltaTime
        cam.frame = cam.avel:exp(DeltaTime) * cam.frame
        -- cam.vel = cam.vel + (planePos + planeQ:Position(camOffset[View]) - ca) * DeltaTime - 1.5*cam.vel*DeltaTime
        cam.vel = (planePos + planeQ:Position(camOffset[View]) - ca)
        -- cam.avel = cam.avel + (planeQ:rotation() * cq^""):log()*DeltaTime - 1.5*cam.avel*DeltaTime
        cam.avel = (planeQ:rotation() * cq^""):log() --*DeltaTime
    end
    local up,look = vec3(0,1,0)^cam.frame,planePos + planeQ:Position(camDir[View])
    camera(cam.at,look,up)
    pushMatrix()
    DrawOtherStuff()

    translate(planePos)
    planeQ:Rotate()
    plane:draw()
    popMatrix()
    
end
 
--** Utility functions - joystick and touch ***
 
--manage joystick
function touched(t)
    if not rotjoy:touched(t) and not mvjoy:touched(t) and t.state == BEGAN then
        print(planeQ:rotation())
        print(cam.frame)
        print(cam.avel)
        print(cam.avel:exp())
        print(cam.avel:exp(DeltaTime^2))
        print(DeltaTime^2)
    end
end
 
function DrawOtherStuff()  --XXXXX added just so we have frame of reference to see how the camera rotates
    pushMatrix() 
        translate(0,0,-1000)
    -- sprite("Planet Cute:Character Princess Girl",0,0,400)
    popMatrix()
    planet1:draw(planePos)
end

Plane.lua

--plane
 
Plane=class()
 
function Plane:init()
    a={}
    c1=color(107, 139, 184, 255)
    c2=color(72, 148, 195, 255)
    c3=color(176, 159, 101, 255)
    c4=color(255,0,0)
    a[1]=Block(3,3,25,c1,vec3(0,0,0))   
    a[2]=Block(15,1,5,c2,vec3(-9,1,-6))
    a[3]=Block(15,1,5,c2,vec3(9,1,-6))
    a[4]=Block(1,4,3,c2,vec3(0,3.5,11))
    a[5]=Block(4,1,2,c2,vec3(-3.5,1,11))
    a[6]=Block(4,1,2,c2,vec3(3.5,1,11))
    a[7]=Block(3,2,5,c3,vec3(0,2.5,-6))
    a[8]=Block(3,3,1,c4,vec3(0,0,-13))
    self.blocks=a 
end
 
function Plane:draw()
    for i=1,#self.blocks do
        self.blocks[i]:draw()
    end 
end
 
Block = class() --taken from 3D lab project
 
function Block:init(w,h,d,c,p,t,r) --width,height,depth,colour,position,texture, (optional) texture range (see above)
    self.width=w
    self.height=h
    self.depth=d  
    self.color=c  
    self.pos=p
    self.tex=t 
    --if no limits specified on which part of image to draw, set to 0,1
    if r~=nil then self.texR=r else self.texR={0,0,1,1} end
    self.blk=self:createBlock()
end
 
function Block:createBlock()
    -- all the unique vertices that make up a block
    --There are only 8 corners in a cube - we define them as vertices
    --all measurements are taken from the centre of the block
    --so bottom left front has x of -1/2 width, y of -1/2 height, and z of 1/2 depth
    local w,h,d=self.width,self.height,self.depth
    local x,y,z=self.pos.x,self.pos.y,self.pos.z
    local v = {
      vec3(x-0.5*w, y-0.5*h, z+0.5*d), -- Left  bottom front
      vec3(x+0.5*w, y-0.5*h, z+0.5*d), -- Right bottom front
      vec3(x+0.5*w, y+0.5*h, z+0.5*d), -- Right top    front
      vec3(x-0.5*w, y+0.5*h, z+0.5*d), -- Left  top    front
      vec3(x-0.5*w, y-0.5*h, z-0.5*d), -- Left  bottom back
      vec3(x+0.5*w, y-0.5*h, z-0.5*d), -- Right bottom back
      vec3(x+0.5*w, y+0.5*h, z-0.5*d), -- Right top    back
      vec3(x-0.5*w, y+0.5*h, z-0.5*d), -- Left  top    back
    }
 
    local cubeverts = {
      -- Front, Right, Back, Left, Top, Bottom
      v[1], v[2], v[3], v[1], v[3], v[4],
      v[2], v[6], v[7], v[2], v[7], v[3],
      v[6], v[5], v[8], v[6], v[8], v[7],
      v[5], v[1], v[4], v[5], v[4], v[8],
      v[4], v[3], v[7], v[4], v[7], v[8],
      v[5], v[6], v[2], v[5], v[2], v[1],
    }
    local cc={}
     --assign colors
        local c=self.color
        local x=0.5
        local BL=color(c.r,c.g,c.b) --bottom left
        local BR=color(c.r,c.g,c.b) --bottom right  
        local TR=color(c.r*x,c.g*x,c.b*x) --top right
        local TL=color(c.r*x,c.g*x,c.b*x) --top left
        for i=1,6 do
            cc[#cc+1]=BL
            cc[#cc+1]=BR
            cc[#cc+1]=TR
            cc[#cc+1]=BL
            cc[#cc+1]=TR
            cc[#cc+1]=TL
        end
  
    --put it all together
    local ms = mesh()
    ms.vertices = cubeverts
    if self.tex then
        ms.texture = self.tex
        ms.texCoords = cubetexCoords
    else
        ms.colors=cc
    end
    --ms:setColors(self.color)
    return ms
end  
    
function Block:draw()
    self.blk:draw() 
end

Quaternion.lua

-- Quaternion library wherein vec4s are promoted to quaternions
-- Author: Andrew Stacey
-- Website: http://www.math.ntnu.no/~stacey/HowDidIDoThat/iPad/Codea.html
-- Licence: CC0 http://wiki.creativecommons.org/CC0
 
--[[
This provides code for handling rotations.  Internally, the rotations
are implemented as quaternions which are in turn implemented by
extending the methods available for vec4 userdata.
 
Due to how vec4s are implemented, we cannot have methods beginning with
letters a, b, g, r, w, x, y, z
--]]
 
-- Simplistic error handling
local error = error or print
 
-- Localise the maths functions/constants that we use for faster lookup.
local abs = math.abs
local pow = math.pow
local sqrt = math.sqrt
local sin = math.sin
local cos = math.cos
local acos = math.acos
local asin = math.asin
local pi = math.pi
local floor = math.floor

--[[
The function "is_a" extends the capabilities of the method "is_a" which is automatically defined by Codea for classes.
 
Parameters:
a: object to be tested
b: test
 
The tests work as follows.
 
1. If the type of b is a string, it is taken as the name of a type to test a against.
2. If the type of b is a table, it is assumed to be a class to test if a is an instance thereof.
3. If the type of b is a userdata, the test is to see if a is the same type of object.
4. If b is a function, then it is replaced by the value of that function.
--]]
 
local function is_a(a,b)
    if type(b) == "function" then
        b = b()
    end
    if type(b) == "string" then
        return type(a) == b
    end
    if type(b) == "table" then
        if type(a) == "table"
            and a.is_a
            and a:is_a(b) then
                return true
        else
            return false
        end
    end
    if type(b) == "userdata" then
        if type(a) == "userdata" then
            a = getmetatable(a)
            b = getmetatable(b)
            return a == b
        else
            return false
        end
    end
    return false
end
 
-- Import the metatable of vec4s for extending
local mtq = getmetatable(vec4())
 
-- We run into difficulties if our vec4s contain NaNs or infs.  This
-- tests for those.
mtq["is_finite"] = function(q)
    if q.x < math.huge and q.x > -math.huge 
        and q.y < math.huge and q.y > -math.huge 
        and q.z < math.huge and q.z > -math.huge 
        and q.w < math.huge and q.w > -math.huge 
    then
        return true
    end
    return false
end
 
--[[
Test if we are real.
--]]
 
mtq["is_real"] = function (self)
    if self.y ~= 0 or self.z ~= 0 or self.w ~= 0 then
        return false
    end
    return true
end
 
--[[
Test if the real part is zero.
--]]
 
mtq["is_imaginary"] = function (self)
    if self.x ~= 0 then
        return false
    end
    return true
end
 
--[[
Normalise a quaternion to have length 1, safely.  Here "safely" means
that we ensure that we do not have NaNs or infs.  If we do, the
returned quaternion is the unit.
--]]
 
mtq["normalise"] = function (q)
    q = q:normalize()
    if q:is_finite() then
        return q
    else
        return vec4(1,0,0,0)
    end
end

--[[
Spherical length and distance.
--]]

mtq["slen"] = function(q)
    q = q:normalise()
    q.x = q.x - 1
    return 2*asin(q:len()/2)
end

mtq["sdist"] = function(q,qq)
    q = q:normalise()
    qq = qq:normalise()
    return 2*asin(q:dist(qq)/2)
end
 
--[[
Add two quaternions inline, including promotion of a number.
 
q + p
--]]
 
local __add = mtq["__add"]
 
mtq["__add"] = function (a,b)
    if is_a(a,"number") then
        a = vec4(a,0,0,0)
    end
    if is_a(b,"number") then
        b = vec4(b,0,0,0)
    end
    return __add(a,b)
end
 
--[[
Same for inline subtraction.
 
q - p
--]]
 
local __sub = mtq["__sub"]
 
mtq["__sub"] = function (a,b)
    if is_a(a,"number") then
        a = vec4(a,0,0,0)
    end
    if is_a(b,"number") then
        b = vec4(b,0,0,0)
    end
    return __sub(a,b)
end
 
--[[
For inline multiplication, we also allow multiplication by a matrix.
In this case, we use the fact that we're viewing quaternions as
rotations and matrices as affine transformations, so the best result
of multiplication is as the matrix representing the appropriate
composition.
 
q * p
--]]
 
local __mul = mtq["__mul"]
 
mtq["__mul"] = function (a,b)
    if is_a(a,"number") then
        return __mul(a,b)
    end
    if is_a(b,"number") then
        return __mul(a,b)
    end
    if is_a(a,matrix) then
        return a:__mul(b:tomatrixleft())
    end
    if is_a(b,matrix) then
        a = a:tomatrixleft()
        return a:__mul(b)
    end
    local x,y,z,w
    x = a.x * b.x - a.y * b.y - a.z * b.z - a.w * b.w
    y = a.x * b.y + a.y * b.x + a.z * b.w - a.w * b.z
    z = a.x * b.z - a.y * b.w + a.z * b.x + a.w * b.y
    w = a.x * b.w + a.y * b.z - a.z * b.y + a.w * b.x
    return vec4(x,y,z,w)
end
 
--[[
Conjugation (corresponds to inverting a rotation).
--]]
 
mtq["conjugate"] = function (q)
    return vec4(q.x, - q.y, - q.z, - q.w)
end
 
mtq["co"] = mtq["conjugate"]
 
--[[
Inline division, including division of and by a number.
--]]
 
local __div = mtq["__div"]
 
mtq["__div"] = function (a,b)
    if is_a(b,"number") then
        return __div(a,b)
    end
    local l = b:lenSqr()
    b = vec4(b.x/l,-b.y/l,-b.z/l,-b.w/l)
    if is_a(a,"number") then
        return vec4(a*b.x,a*b.y,a*b.z,a*b.w)
    end
    local x,y,z,w
    x = a.x * b.x - a.y * b.y - a.z * b.z - a.w * b.w
    y = a.x * b.y + a.y * b.x + a.z * b.w - a.w * b.z
    z = a.x * b.z - a.y * b.w + a.z * b.x + a.w * b.y
    w = a.x * b.w + a.y * b.z - a.z * b.y + a.w * b.x
    return vec4(x,y,z,w)
end
 
--[[
Powers.  Integer powers are defined iteratively.  Non-integer powers
use the slerp function.
 
TODO: test whether the slerp is faster than the iterative method for
integer powers as well.
--]]
 
local function intpower(q,n)
    if n ~= floor(n) then
        error("Only able to do integer powers")
        return false
    end
    if n == 0 then
        return vec4(1,0,0,0)
    elseif n > 0 then
       return q:__mul(intpower(q,n-1))
    elseif n < 0 then
       local l = q:lenSqr()
       q = vec4(q.x/l,-q.y/l,-q.z/l,-q.w/l)
       return q:intpower(-n)
    end
end
 
local function power(q,n)
   if n == floor(n) then
      return intpower(q,n)
   end
   local l = q:len()
   q = q:normalise()
   return l^n * q:slerp(n)
end
 
--[[
Inline exponentiation.
 
q^n
 
The behaviour depends on the type of the exponent:
 
* number: compute the power.
* vec4: conjugate by the exponent.
* other: return the conjugate.
--]]
 
mtq["__pow"] = function (q,n)
    if is_a(n,"number") then
        return power(q,n)
    elseif is_a(n,vec4) then
        return n:__mul(q):__div(n)
    else
        return q:conjugate()
    end
end
 
--[[
Interpolation functions, we assume the input to be already normalised
for speed.  If you cannot guarantee this, renormalise the input first.
The constructor functions do do the renormalisation.
 
Parameters:
q initial rotation
qq final rotation
t parameter (only for the direct functions)
 
If the input does not provide enough parameters, it is assumed that
the missing one is the initial rotation and that this should be taken
to be the identity rotation (represented by vec4(1,0,0,0)).
--]]
 
 
--[[
Linear interpolation, renormalised.
--]]
 
mtq["lerp"] = function (q,qq,t)
    if not t then
        return vec4(1,0,0,0):lerp(q,qq)
    end
    local v
    if (q + qq):len() == 0 then
        -- antipodal points, need a midpoint
        v = vec4(q.y,-q.x,q.w,-q.z)
        v = (1 - 2*t)*q + (1-abs(2*t-1))*v
    else
        v = (1-t)*q + t*qq
    end
    return v:normalise()
end
 
--[[
Spherical interpolation.
 
We have to be quite careful here not to emit anything with NaNs or
infs as there are several opportunities to create them due to the
limits of finite precision mathematics.
--]]
 
mtq["slerp"] = function (q,qq,t)
    if not t then
        return vec4(1,0,0,0):slerp(q,qq)
    end
    local v
    if (q + qq):len() == 0 then
        -- antipodal points, need a midpoint
        v = vec4(q.y,-q.x,q.w,-q.z)
        t = 2*t
    elseif (q - qq):len() == 0 then
        return q
    else
        v = qq
    end
    local ca = q:dot(v)
    local sa = sqrt(1 - pow(ca,2))
    if sa == 0 or sa ~= sa then
        return q
    end
    local a = acos(ca)
    sa = sin(a*t)/sa
    v = (cos(a*t) - ca*sa)*q+ sa*v
    return v
end
 
--[[
Constructor for normalised linear interpolation.
--]]
 
mtq["make_lerp"] = function (q,qq)
    if not qq then
        return vec4(1,0,0,0):make_lerp(q)
    end
    local v,w
    w = q:normalise()
    if (q + qq):len() == 0 then
        -- antipodal points, need a midpoint
        v = vec4(w.y,-w.x,w.w,-w.z)
        return function(t)
            local u = (1 - 2*t)*w + (1-abs(2*t-1))*v
            return u:normalise()
        end
    else
        v = qq:normalise()
        return function(t)
            local u = (1-t)*w + t*v
            return u:normalise()
        end
    end
end
 
--[[
Spherical interpolation
--]]
 
mtq["make_slerp"] = function (q,qq)
    if not qq then
        q,qq = vec4(1,0,0,0),q
    end
    local v,f,u
    if (q + qq):len() == 0 then
        -- antipodal points, need a midpoint
        v = vec4(q.y,-q.x,q.w,-q.z)
        f = 2
    elseif (q - qq):len() == 0 then
        return function(t)
            return q
        end
    else
        v = qq
        f = 1
    end
    v = v:normalise()
    u = q:normalise()
    local ca = u:dot(v)
    local sa = sqrt(1 - pow(ca,2))
    if sa == 0 or sa ~= sa then
        return function(t)
            return q
        end
    end
    local a = acos(ca)
    v = (v - ca*q)/sa
    return function(t)
        return cos(a*f*t)*q + sin(a*f*t)*v
    end
end
 
--[[
Returns the real part.
--]]
 
mtq["toreal"] = function (q)
    return q.x
end
 
--[[
Returns the vector (imaginary) part as a vec3 object.
--]]
 
mtq["vector"] = function (q)
    return vec3(q.y, q.z, q.w)
end
 
mtq["tovector"] = mtq["vector"]

mtq["log"] = function (q)
    local l = q:slen()
    local v = q:tovector()
    v = v:normalize()
    if not v:is_finite() then
        return vec3(0,0,0)
    else
        return v * l
    end
end
                                    
--[[
Represents a quaternion as a string.
--]]
 
mtq["tostring"] = function (q)
    local s
    local im = {{q.y,"i"},{q.z,"j"},{q.w,"k"}}
    if q.x ~= 0 then
        s = string.format("%.3f",q.x)
    end
    for k,v in pairs(im) do
    if v[1] ~= 0 then
        if s then 
                if v[1] > 0 then
                    if v[1] == 1 then
                        s = s .. " + " .. v[2]
                    else
                        s = s .. " + " .. 
                        string.format("%.3f",v[1]) .. v[2]
                    end
                else
                    if v[1] == -1 then
                        s = s .. " - " .. v[2]
                    else
                        s = s .. " - " .. 
                        string.format("%.3f",-v[1]) .. v[2]
                    end
                end
        else
                if v[1] == 1 then
                    s = v[2]
                elseif v[1] == - 1 then
                    s = "-" .. v[2]
                else
                    s = string.format("%.3f",v[1]) .. v[2]
                end
        end
    end
    end
    if s then
        return s
    else
        return "0"
    end
end
 
 
mtq["__concat"] = function (q,s)
    if is_a(s,"string") then
        return q:tostring() .. s
    else
        return q .. s:tostring()
    end
end
 
--[[
Converts the quaternion to a matrix.
 
We distinguish between left and right actions.  A rotation matrix
acting on the right is the transpose of the matrix on the left.
--]]
 
mtq["tomatrixleft"] = function (q)
    q = q:normalise()
    local a,b,c,d = q.x,q.y,q.z,q.w
    local ab = 2*a*b
    local ac = 2*a*c
    local ad = 2*a*d
    local bb = 2*b*b
    local bc = 2*b*c
    local bd = 2*b*d
    local cc = 2*c*c
    local cd = 2*c*d
    local dd = 2*d*d
    return matrix(
        1 - cc - dd,
        bc - ad,
        ac + bd,
        0,
        bc + ad,
        1 - bb - dd,
        cd - ab,
        0,
        bd - ac,
        cd + ab,
        1 - bb - cc,
        0,
        0,0,0,1
    )
end
 
mtq["tomatrixright"] = function (q)
    q = q:normalise()
    local a,b,c,d = q.x,-q.y,-q.z,-q.w
    local ab = 2*a*b
    local ac = 2*a*c
    local ad = 2*a*d
    local bb = 2*b*b
    local bc = 2*b*c
    local bd = 2*b*d
    local cc = 2*c*c
    local cd = 2*c*d
    local dd = 2*d*d
    return matrix(
        1 - cc - dd,
        bc - ad,
        ac + bd,
        0,
        bc + ad,
        1 - bb - dd,
        cd - ab,
        0,
        bd - ac,
        cd + ab,
        1 - bb - cc,
        0,
        0,0,0,1
    )
end
 
mtq["tomatrix"] = mtq["tomatrixright"]
 
--[[
Converts the quaternion to an "angle-axis" representation.
--]]
 
mtq["toangleaxis"] = function (q)
    q = q:normalise()
    local a = 2*acos(q.x)
    local v = vec3(q.y,q.z,q.w)
    if v == vec3(0,0,0) then
        return 0,vec3(0,0,1)
    end
    return a,v:normalise()
end
 
--[[
The following code modifies various of Codea's functions to enable
them to take a quaternion as input.  We have to be careful as for some
passing nil is distinct to passing an empty input.
 
The other issue is as to the distinction between left and right
actions of matrices.  In OpenGL, matrices act on the right.  However,
in standard linear algebra, the convention is for matrices to act on
the left.
--]]
 
local __modelMatrix = modelMatrix
 
function modelMatrix(m)
    if m then
        if is_a(m,vec4) then
            m = m:tomatrixright()
        end
        return __modelMatrix(m)
    else
        return __modelMatrix()
    end
end
 
local __applyMatrix = applyMatrix
 
function applyMatrix(m)
    if m then
        if is_a(m,vec4) then
            m = m:tomatrixright()
        end
        return __applyMatrix(m)
    else
        return __applyMatrix()
    end
end
 
local __viewMatrix = viewMatrix
 
function viewMatrix(m)
    if m then
        if is_a(m,vec4) then
            m = m:tomatrixright()
        end
        return __viewMatrix(m)
    else
        return __viewMatrix()
    end
end
 
local __projectionMatrix = projectionMatrix
 
function projectionMatrix(m)
    if m then
        if is_a(m,vec4) then
            m = m:tomatrixright()
        end
        return __projectionMatrix(m)
    else
        return __projectionMatrix()
    end
end
 
local __rotate = rotate
 
function rotate(a,x,y,z)
    if is_a(a,vec4) then
        local v
        a,v = a:toangleaxis()
        x,y,z = v.x,v.y,v.z
        a = a*180/pi
    end
    __rotate(a,x,y,z)
end
 
local __translate = translate
 
function translate(x,y,z)
    if not y then
        x,y,z = x.x,x.y,x.z
    end
    __translate(x,y,z)
end
                                    

local __scale = scale

function scale(a,b,c)
    if is_a(a,vec3) then
        a,b,c = a.x,a.y,a.z
    end
    if c then
        return __scale(a,b,c)
    end
    if b then
        return __scale(a,b)
    end
    if a then
        return __scale(a)
    end
end
                                    
local __camera = camera

function camera(a,b,c,d,e,f,g,h,i)
    if is_a(a,vec3) then
        a,b,c,d,e,f,g,h,i = a.x,a.y,a.z,b,c,d,e,f,g                              
    end
    if is_a(d,vec3) then
        d,e,f,g,h,i = d.x,d.y,d.z,e,f,g
    end
    if is_a(g,vec3) then
        g,h,i = g.x,g.y,g.z
    end
    if g then
        return __camera(a,b,c,d,e,f,g,h,i)
    elseif d then
        return __camera(a,b,c,d,e,f)
    elseif a then
        return __camera(a,b,c)
    else
        return __camera()
    end
end
 
--[[
We also define some extensions to the vec3 type.
--]]
 
local mt = getmetatable(vec3())
 
-- Same as for vec4.
mt["is_finite"] = function(v)
    if v.x < math.huge and v.x > -math.huge 
        and v.y < math.huge and v.y > -math.huge 
        and v.z < math.huge and v.z > -math.huge 
    then
        return true
    end
    return false
end
 
--[[
Promote to a quaternion with 0 real part.
--]]
 
mt["toQuaternion"] = function (v)
    return vec4(0,v.x,v.y,v.z)
end
 
--[[
Apply a quaternion as a rotation (assumes unit quaternion for speed)
using conjugation.
--]]
 
mt["applyQuaternion"] = function (v,q)
   v = v:toQuaternion()
    v = q:__mul(v)
    v = v:__mul(q:conjugate())
    return v:vector()
end
 
-- There is no native rotate method for a vec3.
mt["rotate"] = function(v,q,x,y,z)
    if is_a(q,"number") then
        q = qRotation(q,x,y,z)
    end
    return v:applyQuaternion(q)
end
 
-- We use the exponential notation for writing the action of a
-- quaternion on a vector (this is consistent with group theory
-- notation).
mt["__pow"] = function (v,q)
    if is_a(q,vec4) then
        return v:applyQuaternion(q)
    end
    return false
end
 
mt["__concat"] = function (u,s)
    if is_a(s,"string") then
        return u:__tostring() .. s
    else
        return u .. s:__tostring()
    end
end
 
--[[
The rotateTo method produces a rotation that rotates the first vector
to the second about an axis that is perpendicular to both.  So long as
the two are not collinear, the axis is unique.  The angle is taken to
be the smallest angle.  If the vectors point in opposite directions,
an orthogonal axis is chosen in such a way as to minimise precision
error.
 
TODO: Allow for a third parameter to specify the axis in case of
ambiguity.
--]]
mt["rotateTo"] = function (u,v)
    if v:lenSqr() == 0 or u:lenSqr() == 0 then
        return vec4(1,0,0,0)
    end
    u = u:normalise()
    v = u + v:normalise()
    if v:lenSqr() == 0 then
        -- Opposite vectors, no canonical direction
        local a,b,c = abs(u.x), abs(u.y), abs(u.z)
        if a < b and a < c then
            v = vec3(0,-u.z,u.y)
        elseif b < c then
            v = vec3(u.z,0,-u.x)
        else
            v = vec3(u.y,-u.x,0)
        end
    end
    v = v:normalise()
    local d = u:dot(v)
    u = u:cross(v)
    return vec4(d,u.x,u.y,u.z)
end
 
--[[
Safe renormalisation, as for quaternions.  Except that there is no
"cannonical" unit vec3, so we return the unit vector in the z
direction.  The rationale for that is that if this is an axis, this
vector represents the axis out of the screen.
--]]
mt["normalise"] = function (v)
    v = v:normalize()
    if v:is_finite() then
        return v
    else
        return vec3(0,0,1)
    end
end
 
--[[
Inline multiplication extended to allow for multiplication by a matrix.
 
We interpret the vec3 as vertical or horizontal depending on whether
the matrix is on the right or left.  Multiplication by a matrix is
viewed as applying an affine transformation.
--]]
local __mulv = mt["__mul"]
 
mt["__mul"] = function(m,v)
    if is_a(m,vec3) and is_a(v,"number") then
        return __mulv(m,v)
    end
    if is_a(m,"number") and is_a(v,vec3) then
        return __mulv(m,v)
    end
    if is_a(m,vec3) and is_a(v,vec3) then
        return vec3(m.x*v.x,m.y*v.y,m.z*v.z)
    end
    if is_a(m,matrix) and is_a(v,vec3) then
        local l = m[13]*v.x + m[14]*v.y + m[15]*v.z + m[16]
        return vec3(
            (m[1]*v.x + m[2]*v.y + m[3]*v.z + m[4])/l,
            (m[5]*v.x + m[6]*v.y + m[7]*v.z + m[8])/l,
            (m[9]*v.x + m[10]*v.y + m[11]*v.z + m[12])/l
        )
    end
    if is_a(m,vec3) and is_a(v,matrix) then
       m,v = v,m
        local l = m[4]*v.x + m[8]*v.y + m[12]*v.z + m[16]
        return vec3(
            (m[1]*v.x + m[5]*v.y + m[9]*v.z + m[13])/l,
            (m[2]*v.x + m[6]*v.y + m[10]*v.z + m[14])/l,
            (m[3]*v.x + m[7]*v.y + m[11]*v.z + m[15])/l
        )
    end
end

local __addv = mt["__add"]

mt["__add"] = function(a,b)
    if is_a(a,"number") then
        a = vec3(a,a,a)
    end
    if is_a(b,"number") then
        b = vec3(b,b,b)
    end
    return __addv(a,b)
end

local __subv = mt["__sub"]

mt["__sub"] = function(a,b)
    if is_a(a,"number") then
        a = vec3(a,a,a)
    end
    if is_a(b,"number") then
        b = vec3(b,b,b)
    end
    return __subv(a,b)
end
                    

--[[
Extensions to the matrix class.
--]]
local mtm = getmetatable(matrix())
 
--[[
Inline multiplication by either quaternion or vector.
--]]
local __mulm = mtm["__mul"]
 
mtm["__mul"] = function (m,mm)
    if is_a(m,matrix) and is_a(mm,matrix) then
        return __mulm(m,mm)
    end
    if is_a(m,matrix) and is_a(mm,vec4) then
        return __mulm(m,mm:tomatrix())
    end
    if is_a(m,vec4) and is_a(mm,matrix) then
        return __mulm(m:tomatrix(),mm)
    end
    if is_a(m,matrix) and is_a(mm,vec3) then
        local l = m[13]*mm.x + m[14]*mm.y + m[15]*mm.z + m[16]
        return vec3(
            (m[1]*mm.x + m[2]*mm.y + m[3]*mm.z + m[4])/l,
            (m[5]*mm.x + m[6]*mm.y + m[7]*mm.z + m[8])/l,
            (m[9]*mm.x + m[10]*mm.y + m[11]*mm.z + m[12])/l
        )
    end
    if is_a(m,vec3) and is_a(mm,matrix) then
        local l = mm[4]*m.x + mm[8]*m.y + mm[12]*m.z + mm[16]
        return vec3(
            (mm[1]*m.x + mm[5]*m.y + mm[9]*m.z + mm[13])/l,
            (mm[2]*m.x + mm[6]*m.y + mm[10]*m.z + mm[14])/l,
            (mm[3]*m.x + mm[7]*m.y + mm[11]*m.z + mm[15])/l
        )
    end
end
 
-- Extending the rotate method to take a quaternion.
local __mrotate = mtm["rotate"]
 
mtm["rotate"] = function(m,a,x,y,z)
    if is_a(a,vec4) then
        a,x = a:toangleaxis()
        x,y,z = x.x,x.y,x.z
    end
    return __mrotate(m,a,x,y,z)
end
 
 
--[[
The following functions are intended to be more "user friendly" and
provide solutions to common problems.
 
The primary functions construct a rotation dependent on some initial
data.  The simplest are to define a rotation from some other method of
specifying rotations: angle-axis or Euler angles.  More complicated
are methods to define a rotation by giving two (orthogonal) frames and
computing the rotation to get from the first to the last.  In
particular, there are certain common frames that might be expected to
be used frequently:
 
1. The initial orientation of the iPad.
 
2. A gravitational frame wherein the Gravity vector is "straight
down".  This is not completely defined as there is currently no way to
choose a corresponding horizontal direction (access to the compass
would provide this).  Two possibilities are: to choose the x-axis so
that it is always in the plane of the iPad, and to use RotationRate to
try to keep track of the initial x-axis.
 
3. A "DeltaOrientation" in which the change in orientation from one
frame to the next is used (this uses RotationRate).
 
There are obvious limits in accuracy in using these.
--]]
 
--[[
Returns a quaternion corresponding to the current gravitational vector
so that after applying the corresponding rotation, the y-axis points
in the gravitational direction and the x-axis is in the plane of the
iPad screen.
 
When we have access to the compass, the x-axis behaviour might change.
--]]
 
local function qGravity()
    local gxy, gy, gygxy, a, b, c, d
    if Gravity.x == 0 and Gravity.y == 0 then
        return vec4(1,0,0,0)
    else
        gy = - Gravity.y
        gxy = sqrt(pow(Gravity.x,2) + pow(Gravity.y,2))
        gygxy = gy/gxy
        a = sqrt(1 + gxy - gygxy - gy)/2
        b = sqrt(1 - gxy - gygxy + gy)/2
        c = sqrt(1 - gxy + gygxy - gy)/2
        d = sqrt(1 + gxy + gygxy + gy)/2
        if Gravity.y > 0 then
                a = a
                b = b
        end
        if Gravity.z < 0 then
                b = - b
                c = - c
        end
        if Gravity.x > 0 then
                c = - c
                d = - d
        end
        return vec4(a,b,c,d)
    end
end
 
function qrGravity(q)
    local qg,qx
    if not q then
        qx = ReferenceFrame()
    else
        qx = q
    end
    local y = vec3(0,-1,0)^qx
    qg = y:rotateTo(Gravity)
    return qg*qx
end
 
mtq["gravity"] = qrGravity
 
local frame = {
        vec3(1,0,0),
        vec3(0,1,0),
        vec3(0,0,1)
    }
local qzyx = vec4(1,0,0,0)
 
--[[
Needs to be run once every frame!
--]]
 
function updateReferenceFrame(q)
    local x,y,z
    --[[
    if CurrentOrientation == PORTRAIT then
        x,y,z = -frame[1],-frame[2],-frame[3]
    elseif CurrentOrientation == PORTRAIT_UPSIDE_DOWN then
        x,y,z = frame[1],frame[2],-frame[3]
    elseif CurrentOrientation == LANDSCAPE_LEFT then
        x,y,z = frame[2],-frame[1],-frame[3]
    elseif CurrentOrientation == LANDSCAPE_RIGHT then
        x,y,z = -frame[2],frame[1],-frame[3]
    end
    --]]
    x,y,z = unpack(frame)
    local qz = qRotation(RotationRate.z*DeltaTime,z.x,z.y,z.z)
    local qy = qRotation(RotationRate.y*DeltaTime,y.x,y.y,y.z)
    local qx = qRotation(RotationRate.x*DeltaTime,x.x,x.y,x.z)
    if q then
        local qq = qz * qy * qx * q
        q.x,q.y,q.z,q.w = qq.x,qq.y,qq.z,qq.w
    else
        qzyx = qz * qy * qx * qzyx
        return qzyx
    end
end
 
function ReferenceFrame(q)
    return q or qzyx
end
 
mtq["updateReferenceFrame"] = updateReferenceFrame
mtq["ReferenceFrame"] = ReferenceFrame
 
--[[
Converts a rotation to a quaternion.  The first argument is the angle
to rotate, the rest must specify an axis, either as a Vec3 object or
as three numbers.
--]]
 
function qRotation(a,x,y,z)
    local q,c,s
    if not y then
        x,y,z = x.x,x.y,x.z
    end
    q = vec4(0,x,y,z)
    q = q:normalise()
    if q == vec4(1,0,0,0) then
        return q
    end
    c = cos(a/2)
    s = sin(a/2)
    q = q:__mul(s)
    q = q:__add(c)
    return q
end
 
--[[
The qEuler function handles conversion from Euler angles.  As there
are many ways to specify Euler angles, we have to allow for quite a
variety.  The table __euler contains the code necessary for allowing
all of this flexibility.
 
The last parameter is the specification, which is a table taken from
the alphabet "xXyYzZ".  This ought to correspond to the axes of
rotation, using lowercase to mean the absolute frame and uppercase to
mean the transformed frame.
 
TODO: Test this.
 
If only two parameters are passed, the first is taken to be a vec3 or
a table containing the Euler angles.
--]]
 
local __euler = {}
__euler.x = function(q)
    return vec3(1,0,0)^q
end
__euler.X = function(q)
    return vec3(1,0,0)
end
__euler.y = function(q)
    return vec3(0,1,0)^q
end
__euler.Y = function(q)
    return vec3(0,1,0)
end
__euler.z = function(q)
    return vec3(0,0,1)^q
end
__euler.Z = function(q)
    return vec3(0,0,1)
end
 
function qEuler(a,b,c,v)
    if c then
        a = {a,b,c}
    else
        if is_a(a,vec3) then
            a = {a.x,a.y,a.z}
        end
        v = b
    end
    v = v or {"x","y","z"}
    local q = vec4(1,0,0,0)
    local w
    for k,u in ipairs(v) do
        w = __euler[u](q)
        q = q * qRotation(a[k],w)
    end
    return q
end
 
-- v is a tangent vector at the identity
function qTangent(x,y,z,t)
    local q
    if is_a(x,"number") then
        q = vec4(0,x,y,z)
        t = t or 1
    else
        q = vec4(0,x.x,x.y,x.z)
        t = y or 1
    end
    local qn = q:normalise()
    if qn == vec4(1,0,0,0) then
        return qn
    end
    local a = t * q:len() --/2
    return cos(a)*vec4(1,0,0,0) + sin(a)*qn
end

mt["exp"] = qTangent
 
function qRotationRate()
    return qTangent(DeltaTime * RotationRate)
end
 
local iGravity = vec3(0,-1,0)
Frame = {
    Roll = vec4(1,0,0,0),
    Pitch = vec4(1,0,0,0),
    Yaw = vec4(1,0,0,0),
    Gravity = vec4(1,0,0,0),
    Rotation = vec4(1,0,0,0),
    AdjustedRotation = vec4(1,0,0,0),
    RotationRate = vec4(1,0,0,0)
}
 
local update_fn 
update_fn = function()
    local q = vec3(iGravity.x,iGravity.y,0):rotateTo(
            vec3(Gravity.x,Gravity.y,0))
    Frame.Roll.x = q.x
    Frame.Roll.y = q.y
    Frame.Roll.z = q.z
    Frame.Roll.w = q.w
    q = vec3(0,iGravity.y,iGravity.z):rotateTo(
            vec3(0,Gravity.y,Gravity.z))
    Frame.Pitch.x = q.x
    Frame.Pitch.y = q.y
    Frame.Pitch.z = q.z
    Frame.Pitch.w = q.w
    q = vec3(iGravity.x,0,iGravity.z):rotateTo(
            vec3(Gravity.x,0,Gravity.z))
    Frame.Yaw.x = q.x
    Frame.Yaw.y = q.y
    Frame.Yaw.z = q.z
    Frame.Yaw.w = q.w
    q = iGravity:rotateTo(Gravity)
    Frame.Gravity.x = q.x
    Frame.Gravity.y = q.y
    Frame.Gravity.z = q.z
    Frame.Gravity.w = q.w
    q = vec4(0,RotationRate.x,RotationRate.y,RotationRate.z)
    local qn = q:normalise()
    if qn ~= vec4(1,0,0,0) then
        local a = q:len()*DeltaTime/2
        qn = cos(a)*vec4(1,0,0,0) + sin(a)*qn
    end
    Frame.RotationRate.x = qn.x
    Frame.RotationRate.y = qn.y
    Frame.RotationRate.z = qn.z
    Frame.RotationRate.w = qn.w
    q = qn*Frame.Rotation
    Frame.Rotation.x = q.x
    Frame.Rotation.y = q.y
    Frame.Rotation.z = q.z
    Frame.Rotation.w = q.w
    q = qn*Frame.AdjustedRotation
    local g = iGravity^(q)
    qn = g:rotateTo(iGravity)
    q = q*qn
    Frame.AdjustedRotation.x = q.x
    Frame.AdjustedRotation.y = q.y
    Frame.AdjustedRotation.z = q.z
    Frame.AdjustedRotation.w = q.w
    tween.delay(0,update_fn)
end
    
 
tween.delay(.5,function()
     iGravity = vec3(Gravity.x,Gravity.y,Gravity.z)
     tween.delay(0,update_fn)
     end)
 
--[[
A suite of test functions.
--]]
function testRotations()
    print("Rotation tests:")
    local tolerance = 10^(-6)
    local vars = [[local q1,q2,i,j,k
    q1 = vec4(1,2,3,4)
    q2 = vec4(.5,.5,-.5,-.5)
    i = vec4(0,1,0,0)
    j = vec4(0,0,1,0)
    k = vec4(0,0,0,1)
    q3 = vec4(1/math.sqrt(2),0,0,1/math.sqrt(2))
    v1 = vec3(1,0,0)
    v2 = vec3(0,1,0)
    v3 = vec3(1,-1,2)
    m = matrix(
        10,10,5,1,
        4,6,4,1,
        1,3,3,1,
        0,1,2,1
    )
    return ]]
    local tests = {
        {"q1 * q2", vec4(3,2,4,-1)},
        {"q1 + q2", vec4(1.5,2.5,2.5,3.5)},
        {"q1 - q2", vec4(.5,1.5,3.5,4.5)},
        {"q1 / q2", vec4(-2,0,-1,5)},
        {"q1^q2", vec4(1,-4,-2,3)},
        {"q1^\"\"", vec4(1,-2,-3,-4)},
        {"i * j", vec4(0,0,0,1)},
        {"j * k", vec4(0,1,0,0)},
        {"k * i", vec4(0,0,1,0)},
        {"v1^q3", vec3(0,1,0)},
        {"q3:tomatrixleft()*v1", vec3(0,1,0)},
        {"v1:rotate(q3)", vec3(0,1,0)},
        {"v2^q3", vec3(-1,0,0)},
        {"q3:tomatrixleft()*v2", vec3(-1,0,0)},
        {"v2:rotate(q3)", vec3(-1,0,0)},
        {"qRotation(math.pi/2,0,0,1)", vec4(1/math.sqrt(2),0,0,1/math.sqrt(2))},
        {"v1:rotateTo(v2)", vec4(1/math.sqrt(2),0,0,1/math.sqrt(2))},
        {"v3 * m", vec3(8,11,9)/3},
        {"m * v3", vec3(11,7,5)/4}
    }
    local t,a,m,n,tn
    tn,n = 0,0
    for k,u in ipairs(tests) do
        tn = tn + 1
        t = loadstring(vars .. u[1])
        a = t()
        if a == u[2] then
            m = "OK"
            n = n + 1
        elseif a.dist and a:dist(u[2]) < tolerance then
            m = "OK (" .. a:dist(u[2]) .. ")"
            n = n + 1
        else
            m = "not OK (expected " .. u[2] .. ")"
        end
        print(u[1] .. " = " .. a .. " : " .. m)
    end
    print(n .. " tests passed out of " .. tn)
end
 
 
 
 

Sky.lua

-- Sky globe
--displayMode(FULLSCREEN)
-- supportedOrientations(LANDSCAPE_ANY)
  
function SetupSky()
    img1=readImage("Documents:SkyDome")
    img2=image(img1.width,img1.width/2)
    setContext(img2)
    spriteMode(CORNER)
    sprite(img1,0,img2.height-img1.height)
    sprite(img1,0,img2.height-img1.height-img1.height)
    setContext()
 
    local color1 = color(255, 255, 255, 255)
    --this sphere code comes from Jmv38, see bottom
    planet1 = Sphere({  
                    nx = 40, ny = 20 ,            -- mesh definition
                    meshOptimize = true,           -- optimize mesh for sphere
                    c1 = color1 , c2 = color1 ,    -- mesh colors
                    centre = vec3(0,0,0)  ,         -- sphere center    
                    r = 500   ,         -- radius of the sphere
                    rotTime1 = 20 ,   -- rotation time in s             
                    hflip = true,    -- to flip image horozontally
                  })
    cam = vec3(0, 50, 300)
    planet1.ms.texture=img2
    --parameter.number("FPS",0,60,60)
end
 
--the code below comes from Jmv38, who explains it here
--http://jmv38.comze.com/CODEAbis/server.php (look for 3D tutorial)
 
Sphere = class()
 
function Sphere:init(input)
    -- spatial position of sphere
    self.centre = input.centre or vec3(0,0,0)
    -- angular position of sphere, defined by angles around x,y,z axis
    self.angle = vec3(0,0,0)
    -- sphere radius and rotation
    self.radius = input.r
    self.tRot = input.rotTime1
    -- sphere rotation 2
    self.tRot2 = input.rotTime2
    self.centre2 = input.centre2 or vec3(0,0,0) -- center of rotation 2
    self.axis2 = input.angle2 or vec3(0,1,0)   -- axis of rotation 2
    -- mesh definition
    self.nx = input.nx    -- number of triangles in x
    self.ny = input.ny    -- and in y
    self.c1 = input.c1    -- 2 color() objects, to see the triangles
    self.c2 = input.c2
    self.optimized = input.meshOptimize    -- boolean
    -- sphere decoration
    self.url = input.url    -- texture as a url (text)
    self.hflip = input.hflip    -- to flip image horizontally
    if input.lightDir then
        self.lightDir = input.lightDir:normalize()   -- a vec3 pointing to the sun
    end
    self.shadowRatio = input.shadowRatio or 1.05   -- close to 1.05
    
    -- create mesh and colors
    local vertices,colors,tc = {},{},{}
    if self.optimized then
        vertices,colors,tc = self:optimMesh({ nx=self.nx, ny=self.ny, c1=self.c1, c2=self.c2 })
    else
        vertices,colors,tc = self:simpleMesh({ nx=self.nx, ny=self.ny, c1=self.c1, c2=self.c2 })
    end
 
    -- if a radius is given, warp to a sphere
    if self.radius then 
    vertices = self:warpVertices({
            verts=vertices, 
            xangle=180, 
            yangle=180 
        }) end
 
    -- create the mesh itself
    self.ms = mesh()
    self.ms.vertices = vertices
    self.ms.colors = colors
    -- add the texture from internet
    --if self.url then 
    --    self:load( self.url ) -- this will not be instantaneous!
    --end
    self.ms.texture = self.url
    self.ms.texCoords = tc
    
    -- add some shadows
    if self.lightDir then self:shadows() end
 
end
 
function Sphere:shadows()
        self.ms2 = mesh()
        local dir = self.lightDir
        local vertices2,colors2 = {},{}
        local d = 0
        for i,v in ipairs(self.ms.vertices) do
            vertices2[i] = v
            d = v:dot(dir)
            d = 128 - 4*(d-0.1)*128
            if d<0 then d=0 end
            if d>255 then d=255 end
            colors2[i] = color(0,0,0,d)
        end
        self.ms2.vertices = vertices2
        self.ms2.colors = colors2
end
 
function Sphere:simpleMesh(input)
    -- create the mesh tables
    local vertices = {}
    local colors = {}
    local texCoords = {}
    --local w,h = img.width/10, img.height/10
    local k = 0
    local s = 1
    -- create a rectangular set of triangles
    local x,y
    local nx,ny = input.nx,input.ny
    local opt = input.opt
    local sx, sy = 1/ny, 1/ny
    local color1 = input.c1
    local color2 = input.c2
    local center = vec3(1,0.5,0)
    for y=0,ny-1 do
      for x=0,nx-1 do
        vertices[k+1] = vec3( sx*x    , sy*y    , 1) - center
        vertices[k+2] = vec3( sx*(x+1), sy*y    , 1) - center
        vertices[k+3] = vec3( sx*(x+1), sy*(y+1), 1) - center 
        vertices[k+4] = vec3( sx*x    , sy*y    , 1) - center 
        vertices[k+5] = vec3( sx*x    , sy*(y+1), 1) - center 
        vertices[k+6] = vec3( sx*(x+1), sy*(y+1), 1) - center 
        colors[k+1] = color1 
        colors[k+2] = color1 
        colors[k+3] = color1 
        colors[k+4] = color2 
        colors[k+5] = color2 
        colors[k+6] = color2 
        k = k + 6    
      end
    end   
    return vertices,colors
end
 
function Sphere:optimMesh(input)
    -- create the mesh tables
    local vertices = {}
    local colors = {}
    local texCoords = {}
    --local w,h = img.width/10, img.height/10
    local k = 0
    local s = 1
    -- create a set of triangles with approx constant surface on a sphere
    local x,y
    local x1,x2 = {},{}
    local i1,i2 = 0,0
    local nx,ny = input.nx,input.ny
    local sx, sy = nx/ny, 1/ny
    local color1 = input.c1
    local color2 = input.c2
    local center = vec3(1,0.5,0)
    local m1,m2,c
    local flip = 1
    if self.hflip then flip=-1 end
    for y=0,ny-1 do -- for each horizontal band
        -- number of points on each side of the band
        local nx1 = math.floor( nx * math.abs(math.cos(    ( y*sy-0.5)*2 * math.pi/2)) )
        if nx1<6 then nx1=6 end
        local nx2 = math.floor( nx * math.abs(math.cos( ((y+1)*sy-0.5)*2 * math.pi/2)) ) 
        if nx2<6 then nx2=6 end
        -- points on each side of the band
        x1,x2 = {},{}
        for i1 = 1,nx1 do x1[i1] = (i1-1)/(nx1-1)*sx end
        for i2 = 1,nx2 do x2[i2] = (i2-1)/(nx2-1)*sx end
        x1[nx1+1] = x1[nx1] -- just a trick to manage last triangle without thinking
        x2[nx2+1] = x2[nx2]
        -- start on the left
        local i1,i2 = 1,1
        c = 1    -- starting color
        local continue = true
        local n,nMax = 0,0
        nMax = nx*2+1
        while continue do
            -- center of the 2 current segments
            m1 = (x1[i1]+x1[i1+1])/2
            m2 = (x2[i2]+x2[i2+1])/2
            if m1<=m2 then -- the less advanced base makes the triangle
        vertices[k+1] = vec3(   x1[i1], sy*y    , 1)  - center
        vertices[k+2] = vec3( x1[i1+1], sy*y    , 1)  - center
        vertices[k+3] = vec3(   x2[i2], sy*(y+1), 1)  - center
        texCoords[k+1] = vec2(   x1[i1]/2*flip, sy*y    ) 
        texCoords[k+2] = vec2( x1[i1+1]/2*flip, sy*y    )
        texCoords[k+3] = vec2(   x2[i2]/2*flip, sy*(y+1))
        if i1<nx1 then i1 = i1 +1 end
            else
        vertices[k+1] = vec3(   x1[i1], sy*y    , 1) - center
        vertices[k+2] = vec3(   x2[i2], sy*(y+1), 1) - center
        vertices[k+3] = vec3( x2[i2+1], sy*(y+1), 1) - center
        texCoords[k+1] = vec2(   x1[i1]/2*flip, sy*y    )
        texCoords[k+2] = vec2(   x2[i2]/2*flip, sy*(y+1))
        texCoords[k+3] = vec2( x2[i2+1]/2*flip, sy*(y+1))
        if i2<nx2 then i2 = i2 +1 end        
            end
            -- set the triangle color
            if c==1 then col=color1 else col=color2 end
        colors[k+1] = col
        colors[k+2] = col
        colors[k+3] = col
            if c==1 then c=2 else c=1 end
            if i1==nx1 and i2==nx2 then continue=false end
        -- increment index for next triangle
        k = k + 3
        n = n + 1
        if n>nMax then continue=false  end -- just in case of infinite loop
        end
    end   
    return vertices,colors,texCoords
end
 
 
function Sphere:warpVertices(input)
    -- move each vector to its position on sphere
    local verts = input.verts
    local rad = 180/math.pi
    local xangle = -input.xangle/rad
    local yangle = -input.yangle/rad
    for i,v in ipairs(verts) do
        verts[i] = vec3(0,0,1):rotate(xangle*v[2],1,0,0):rotate(yangle*v[1],0,1,0)
    end
    return verts
end
 
function Sphere:draw(c,s)
    c = c or self.centre
    s = s or self.radius or 100
    pushMatrix()
    translate(c)
    scale(s)
    self.ms:draw()  
    popMatrix()    
end
 
function SetupCloud()
    cloud=mesh()
    local x,y=300,200
    cloud:addRect(0,0,x,y)
    cc=1
    counter=125
end
 
function DrawCloud()
    if counter+cc>240 or counter+cc<125 or math.random()<0.01 then cc=-cc end 
    counter=counter+cc
    cloud:setColors(color(255,255,255,counter))
    pushMatrix() translate(0,0,-40) cloud:draw() popMatrix()
    pushMatrix() translate(0,0,10) cloud:draw() popMatrix()
end