327 lines
7.4 KiB
Lua
327 lines
7.4 KiB
Lua
--Render and simulate 1D wave equation.
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local love = love
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local N = 33
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local SOUNDSPEED
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local IMPULSESIZE
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local DAMPING
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local old, cur, new --States add beginning of last tick, current tick, and next tick respectively.
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local function Current() return cur end
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local function Next() return new end
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local Integrate
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local Interpolate
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local Derivative
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local SecondDerivative
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local DFT
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local shader = love.graphics.newShader([[
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uniform float re[33];
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uniform float im[33];
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uniform float score;
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//Slow IDFT
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float r( float x )
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{
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float r = re[0];
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for( int k = 1; k < 17; k++ )
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{
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float c = cos( x * float( k ) );
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float s = sin( x * float( k ) );
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r +=
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+ c * re[k]
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- s * im[k]
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+ c * re[33 - k]
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+ s * im[33 - k];
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}
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return r;
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}
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vec4 effect(vec4 color, Image tex, vec2 texture_coords, vec2 screen_coords)
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{
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vec2 p = (3.0 * screen_coords - 1.5 * love_ScreenSize.xy ) / love_ScreenSize.y;
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p.y = -p.y;
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float r = r( atan(p.y, p.x) ) - length( p );
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float q = float( r < 0.05 ) * clamp( 1.0 - score, 0.0, 1.0 ) ;
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return vec4( q + (1.0 + clamp( score, 0.0, 1.0 ) * r * r * 0.3) * color.xyz, float(r > 0.0) ) ;
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}
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]])
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--Calculate discrete fourier transform of radius function.
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do
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local twiddlec = {}
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local twiddles = {}
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for i = 1, N do
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twiddlec[i], twiddles[i] = math.cos( - 2.0 * ( i - 1 ) * math.pi / N ), math.sin( - 2.0 * (i - 1) * math.pi / N )
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end
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local function Twiddle( j )
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j = 1 + j % N
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return twiddlec[j], twiddles[j]
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end
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--Slow Discrete Fourier Transform of a real sequence.
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DFT = function( wave )
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local seq = wave.radii
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local dre, dim = wave.dftre, wave.dftim
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for k = 0, N - 1 do
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local x, y = 0, 0 --Fourier coefficients in bin k.
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for n = 0, N - 1 do
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local cos, sin = Twiddle( n * k )
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x = x + seq[n + 1] * cos
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y = y + seq[n + 1] * sin
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end
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dre[k + 1], dim[k + 1] = x / N , y / N
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end
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end
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end
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--Minimal-oscillation interpolant of a real function from its discrete Fourier coefficients.
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Interpolate = function( wave, x )
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local re, im = wave.dftre, wave.dftim
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local y = re[1]
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for k = 1, N / 2 do
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local c, s = math.cos( x * k ), math.sin( x * k )
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y = y
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+ c * re[k + 1]
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- s * im[k + 1]
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+ c * re[N - k + 1]
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+ s * im[N - k + 1]
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end
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return y
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end
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--Derivative of the interpolation.
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Derivative = function( wave, x )
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local re, im = wave.dftre, wave.dftim
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local y = 0
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for k = 1, N / 2 do
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local c, s = k * math.cos( x * k ), k * math.sin( x * k )
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y = y
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- c * im[k + 1]
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- s * re[k + 1]
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+ c * im[N - k + 1]
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- s * re[N - k + 1]
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end
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return y
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end
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--Second derivative of the interpolation.
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SecondDerivative = function( wave, x )
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local re, im = wave.dftre, wave.dftim
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local y = 0
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for k = 1, N / 2 do
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local c, s = -k * k * math.cos( x * k ), -k * k * math.sin( x * k )
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y = y
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+ c * re[k + 1]
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- s * im[k + 1]
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+ c * re[N - k + 1]
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+ s * im[N - k + 1]
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end
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return y
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end
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--Bandlimited impulse located at angle theta.
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local function AliasedSinc( theta, x )
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local n, d = math.sin( N * 0.5 * ( x - theta ) ), N * math.sin( 0.5 * ( x - theta ) )
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if d < 0.0001 and d > -0.0001 then return 1.0 end
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return n / d
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end
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--Apply bandlimited impulse to wave, adjust free parameters according to game state.
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local function OnImpact( impact, level )
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IMPULSESIZE = 10.0 * ( level - 2.0) / 120.0
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SOUNDSPEED = 25 - 10 * level / 120
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DAMPING = 0.01 * ( 1.0 - 0.4 * level / 120 )
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--Apply bandlimited impulse.
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local r = cur.radii
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local theta = impact.th
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local magnitude = IMPULSESIZE * impact.speed
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local dt = 1.0 / 120.0
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for i = 0, N - 1 do
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r[ i + 1 ] = r[ i + 1 ] + dt * magnitude * AliasedSinc( theta, 2.0 * math.pi * i / N )
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end
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--We've updated the positions, now we need to take a DFT
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--in order to get the bandlimited second spatial derivative.
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cur:DFT()
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return Integrate( dt )
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end
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local mt = { __index = {
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DFT = DFT,
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Interpolate = Interpolate,
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Derivative = Derivative,
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SecondDerivative = SecondDerivative,
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Impulse = Impulse } }
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local function Wave( )
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local t = {
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--radii[k] = radius of point on curve at angle (k - 1) / N
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radii = {},
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--SPACE DFT of radius function (which is periodic)
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dftre = {},
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dftim = {},
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}
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for i = 1, N do
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t.radii[i] = 1.0
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end
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DFT( t )
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return setmetatable(t, mt)
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end
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local function Draw( score )
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-- Blue circle.
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love.graphics.setColor( 91 / 255, 206 / 255, 250 / 255 )
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--Black circle.
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if options["high contrast"].value then
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love.graphics.setColor( 0, 0, 0 )
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end
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shader:send( "re", unpack( cur.dftre ) )
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shader:send( "im", unpack( cur.dftim ) )
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shader:send( "score", score )
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love.graphics.setShader( shader )
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love.graphics.circle("fill", 0, 0, 1.5)
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local t = love.timer.getTime()
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love.graphics.setShader( )
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-- Debug dots.
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--[[
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for i = 1, N do
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local th = ( i - 1 ) * 2.0 * math.pi / N
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local cx, cy = cur.radii[i] * math.cos( th ), cur.radii[i] * math.sin( th )
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love.graphics.setCanvas()
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love.graphics.setColor( 0, 0, 0, 0.5 )
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love.graphics.circle( "fill", cx, cy, 0.02 )
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for k = 0.1, 1.0, 0.1 do
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--Interpolant.
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love.graphics.setColor( 1.0, 0, 0, 0.7 )
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th = ( i - 1 + k ) * 2.0 * math.pi / N
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local r = cur:Interpolate( th )
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local x, y = r * math.cos( th ), r * math.sin( th )
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love.graphics.circle( "fill", x, y, 0.01 )
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--love.graphics.circle( "fill", th / math.pi - 1.0 , r, 0.01)
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--First derivative.
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--love.graphics.setColor( 0, 1.0, 0, 0.7 )
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r = cur:Derivative( th )
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x, y = r * math.cos( th ), r * math.sin( th )
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--love.graphics.circle( "fill", x, y, 0.01 )
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love.graphics.circle( "fill", th / math.pi - 1.0, r, 0.01)
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--Second derivative.
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love.graphics.setColor( 0, 0, 1.0, 0.7 )
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r = cur:SecondDerivative( th )
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x, y = r * math.cos( th ), r * math.sin( th )
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--love.graphics.circle( "fill", x, y, 0.01 )
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love.graphics.circle( "fill", th / math.pi - 1.0, r, 0.01)
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love.graphics.setColor( 1.0, 1.0, 1.0, 0.2 )
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love.graphics.circle( "fill", 2.0 * ( i + k ) / N - 1.2, 0, 0.02 )
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end
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end
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love.graphics.setColor( 1, 1, 1, 0.5 )
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local r = cur:Interpolate( t )
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local x, y = r * math.cos( t ), r * math.sin( t )
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love.graphics.circle( "fill", x, y, 0.02 )]]
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end
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local function Update( )
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--Deep copy of current state to old state.
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for name, t in pairs( cur ) do
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for i = 1, N do
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old[name][i] = t[i]
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end
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end
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--Deep copy of new state to current state.
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for name, t in pairs( new ) do
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for i = 1, N do
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cur[name][i] = t[i]
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end
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end
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end
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Integrate = function( step )
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for i = 1, N do
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local rxx = cur:SecondDerivative( math.pi * 2.0 * ( i - 1 ) / N )
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local r = ( 1.0 - DAMPING ) * ( 2.0 * cur.radii[i] - old.radii[i] + step * step * SOUNDSPEED * rxx ) --Verlet
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+ DAMPING --Damping: oscillate toward 1.
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--Avoid explosions.
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r = math.max( 0.5, math.min( r, 4.0 ) )
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new.radii[i] = r
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end
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new:DFT( )
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end
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local function DetectCollision( px, py, vpx, vpy )
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local xi, xf = px + vpx * step, py + vpy * step
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end
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local function Reset()
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IMPULSESIZE = 1 / 10.0
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SOUNDSPEED = 5.0
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DAMPING = 0.1 / 1
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old = Wave()
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cur = Wave()
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new = Wave()
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end
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Reset()
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return {
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Reset = Reset,
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Update = Update,
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Integrate = Integrate,
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OnImpact = OnImpact,
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DetectCollision = DetectCollision,
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Draw = Draw,
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Current = Current,
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Next = Next,
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} |