wow nice! here is a nice screensaver us newbs here can appreciate.  Now are the momenta and velocities gaussian distributed?

You can't use a circle to solve that problem because the slope of the slope of the slope won't slope the slope of the sloping, making a sloping mess!  All you can do is use a bezier curve to find the solution, and that's not possible with circles.

Imagine having to mow the lawn or run a tractor/combine harvester in a bezier curve, Bill needs to learn to face reality and think outside the box every once in a while.

This is QB64 man! You don't have to Declare Sub nor "Call" Subs. :)

The IDE doesn't even cap Declare Sub, interesting...

That sure is a lot of liberty, but where is the justice?

Is that not a JustBasic user?  What is the difference between JB and LB -- where is the line drawn between liberty and justice?  Which one would you recommend to a beginner?  In an ideal world, would you prefer that everyone here used JB instead of QB64 instead, the same as I would but for FB?

I am not seeing all that cloud making code being activated which BTW came from here:
https://qb64forum.alephc.xyz/index.php?topic=4615.msg140454#msg140454

Because it happens to be daytime with clear sunny skies

defdbl a-z
dim shared pi as double
pi = 4*atn(1)

sw = 800
sh = 600

screen _newimage(sw, sh, 32)

zoom = 100

a = 0.5
k = 0.09

sx = 2
sy = 1

tx = a*exp(k*16)*cos(16) + sx
ty = a*exp(k*16)*sin(16) + sy

x = a*exp(k*0)*cos(0) + sx - tx
y = a*exp(k*0)*sin(0) + sy - ty

r = sqr(x*x + y*y)
arg = _atan2(y, x)

for tt = r to 0 step -1
xl = tt*cos(arg)
yl = tt*sin(arg)

for yy=0 to sh
for xx=0 to sw
u = (xx - sw/2)/zoom
v = (sh/2 - yy)/zoom

cdiv uu, vv, u - xl, v - yl, u + xl, v + yl
clog uu, vv, uu, vv

mm = sqr(uu*uu + vv*vv)
aa = (pi + _atan2(vv, uu))/(2*pi)
pset (xx, yy), hrgb(aa, mm)
next
next

sleep
next

dim pp(5)
pp(5) = 0
pp(4) = 7
pp(3) = 10
pp(2) = 12
pp(1) = 15
pp(0) = 15.9

for ttt= 0 to 5
tt = pp(ttt)
for yy=0 to sh
for xx=0 to sw
u = (xx - sw/2)/zoom
v = (sh/2 - yy)/zoom

uu = 0
vv = 0
for t=tt to 16 step 0.1
x = a*exp(k*t)*cos(t) + sx - tx
y = a*exp(k*t)*sin(t) + sy - ty

cdiv p, q, 1, 0, x - u, y - v

dx = a*exp(k*t)*(k*cos(t) - sin(t))
dy = a*exp(k*t)*(k*sin(t) + cos(t))

cmul p, q, p, q, dx, dy

uu = uu + 0.1*p
vv = vv + 0.1*q
next

for t=16 - 0.1 to tt step -0.1
x =-a*exp(k*t)*cos(t) - sx + tx
y =-a*exp(k*t)*sin(t) - sy + ty

cdiv p, q, 1, 0, x - u, y - v

dx = a*exp(k*t)*(k*cos(t) - sin(t))
dy = a*exp(k*t)*(k*sin(t) + cos(t))

cmul p, q, p, q, dx, dy

uu = uu + 0.1*p
vv = vv + 0.1*q
next

cmul uu, vv, uu, vv, 0, -1/(2*pi)

mm = sqr(uu*uu + vv*vv)
aa = (pi + _atan2(vv, uu))/(2*pi)
pset (xx, yy), hrgb(aa, mm)
next
next

sleep
next

sleep
system

function hrgb&(h, m)
r =  0.5 - 0.5*sin(2*pi*h - pi/2)
    g = (0.5 + 0.5*sin(2*pi*h*1.5 - pi/2)) * -(h < 0.66)
    b = (0.5 + 0.5*sin(2*pi*h*1.5 + pi/2)) * -(h > 0.33)

    n = 128

    mm = m*10000 mod 500
p = abs((h*n) - int(h*n))

rr = 200*r - 0.15*mm - 35*p
gg = 200*g - 0.15*mm - 35*p
bb = 200*b - 0.15*mm - 35*p

if rr < 0 then rr = 0
if gg < 0 then gg = 0
if bb < 0 then bb = 0

hrgb& = _rgb(rr, gg, bb)
end function

function cosh#(x as double)
    cosh# = 0.5*(exp(x) + exp(-x))
end function

function sinh#(x as double)
    sinh# = 0.5*(exp(x) - exp(-x))
end function

'u + iv = (x + iy)^(a + ib)
sub cexp(u, v, xx, yy, aa, bb)
x = xx
y = yy
a = aa
b = bb
   
    lnz = x*x + y*y
   
    if lnz = 0 then
        u = 0
        v = 0
    else
        lnz = 0.5*log(lnz)
        argz = atan2(y, x)
       
        mag = exp(a*lnz - b*argz)
        ang = a*argz + b*lnz
       
        u = mag*cos(ang)
        v = mag*sin(ang)
    end if
end sub

'u + iv = (x + iy)*(a + ib)
sub cmul(u, v, xx, yy, aa, bb)
x = xx
y = yy
a = aa
b = bb
    u = x*a - y*b
    v = x*b + y*a
end sub

'u + iv = (x + iy)/(a + ib)
sub cdiv(u, v, xx, yy, aa, bb)
x = xx
y = yy
a = aa
b = bb

    d = a*a + b*b
    u = (x*a + y*b)/d
    v = (y*a - x*b)/d
end sub

sub clog(u, v, xx, yy)
x = xx
y = yy

    lnz = x*x + y*y
   
    if lnz = 0 then
        u = 0
        v = 0
    else
u = 0.5*log(lnz)
v = _atan2(y, x)
end if
end sub


I see that you can really improve a lot on this as well. The solution at the beginning of this thread is still sufficient for the purpose for which it was written. I will publish the exact use and the reason why I deal with it soon. It is again just one fragment from a larger program. Thank you all for your many inspiring suggestions!

nejsam to co si myslis, ne, nejsam ja tot

With non-monospaced fonts, the routine has to check and build width pixel by pixel as it goes. A "W" might spread and use 64 pixels, an "i" might use 8...  non-monospaced fonts are *much* slower to calculate print widths with, and that's something I wanted to emphasize with my demo.

Is this a one dimensional form of OCR? the difference in aka screen 0 printing