Curved curve

[honkytonk]

Hello, work:
We have three points aligned.
- If the midpoint = total/2 we want a straight curve
If we have the midpoint > or < total/2 we want a curved curve (exp log ... )
The curvature of which will depend on the percentage deviation of the midpoint from the total/2.
How to do ?

[tsh73]

Draw some examples so folks understand what you want.
(what is "straight curve" anyway?)
so far, there is a single way to put a parabola (ax^2+bx+x) via three points, but midpoint will not be parabola min/max.

[honkytonk]

"Curve" in the sense of graphical representation
A straight curve is a curve non curved (a straight).
The question is: Can we draw an exp or log curve with three points?

[B+]

fANSII graphs of log and exp (e) functions of x over a limit x to create fractions ie,

' graph exp(x) and log(x) up to x = 25  2019-07-28 B+ start mods

for x = 1 to 25
   print x, string65$(log(x) / log(25))
next
Print "OK, that was log(x) / log(25) on a sideways ANSII graph."
Print
input "Press enter to see exp(x) / exp(25)... "; wate$
for x = 1 to 25
   print x, string65$(exp(x) / exp(25))
next
Print "OK, that was exp(x) / exp(25) on a sideways ANSII graph."

function string65$(fraction)
   for i = 1 to 65
       if i/65 < fraction then b$ = b$ + "X"
   next
   string65$ = b$
end function




Attachments:

[honkytonk]

I conclude that either the midpoint is on the curve (log or exp) or it is not there.
If it is not, how to adjust the curve according to the deviation from where it should be to satisfy log or exp.

[B+]

OK here is a look at exp(x):
' Gridlock Graph Dialog.txt for JB [B+=MGA] 2017-03-07
' from graphing fx.txt finished 2017-03-06
' but want grid to align with axis when possible = gridlock!

' from graphing evalB.txt for JB [B+=MGA] 2017-02-22
' and tsh73 Implicit function, x, y screen functions


global pi, rad, deg, Dflag
global xPs, yPs, xmin, ymin, xmax, ymax  'for xScreen, yScreen functions
xPs = 400 : yPs = 400

pi = acs(-1)
rad = pi / 180
deg = 180 / pi
Dflag = 1       'measure angles in degrees else radians

'xmin=-2*pi : xmax=pi : ymin =-2 : ymax=1  'sin(x)
'xmin=-10 : xmax=5 : ymin =-50 : ymax=10  'quadradic x^2 + 2*x - 35
xmin=-2 : xmax=5 : ymin =-1 : ymax=100  'exp(x)

call ShowGraph
print "Done, bye!"

sub ShowGraph
   WindowWidth = 500
   WindowHeight = 500
   UpperLeftX = int((DisplayWidth - WindowWidth) / 2)
   UpperLeftY = int((DisplayHeight - WindowHeight) / 2)
   open "Graph Dialog" for graphics_nsb_nf as #gd
   #gd "font arial 8"
   #gd "trapclose [quitGraphDialog]"
   #gd "down"

   'graph box section
   #gd "color cyan"
   #gd "backcolor lightgray"
   #gd "place ";0;" ";0
   #gd "boxfilled ";xPs;" ";yPs
   #gd "backcolor white"

   x0 = xScreen(0) : y0 = yScreen(0)

   'xgrid lines
   inc = (xmax - xmin) / 20
   #gd "color cyan"
   if 0 <= x0  and x0 <= xPs then
       start = 0
       while start <= xmax
           scan
           #gd "line ";xScreen(start);" ";0;" ";xScreen(start);" ";400
           start = start + inc
       wend
       start = 0
       while start >= xmin
           scan
           #gd "line ";xScreen(start);" ";0;" ";xScreen(start);" ";400
           start = start - inc
       wend
   else
       start = xmin
       for i = 0 to 20
           #gd "line ";i * 20;" ";0;" ";i * 20;" ";400
           start = start + inc
       next
   end if
   #gd "color black"
   #gd "place ";  2;" ";415;";| x min (left) is ";xmin;"."
   #gd "place ";145;" ";430;";| x scale increment is ";inc;"."
   #gd "place ";170;" ";445;";| x middle is ";xmin + 10 * inc;"."
   #gd "place ";325;" ";460;";| x max (right) is ";xmax;"."

   'y grid lines
   inc = (ymax - ymin) / 20
   if 0 <= y0  and y0 <= yPs then
       start = 0
       while start <= ymax
           scan
           #gd "color cyan"
           #gd "line ";0;" ";yScreen(start);" ";400;" ";yScreen(start)
           #gd "color black"
           #gd "place ";405;" ";yScreen(start);";|";start
           start = start + inc
       wend
       start = 0
       while start >= ymin
           scan
           #gd "color cyan"
           #gd "line ";0;" ";yScreen(start);" ";400;" ";yScreen(start)
           #gd "color black"
           #gd "place ";405;" ";yScreen(start);";|";start
           start = start - inc
       wend
   else
       start = ymin
       for i = 0 to 20
           #gd "color cyan"
           #gd "line ";0;" ";i * 20;" ";400;" ";i * 20
           #gd "color black"
           #gd "place ";405;" ";400 -(i*20);";|";start
           start = start + inc
       next
   end if

   #gd "size 2"

   'x, y axis if in graph
   #gd "color yellow"
   if 0 <= y0  and y0 <= yPs then
       #gd "line ";xScreen(xmin);" ";y0;" ";xScreen(xmax);" ";y0
   end if
   if 0 <= x0  and x0 <= xPs then
       #gd "line ";x0;" ";yScreen(ymin);" ";x0;" ";yScreen(ymax)
   end if

   'fx graph
   #gd "color blue"
   stepper = (xmax - xmin)/xPs
   for xfx = xmin to xmax step stepper
       if xfx <= xmin then
           lastx = xScreen(xfx) : lasty = yScreen(fx(xfx))
       else
           x = xScreen(xfx) : y = yScreen(fx(xfx))
           if x > 0 and x < xPs and y > 0 and y < yPs then
               if lastx > 0 and lastx <= xPs and lasty > 0 and lasty <= yPs then
                   #gd "line ";lastx;" ";lasty;" ";x;" ";y
               end if
           end if
       end if
       lastx = x : lasty = y
   next
   wait

[quitGraphDialog]
   close #gd
end sub

'conversions (logical coords to screen) tsh73 2017-03-04
function xScreen(x)
'global xPs, yPs, xmin, ymin, xmax, ymax  'for xScreen, yScreen functions
   xScreen = (x - xmin) / (xmax - xmin) * xPs
end function

function yScreen(y)
'global xPs, yPs, xmin, ymin, xmax, ymax  'for xScreen, yScreen functions
   yScreen = (1 - (y - ymin) / (ymax - ymin)) * yPs
end function

function fx(x) 'any general function of x
   'fx = sin(x)      'xmin=-2*pi : xmax=2*pi : ymin =-2 : ymax=2
   'fx = x^2 +2*x -35 'roots 5, -7  xmin=-10 : xmax=10 : ymin =-10 : ymax=10
   fx = exp(x)
end function


Pick two end points for midpoint are you talking midpoint on x axis or the function exp(midpoint)?

[honkytonk]

Thanks for exp, I'll see what I can do with it, véry thanks

[tsh73]

Hello there
I was able to fit exponent over three points using Excel
exponent presented in a form
a*exp(b*x)+c
so we have 3 parameters to fiddle with.
Excel minimizes sum of square distances from given points to function values
It works if points curve up
If points curve down we can change EXP to LOG and it will work.
Now, for it to work in JB someone have to white "minimize f(x,y,x) by three variables".
Now pictures.
Source one

fitted one

and one with LOG (curved down)


EDIT
Oh, I forgot. I's excel "Solver" add-on. You set it to Minimize of yellow cell (with sum of squared differences)
changing parameter values (cyan ones). Just press "Solve", and it will get parameters (or says it cannot).

[tsh73]

Now, for it to work in JB someone have to white "minimize f(x,y,x) by three variables".

I kind of did it.
Right word is "Coordinate descent" - I did something along it, highly inefficiently - but it does work.
(really should use something better)
It goes by finding minimum along one coordinate (parameter), then next one, ...
(actually I know that supposed minimum is 0 so end condition is "difference < epsilon")

For the first example
Numbers are different (obviously Excel uses bigger precision - smaller epsilon. Also likely more clever algorithm) but on the graph (I put numbers I got into same Excel) difference of 1e-2 or 1e-10 is not visible at all anyway.
Results:
i> x y    f                 df
1> 1 1    3.71828183        2.71828183
2> 2 2    8.3890561         6.3890561
3> 3 4    21.0855369        17.0855369
340.124666> 1 1 1
eps = 0.0001
0.99918639e-4> 0.4664 0.71188 0.05548
accuracy reached
i> x y    f                 df
1> 1 1    1.00591867        0.59186692e-2
2> 2 2    1.99230175        -0.76982508e-2
3> 3 4    4.00237169        0.23716933e-2
fitted function
0.4664*exp(0.71188*x)+0.05548

code:
'Coordinate descent for exponent fitting
'tsh73 Aug 2019
global N
N=3
'x, y pairs to fit
data 1,1, 2,2, 3,4
for i=1 to N
   read x, y
   x(i)=x: y(i)=y
next

for i = 1 to N
   q(i)=1  'initial coeffts
next

call showDiff
call showCoeffts

eps = 0.0001
print "eps = ";eps

h0=.1
d0=d()
k=0 'not valid, will be fixed next line
while 1
   k = (k) mod N +1 'next coordinate to optimize along
   h=h0
   'q(1)=0.1
   while 1
       scan
       'try to fiddle with q(k)
       'is + move any good?
       q(k)=q(k)+h
       d1=d()
       if d1<d0 then 'good move
           d0=d1
           goto [nxt]
       else    '+ is bad move, is - any good?
           q(k)=q(k)-2*h
           d2=d()
           if d2<d0 then 'good move
               d0=d2
               goto [nxt]
           else    'no good moves, reset to initial and decrease h
               q(k)=q(k)+h
               if h < eps then exit while  'no better for q(k)
               'else
               h=h/10
           end if
       end if
   [nxt]
   'call showCoeffts
   'a$=input$(1)
   wend

   locate 1,7  'overprint. Comment to see long list
   call showCoeffts
   'a$=input$(1)   'uncomment for pause
   dd=d()
   'if abs(dd-lastD)<eps^2 then print "stops converging": exit while
   if dd<eps then print "accuracy reached": exit while
   lastD=dd

wend

call showDiff
print "fitted function"
print q(1);"*exp(";q(2);"*x)+";q(3)

end

sub showDiff
   print "i";"> "; "x";" "; "y";"    "; "f",, "df"
   for i=1 to N
       print i;"> "; x(i);" "; y(i);"    ";  f(x(i)), f(x(i))-y(i)
   next
end sub

sub showCoeffts 'so can step over in debug
   print d();"> ";
   for i=1 to N
       print q(i);" ";
   next
   print
end sub

function d()    'sum of differences squared
   for i=1 to N
       d=d+(f(x(i))-y(i))^2
   next
end function

function f(x)   'paramerised function to fit data ponts
   f=q(1)*exp(q(2)*x)+q(3)
end function

Not much success with LOG variant. It tends to loop - condition "stops converging" is against that.
Again, numbers are different; difference is bigger then with EXP, but looks OK on a graph

[honkytonk]

I tried the fitted function

    NOMAINWIN
   WindowWidth = 600: WindowHeight = 400
   UpperLeftX = 200: UpperLeftY = 20
   BUTTON #w.go, "Go !", [go], UL, 20, 20, 40, 20
   GRAPHICBOX #w.m 5, 5, 585, 360
   OPEN "test" FOR window_nf AS #w
   #w, "TRAPCLOSE [closetest]"
   #w.m  "down;fill darkblue;color white;backcolor darkblue"
   #w.m, "size 2"
   wait
[go]
   for x=1 to 500
     y=0.4664*exp(0.71188*x)+0.05548 'fitted function
    ' #w.m, "place ";x;" ";y: #w.m, "\ ."
     #w.m, "set ";x;" ";y
   next x
wait
 [closetest]
   CLOSE  #w
   END


[tsh73]

I tried the fitted function

points to fit are
data 1,1, 2,2, 3,4
so you need X in 1..3 (or 0..4)
Much scaling needed.
You have x going up to 500
exp(500) is VERY big - drawing such Y clips several times (since "comb" on your graph)

Here's I used plot program from
alycesrestaurant.com/lbpe/Graphics101plotting.html
and added points


nomainwin
global winW, winH, xmin, xmax, ymin, ymax
open "test" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down"

#gr "home"
#gr "posxy w h"
winW=2*w: winH=2*h
'f(x)=1.5*x^2-2*sin(5*x), x in [-2,3]
xmin=0: xmax=4

nPoints=winW    'we have only this much screen dots in X range
dx=(xmax-xmin)/nPoints   'so this will be step in math coordinates

'now, to get ymin, ymax we have to loop
ymin=f(xmin)
ymax=ymin
for x=xmin to xmax step dx
   y=f(x)
   if ymin > y then ymin = y
   if ymax < y then ymax = y
next

'now we just - plot function. Note same loop
#gr "color red"
#gr "size 3"
y=f(xmin)
#gr "set ";sx(xmin);" ";sy(y)  'just set first dot
for x=xmin to xmax step dx
   y=f(x)
   #gr "goto ";sx(x);" ";sy(y)  'then connect dots
next

'and finally, add axis
#gr "color black"
#gr "size 1"
#gr "line ";sx(xmin);" ";sy(0);" ";sx(xmax);" ";sy(0)
#gr "line ";sx(0);" ";sy(ymin);" ";sx(0);" ";sy(ymax)

'labeling
#gr "place ";sx(0)+5;" ";sy(0)-5
#gr "\0,0"
#gr "place ";sx(xmax)-20;" ";sy(0)-5
#gr "\X"
#gr "place ";sx(0)+5;" ";sy(ymax)+20
#gr "\Y"

'add points
#gr "color blue"
data 1,1, 2,2, 3,4
for i = 1 to 3
   read x, y
   #gr "place ";sx(x);" ";sy(y)
   #gr "circle 5 "
next

#gr "flush"
#gr, "getbmp drawing 1 1 ";winW;" ";winH
bmpsave "drawing", "graph.bmp"

wait

[quit]
   close #gr
   end

'"any" function. You can change it as you like
function f(x)
   f=0.4664*exp(0.71188*x)+0.05548 'fitted function
end function

'To translate X from interval [a,b] to [c,d] we'll do (X-a)/(b-a)*(d-c)+c.
'create two functions: sx(x) and sy(y)
function sx(x)
   sx=(x-xmin)/(xmax-xmin)*winW
end function
function sy(y)
   sy=winH-(y-ymin)/(ymax-ymin)*winH   'Y is inverted, so winH-...
end function


[honkytonk]

Thank you tsh73.
I have work for a year; thank you so much.