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If you think of yourself walking along a line graph- much as in turtle graphics- then 'curvature' is experienced as how quickly your direction changes. At any given point an 'osculating circle' defines the curvature by its radius.
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The nearer we are to a straight line, the larger this radius is, For a straight line it becomes infinite. So if we take the reciprocal we instead get a number which is zero for a straight line, and increasingly big for rapidly changing lines.
I've had fun drawing graphics involving osculating circles, and also playing with colours code according to radius of curvature or its inverse, the curvature,
This last one uses colour coding of the curvature, which is also size-coded. Not very successfully!
nomainwin
WindowWidth =800
WindowHeight =800
pi =4 *atn( 1)
radius =32
global col$, rd, gn, bu
graphicbox #w.g1, 1, 1, 800, 800
open "Kiss me quick" for graphics_nsb as #w
#w "trapclose quit"
#w.g1 "down ; fill darkblue ; color cyan"
n =0
for th =0 -0.005 to 32.05 *pi step 0.01
x =radius *cos( th)
y =radius *sin( th)
pres$ =str$( x) +" " +str$( y)
#w.g1 "size 1 ; set "; 400 +10 *x; " "; 390 -10 *y
if ( th >0) and ( ( n mod 10) =0) then
print using( "###.##", th); " "; using( "###.##", x) , using( "###.##", y),
result$ =smallCircleThru$( lastButOne$, last$, pres$) ' find oscu. circle centre and radius
xC =int( 400 +10 *val( word$( result$, 1, " ")))
yC =int( 360 -10 *val( word$( result$, 2, " ")))
rC =int( 10 *val( word$( result$, 3, " ")))
call hsv2rgb 359 *th /6, 0.79, 0.99
x1 =val( word$( lastButOne$, 1, " ")): y1 =val( word$( lastButOne$, 2, " ")) ' the three points on curve
x2 =val( word$( last$ , 1, " ")): y2 =val( word$( last$, 2, " "))
x3 =val( word$( pres$, 1, " ")): y3 =val( word$( pres$, 2, " "))
gradChord =( y3 -y1 ) /( x3 -x1)
angRad =atn( 1 /gradChord)
'print using( "###.##", gradChord); " "; using( "###.##", angRad),
radSmall =rC /10
xSmallCirc =x2 -radSmall *cos( angRad)
ySmallCirc =y2 -radSmall *sin( angRad)
'print using( "###.##",xSmallCirc); " "; using( "###.##", ySmallCirc); " "; using( "###.##", radSmall),
#w.g1 "backcolor "; col$
#w.g1 "color black"
'print "( "; col$; ")"
'#w.g1 "size 1 ; place "; 400 +10 *xSmallCirc; " "; 360 -10 *ySmallCirc
#w.g1 "size 1 ; place "; 400 +10 *x; " "; 390 -10 *y
#w.g1 "size 1 ; circlefilled "; radSmall
end if
'print
lastButOne$ =last$
last$ =pres$
radius =radius *0.9998
if radius <=0 then exit for
n =n +1
scan
'timer 100, [drop]
'wait
[drop]
next th
call saveScreen
wait
sub quit h$
close #h$
end
end sub
function smallCircleThru$( a$, b$, c$) ' circle through x1,y1 x2,y2 x3,y3
x1 =val( word$( a$, 1, " ")): y1 =val( word$( a$, 2, " "))
x2 =val( word$( b$, 1, " ")): y2 =val( word$( b$, 2, " "))
x3 =val( word$( c$, 1, " ")): y3 =val( word$( c$, 2, " "))
A =x1 *( y2 -y3) -y1 *(x2 -x3) +x2 *y3 -x3 *y2
B =( x1^2 +y1^2) *( y3 -y2) +( x2^2 +y2^2) *(y1 -y3) +( x3^2 +y3^2) *( y2 -y1)
C =( x1^2 +y1^2) *( x2 -x3) +( x2^2 +y2^2) *(x3 -x1) +( x3^2 +y3^2) *( x1 -x2)
D =( x1^2 +y1^2) *( x3 *y2 -x2 *y3) +( x2^2 +y2^2) *( x1 *y3 -x3 *y1) +( x3^2 +y3^2) *( x2 *y1 -x1 *y2)
xC =0 -B /2 /A
yC =0 -C /2 /A
R =( ( B^2 +C^2 -4 *A *D) /( 4 *A^2))^0.5
smallCircleThru$ =str$( xC) + " " +str$( yC) +" " +str$( R)
end function
sub saveScreen
#w.g1 "flush"
#w.g1 "getbmp scr 0 0 790 770"
bmpsave "scr", "scrJHF" +str$( time$( "seconds")) +".bmp"
#w.g1 "cls ; drawbmp scr 0 0"
end sub
sub hsv2rgb h, s, v ' hue 0-360, saturation 0-1, value 0-1
c =v *s ' chroma
h =h mod 360
x =c *( 1 -abs( ( ( h /60) mod 2) -1))
m =v -c ' matching adjustment
select case
case h < 60
r = c: g = x: b = 0
case h <120
r = x: g = c: b = 0
case h <180
r = 0: g = c: b = x
case h <240
r = 0: g = x: b = c
case h <300
r = x: g = 0: b = c
case else
r = c: g = 0: b = x
end select
rd = abs( int( 256 *( r + m)))
gn = abs( int( 256 *( g + m)))
bu = abs( int( 256 *( b + m)))
col$ =right$( " 0" +str$( rd), 3) +" " +right$( " 0" +str$( gn), 3) +" " +right$( " 0" +str$( bu), 3)
end sub