Three Bezier Functions

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Andy_A	Created : 17 November 2016 System : Windows Language : Blitz Three Bezier Functions Compare functions for speed
I needed a quick cubic Bezier function, and have used all three. The moshplant (website) and forward differencing algos are the faster functions. The forward differencing has a very slight edge over moshplant. At any rate, here they are. ;Comparison of three Bezier curve drawing functions AppTitle "Comparison of Three Bezier Curve Drawing Functions" Graphics 900, 600, 0, 2 SetBuffer BackBuffer() SeedRnd MilliSecs() Dim x(2,3), y(2,3) Local i%, iterations%, numPnts%, st0%, st1%, st2%, et0%, et1%, et2%, iter$ ;gen 4 random control points for the Bezier, store values + offsets in an array For i = 0 To 3 x(0, i) = Rand(0,299) ;points in left third of screen y(0, i) = Rand(100,599) x(1, i) = x(0, i) + 300 ;points in middle third of screen y(1, i) = y(0, i) x(2, i) = x(0, i) + 600 ;points in right third of screen y(2, i) = y(0, i) Next ;Draw control point and tangents For i = 0 To 3 Color 60,60,60 If i < 3 Then Line( x(0,i),y(0,i),x(0,i+1),y(0,i+1) ) Line( x(1,i),y(1,i),x(1,i+1),y(1,i+1) ) Line( x(2,i),y(2,i),x(2,i+1),y(2,i+1) ) End If Color 255,0,0 Oval x(0,i)-2, y(0,i)-2,5,5,True Oval x(1,i)-2, y(1,i)-2,5,5,True Oval x(2,i)-2, y(2,i)-2,5,5,True Next Color 255,255,255 Flip iterations = 5000 ;number of iterations to perform for each Bezier drawing function numPnts = 100 ;number of points calculated to draw Bezier curve ;Regular Beziers LockBuffer() st0 = MilliSecs() For i = 1 To iterations bezier( x(0,0), y(0,0), x(0,1), y(0,1), x(0,2),y(0,2), x(0,3), y(0,3), numPnts) Next UnlockBuffer() Flip et0 = MilliSecs()-st0 ;Re-factored Beziers LockBuffer() st1 = MilliSecs() For i = 1 To iterations mBezier( x(1,0), y(1,0), x(1,1), y(1,1), x(1,2),y(1,2), x(1,3), y(1,3), numPnts) Next UnlockBuffer() Flip et1 = MilliSecs()-st1 ;Forward Differencing Beziers LockBuffer() st2 = MilliSecs() For i = 1 To iterations fwdDiff( x(2,0), y(2,0), x(2,1), y(2,1), x(2,2),y(2,2), x(2,3), y(2,3), numPnts) Next UnlockBuffer() Flip et2 = MilliSecs()-st2 iter = Str(iterations) Text 5, 0, iter+" Regular Bezier function calls, elapsed time: "+et0+"ms ("+(Float(iterations)/Float(et0)1000.)+" Beziers/sec)" Text 5,15, iter+" Moshplant Bezier function calls, elapsed time: "+et1+"ms ("+(Float(iterations)/Float(et1)1000.)+" Beziers/sec)" Text 5,30, iter+" Forward differencing Bezier function calls, elapsed time: "+et2+"ms ("+(Float(iterations)/Float(et2)1000.)+" Beziers/sec)" WaitMouse() End Function bezier (x0%,y0%, x1%,y1%, x2%,y2%, x3%,y3%, segs%=24) ;Source: https://www.wikipedia.org/wiki/Bezier_curve ;The parametric form of the curve is: ;P(t) = A(1 - t)3 + 3Bt(1 - t)2 + 3Ct2(1 - t) + Dt3 Where 0.0 <= t <= 1.0 Local xt#, xp#, xi#, xp2#, xi2#, t1#, t2#, t3#, t4# Local tmpX1%, tmpY1%, tmpX2%, tmpY2% xt = 1.0 / Float(segs) ;segs = number of line segments defining Bezier curve xp = 1.0 - xi tmpX1 = x0 tmpY1 = y0 While xi < 1.0-xt xi = xi + xt xp = 1.0 - xi xp2 = xp xp xi2 = xi * xi t1 = xp2 * xp t2 = xp2 * xi * 3.0 t3 = xi2 * xp * 3.0 t4 = xi2 * xi tmpX2 = (t1 * x0) + (t2 * x1) + (t3 * x2) + (t4 * x3) tmpY2 = (t1 * y0) + (t2 * y1) + (t3 * y2) + (t4 * y3) Line tmpX1, tmpY1, tmpX2, tmpY2 tmpX1 = tmpX2 tmpY1 = tmpY2 Wend End Function Function mBezier(x0%,y0%, x1%,y1%, x2%,y2%, x3%,y3%, segs%=24) ;Darrel Plant - https://www.moshplant.com/direct-or/bezier/math.html ;re-factored Adobe PostScript references Local rate#, t#, t2#, t3#, cx#, bx#, ax#, cy#, by#, ay# Local oldx%, oldy%, newx#, newy# ;segs is the number of line segments used to draw Bezier rate = 1.0/Float(segs) : t = rate oldx = x0 : oldy = y0 ;derive the coefficients cx = 3.0Float(x1-x0) bx = 3.0Float(x2-x1)-cx ax = Float(x3-x0)-cx-bx cy = 3.0Float(y1-y0) by = 3.0Float(y2-y1)-cy ay = Float(y3-y0)-cy-by ;iterate line segments indicated by rate While t<=1.0 t2 = tt : t3 = t2t newx = axt3+bxt2+cxt+x0 newy = ayt3+byt2+cyt+y0 Line oldx, oldy, newx, newy oldx = newx : oldy = newy t = t + rate Wend End Function Function fwdDiff(x0%,y0%, x1%,y1%, x2%,y2%, x3%,y3%, segs% = 24) ;Hannu KankaanpÃ¤Ã¤ - https://www.niksula.hut.fi/~hkankaan/Homepages/bezierfast.html ;Bezier generated using 'Forward Differencing' Local fx#, fdx#, fddx#, fdddx#, fddxPer2#, fdddxPer2#, fdddxPer6# Local fy#, fdy#, fddy#, fdddy#, fddyPer2#, fdddyPer2#, fdddyPer6# Local t#, temp# Local i%, rx%, ry% t = 1.0/Float(segs) temp = t * t fx = x0 fdx = 3. * (x1-x0) * t fddxPer2 = 3. * (x0 - 2. * x1 + x2) * temp fdddxPer2 = 3. * (3. * (x1 - x2) + x3 - x0) * temp * t fdddx = fdddxPer2 + fdddxPer2 fddx = fddxPer2 + fddxPer2 fdddxPer6 = fdddxPer2/3. fy = y0 fdy = 3. * (y1-y0) * t fddyPer2 = 3. * (y0 - 2. * y1 + y2) * temp fdddyPer2 = 3. * (3. * (y1 - y2) + y3 - y0) * temp * t fdddy = fdddyPer2 + fdddyPer2 fddy = fddyPer2 + fddyPer2 fdddyPer6 = fdddyPer2/3. rx = x0: ry = y0 For i = 0 To segs-1 fx = fx + fdx + fddxPer2 + fdddxPer6 fdx = fdx + fddx + fdddxPer2 fddx = fddx + fdddx fddxPer2 = fddxPer2 + fdddxPer2 fy = fy + fdy + fddyPer2 + fdddyPer6 fdy = fdy + fddy + fdddyPer2 fddy = fddy + fdddy fddyPer2 = fddyPer2 + fdddyPer2 Line(rx, ry, fx, fy) rx = fx: ry = fy Next End Function --v

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