From: Subject: Smoothing algorithm using Bezier Curves Date: Mon, 10 Dec 2007 22:51:19 -0500 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C83B7F.2DBB8AD0" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3198 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C83B7F.2DBB8AD0 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.efg2.com/Lab/Graphics/Jean-YvesQueinecBezierCurves.htm Smoothing algorithm using Bezier Curves

Graphics

=

=20


Smoothing Algorithm Using B=E9zier Curves

by Jean-Yves = Qu=E9inec=20


Introduction

I needed a smoothing algorithm for my LQprofil program.=20 LQprofil is designed for scale modelers, who have to draw and = print=20 airfoils templates, i.e., a plate cut to the shape used to build scaled = airfoils=20 sections. An airfoils section (such as NACA airfoils) is defined by an = array of=20 points. I tried existing routines, but none gave good results for me. = Some of=20 them produced less or excessive smoothed curves, and frequently the = smoothed=20 curves didn't go through the defined points. I decided to create my = own=20 algorithm, with a smoothing factor, and draw curves exactly through the = points.=20 B=E9zier curves seemed to be suitable to solve the problem.=20

B=E9zier Curves

Pierre B=E9zier was a French engineer who was working the Renault = company.=20 B=E9zier equations uses 4 points: two points of the curve (P1 and P4) = and two=20 control points (see the diagram below).  P2 is located at the = "right" side=20 of P1, and P3 is located at the "left" side of P4.=20

B=E9zier equations are parametric equations  in a variable = t, and=20 are symmetrical relative to x and y:=20

Bx(t) =3D (1-t)3 =  x P1x + 3 x=20 (1-t)2 x t x P2x + 3 x (1-t) x t2 x P3x + = t3 x=20 P4x
By(t) =3D (1-t)3  x P1y + 3 x (1-t)2 = x t x P2y=20 + 3 x (1-t) x t2 x P3y + t3 x P4y
=20

=20

The above figure shows a geometrical construction for points P1, P2, = P3, and=20 P4 with t =3D 0.5.=20

The t parameter (varying from 0 to 1) cuts the P1-P4 segment = to=20 intervals, according to the wanted accuracy.
When t =3D 0 = results are=20 B.x =3D P1.x and By=3D P1.y
When t =3D 1 results are B.x =3D = P4.x and By=3D P4.y=20

If one has NB1 points for the curve, one needs a total of NB2 =3D = NB1*3 +1=20 points, because between each point P of the curve we have to build two = control=20 points named PC1 and PC2. The points of the curve, and associated = control=20 points, are stored in the BB array of Tpoints BB :=3D array[0..NB2] of = Tpoints.=20

One can say that a P point of the curve, which is not an end point, = has a=20 prior "left" control point of P2 type, and has a next "right" control = point of=20 P1 type. The B=E9zier curve goes through the P1 and P4 points, and the = curve=20 appearance depends on the two control points, P2 and P3. To solve the = curve=20 smoothing problem, we have to find a method for building the control = points P2=20 and P3.=20

First approach

=20

To smooth the Pn edges of the green polyline, one can intuitively = imagine=20 that the smoothing curve should be tangent to a segment, which is = parallel to=20 the segment (Pn-3, Pn+3). It joins the prior and next points relative to = the=20 current point. The above figure shows how to draw the PC1 control point, = located=20 to the "right" of the curve point Pn. S1a is the middle of the Pn = S1=20 segment and PC1 is located on the segment (S1a , Pn+3). The length of = the (S1a ,=20 PC1) segment is about 1/3 of the (S1, Pn+3) segment length.=20

In brief :=20

The particular case of the end points (n=3D 0 and n =3D NB2) offers 2 = possibilities depending on whether the curve is closed or open. When the = curve=20 is closed, P(0) =3D P(NB), one can execute the smoothing algorithm = because the=20 next and prior points are known. When the curve is open P(0) has no = prior PC2=20 point. We shall consider that PC2 and P(0) are the same. Idem for the = PC1 conrol=20 point of the last point P(NB2).

This method gives some good results with simple curves, but doesn't = work fine=20 in all cases. For example, a hexagon is transformed into a potato=EFd = (looks like=20 a potato). The next step will try find a more flexible solution. For = example,=20 the algorithm should be able to transform progressively a square to a = circle.=20

Geometrical control points=20 construction

This geometrical solution is completely empirical, but = gives=20 excellent results.=20

=20

  1. S1 is the symmetrical point of P[n-1] in relation to P[n]=20
  2. S2 is the symmetrical point of P[n+2] in relation to P[n]=20
  3. S1a and S2a are the homothetical points of S1 and S2 with a = smooth=20 factor varying from 0.0 to 1.5. This allows to weight the prior = and next=20 points positions before drawing the curve. A smooth factor of zero = produces a=20 straight line P[n], P[n+1]. the figure above shows a smooth factor of=20 0.75.
    A smooth of factor excessively curves the line. A smooth = factor of=20 1.05 produces a circle from a square.=20
  4. M is the middle point of the segment P[n-1] , P[n]=20
  5. S1b is the middle point of the segment S1a , P[n-1]=20
  6. S2b is the middle point of the segment S2a , P[n]=20
  7. The PC1 control point is the middle point of the segment M , S1b=20
  8. The PC2 control point is the middle point of the segment M , S2b =

 This method is coded in the PB1 Delphi project.=20

A screen dump of the program appears below:=20

=20

 =20
Program Description Keywords

PB1=20

Draw and distort polygons using B=E9zier curves and an original=20 smoothing algorithm.=20

This algorithm provides an adjustable smooth parameter, able to = transform a square to a circle.  

Bezier curves are drawn with a NOTXOR pen, option not available = with=20 the API windows function Polybezier.  Simply by clicking on = the=20 polygon edges or on the B=E9zier control points, you can obtain = many complex=20 shapes, such as flowers. The areas may be coloured.

Smoothing algorithm with smooth factor,
B=E9zier = curves,
Square to=20 circle transformation,
Grid pattern,
Drawing flowers.=20

Download:  Jean-YvesQueincBezierCurves.zip=20

Updated
26=20 Feb 2005


since 12 Jan 2001

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