请教用少数点画出波形的平滑算法.(100分)

我用lingto的办法对数据点进行连接，来得到波形，但是所取的点数太少，所以得到的波形不够平滑,如显示正弦波时，波形像梯形似的，哪位高手知道在数据点比较少的情况下，用什么算法能得到平滑的曲线？

listen,前几日有一个问类似问题的,好像是要得到"光滑曲线",
有yysun回答的,可以到yysun的信息里查一下

借花献佛 http://202.120.85.61/delphibbs/DispQ.asp?LID=202694
来自：yysun 时间：00-3-22 21:59:57 ID:204035
将离散的点连接成为光滑曲线是我以前教《计算机地图制图》的主要内容之一。
其中有2个要点：
1、什么是光滑？光滑就是用细密的折线段来模拟曲线。
2、光滑后的曲线需要经过原始离散点。
B样条函数(PolyBezierTo，PolyBezier)不符合这个条件。
常用的方法叫5点光滑法(数学原理请参加测绘出版社的《计算机地图制图》)。
以下是程序：
unit Unit3;
interface
uses
SysUtils, WinTypes, WinProcs, Messages, Classes, Graphics, Controls,
Forms, Dialogs, ExtCtrls, StdCtrls;
type
TForm1 = class(TForm)
Panel1: TPanel;
PaintBox1: TPaintBox;
Button1: TButton;
Button2: TButton;
Button3: TButton;
Button4: TButton;
procedure Button1Click(Sender: TObject);
procedure FormResize(Sender: TObject);
procedure Button2Click(Sender: TObject);
procedure Button3Click(Sender: TObject);
procedure Button4Click(Sender: TObject);
private
{ Private declarations }
public
{ Public declarations }
end;

var
Form1: TForm1;
implementation
{$R *.DFM}
type
rpoint = record
x,y:single;
end;
rline=array[0..100] of rpoint;
xys = array[0..100] of single;
Const xWorldMin: single = 0;
yWorldMin : single = 0;
xWorldMax: single = 100;
yWorldMax : single = 100;
ScreenWidth: integer = 640;
ScreenHeight : integer = 480;
var sScale: single;
procedure AdjustScale;
var sx, sy: single;
begin
sx := (xWorldMax - xWorldMin)/ ScreenWidth;
sy := (yWorldMax - yWorldMin)/ ScreenHeight;
if sx < sy then
sScale := 1.0/sy
else
sScale := 1.0/sx;
xWorldMax := xWorldMin + ScreenWidth / sScale;
yWorldMin := yWorldMax - ScreenHeight / sScale ;
end;

procedure XYWorld2Screen(x, y: single;
var xx, yy: integer);
begin
xx := trunc ((x - xWorldMin) *sScale);
yy := trunc ((yWorldMax - y) *sScale);
xx := trunc( (x - xWorldMin) / (xWorldMax - xWorldMin) * ScreenWidth);
yy := trunc( (yWorldMax - y) / (yWorldMax - yWorldMin) * ScreenHeight);
end;

procedure pMoveTo(x, y: single);
var sx, sy: integer;
begin
XYWorld2Screen(x, y, sx, sy);
Form1.Paintbox1.Canvas.MoveTo(sx, sy);
end;

procedure pLineTo(x, y: single);
var sx, sy: integer;
begin
XYWorld2Screen(x, y, sx, sy);
Form1.Paintbox1.Canvas.LineTo(sx, sy);
end;

procedure pPolyline(rl: rline;
no:integer);
var ii: integer;
begin
pMoveto(rl[0].x, rl[0].y);
for ii := 1 to no-1do
pLineto(rl[ii].x, rl[ii].y);
end;

procedure smoothline(rl: rline;
no:integer);
const step = 0.1;
dist = 0.05;
var
k,m,l,j,cir1,ctl:integer;
bb:char;
a,b,x1,y1:array[1..5] of single;
sn,sn1,sn2,cn,cn1,cn2,w2,w3,z,z1,a0,b0,sn4,cn4,
p0,p1,p2,p3,q0,q1,q2,q3,xt,yt,xt1,yt1,lastx,lasty,p11,p21,q11,q21:single;
r,r1:single;
x,y : ^xys;
begin
ctl:=1;sn2:=0;cn2:=0;sn1:=0;cn1:=0;
if (no<3) then
begin
pMoveTo(rl[0].x,rl[0].y);
pLineTo(rl[1].x,rl[1].y);
end;
new(x);new;
pMoveto(rl[0].x,rl[0].y);
for k:=0 to no-1do
begin
x^[k+2]:=rl[k].x;
y^[k+2]:=rl[k].y;
end;
if ((abs(rl[0].x-rl[no-1].x)<0.05) and (abs(rl[0].y-rl[no-1].y)<0.05)) then
begin
x^[1]:=rl[no-1].x;
y^[1]:=rl[no-1].y;
x^[0]:=rl[no-2].x;
y^[0]:=rl[no-2].y;
x^[no+3]:=rl[1].x;
y^[no+3]:=rl[1].y;
x^[no+2]:=rl[0].x;
y^[no+2]:=rl[0].y;
end
else
begin
p0:=x^[2];
p1:=x^[3];
p2:=x^[4];
q0:=y^[2];
q1:=y^[3];
q2:=y^[4];
x^[1]:=3*(p0-p1)+p2;
y^[1]:=3*(q0-q1)+q2;
p0:=x^[1];
p1:=x^[2];
p2:=x^[3];
q0:=y^[1];
q1:=y^[2];
q2:=y^[3];
x^[0]:=3*(p0-p1)+p2;
y^[0]:=3*(q0-q1)+q2;
p0:=x^[no+1];
p1:=x^[no-1];
p2:=x^[no];
q0:=y^[no+1];
q1:=y^[no-1];
q2:=y^[no];
x^[no+2]:=3*(p0-p2)+p1;
y^[no+2]:=3*(q0-q2)+q1;
p0:=x^[no+2];
p1:=x^[no+1];
p2:=x^[no];
q0:=y^[no+2];
q1:=y^[no+1];
q2:=y^[no];
x^[no+3]:=3*(p0-p1)+p2;
y^[no+3]:=3*(q0-q1)+q2;
end;

for m:=3 to no+1do
begin
j:=m-2;
for k:=1 to 5do
begin
x1[k]:=x^[j];
y1[k]:=y^[j];
j:=j+1;
end;
for k:=1 to 4do
begin
a[k]:=x1[k+1]-x1[k];
b[k]:=y1[k+1]-y1[k];
end;
if m>=5 then
begin
sn1:=sn2;
cn1:=cn2;
end;
if ((a[2]=b[2]) and (a[2]=0)) then
begin
a[2]:=a[1];
b[2]:=b[1];
end;
if ((a[3]=b[3]) and (a[3]=0)) then
begin
a[3]:=a[2];
b[3]:=b[2];
end
else
begin
if ((a[4]=b[4]) and (a[4]=0)) then
begin
a[4]:=a[2];
b[4]:=b[2];
end;
end;

w2:=abs(a[3]*b[4]-a[4]*b[3]);
w3:=abs(a[1]*b[2]-a[2]*b[1]);
if ((w3=w2) and (w3=0)) then
begin
w2:=1;
w3:=1;
end;
a0:=w2*a[2]+w3*a[3];
b0:=w2*b[2]+w3*b[3];
r1:=a0*a0+b0*b0;
r:=sqrt(r1)+0.0001;
sn:=b0/r;
cn:=a0/r;
if (m=3) then
begin
sn1:=sn;
cn1:=cn;
end
else
begin
sn2:=sn;
cn2:=cn;
r1:=a[2]*a[2]+b[2]*b[2];
r:=sqrt(r1);
p0:=x1[2];
q0:=y1[2];
p1:=r*cn1;
q1:=r*sn1;
p2:=3*a[2]-r*(cn2+2*cn1);
q2:=3*b[2]-r*(sn2+2*sn1);
p3:=(-2)*(x1[3]-x1[2])+r*(cn2+cn1);
q3:=(-2)*(y1[3]-y1[2])+r*(sn2+sn1);
z:=1;
repeat
z:=z*0.9;
xt:=p0+z*(p1+z*(p2+z*p3))-x1[2];
yt:=q0+z*(q1+z*(q2+z*q3))-y1[2];
r:=xt*xt+yt*yt;
r1:=sqrt(r);
until (r1<=step);
z1:=z;
while(z<=1)do
begin
xt:=p0+z*(p1+z*(p2+z*p3));
yt:=q0+z*(q1+z*(q2+z*q3));
plineto(xt,yt);
z:=z+z1;
end;
end;
end;
dispose(x);dispose;
end;

proceduredo
ubleline(rl:rline;nn:integer;ww:single);
var l1,l2:rline;
ii:integer;
xt,yt,r,SN,CN:single;
begin
for ii:=0 to nn-2do
begin
xt:=rl[ii+1].x-rl[ii].x;
yt:=rl[ii+1].y-rl[ii].y;
r:=sqrt(xt*xt+yt*yt);
SN:=yt/r;CN:=xt/r;
l1[ii].x:=rl[ii].x+ww*SN;
l1[ii].y:=rl[ii].y-ww*CN;
l2[ii].x:=rl[ii].x-ww*SN;
l2[ii].y:=rl[ii].y+ww*CN;
end;
xt:=rl[nn-2].x-rl[nn-1].x;
yt:=rl[nn-2].y-rl[nn-1].y;
r:=sqrt(xt*xt+yt*yt);
SN:=yt/r;CN:=xt/r;
l1[nn-1].x:=rl[nn-1].x-ww*SN;
l1[nn-1].y:=rl[nn-1].y+ww*CN;
l2[nn-1].x:=rl[nn-1].x+ww*SN;
l2[nn-1].y:=rl[nn-1].y-ww*CN;
smoothline(l1,nn);
smoothline(l2,nn);
end;

procedure Thickline(rl:rline;nn:integer;ww:single);
const penwide=0.001;
var k,j:integer;
begin
k:=trunc(ww/penwide);
for j:=1 to kdo
do
ubleline(rl,nn,j*penwide);
do
ubleline(rl,nn,ww);
end;

procedure TForm1.FormResize(Sender: TObject);
begin
ScreenWidth := Paintbox1.Width-1;
ScreenHeight := Paintbox1.Height-1;
AdjustScale;
end;

var ALine: rLine;
procedure TForm1.Button1Click(Sender: TObject);
begin
ALine[0].x := 10;
ALine[0].y := 10;
ALine[1].x := 40;
ALine[1].y := 60;
ALine[2].x := 60;
ALine[2].y := 70;
ALine[3].x := 80;
ALine[3].y := 50;
ALine[4].x := 120;
ALine[4].y := 30;
ALine[5].x := 130;
ALine[5].y := 30;
ALine[6].x := 150;
ALine[6].y := 30;
Paintbox1.Canvas.Pen.Color := clBlack;
pPolyline(ALine, 7);
end;

procedure TForm1.Button2Click(Sender: TObject);
begin
Paintbox1.Canvas.Pen.Color := clRed;
smoothline(ALine, 7);
end;

procedure TForm1.Button3Click(Sender: TObject);
begin
Paintbox1.Canvas.Pen.Color := clBlue;
do
ubleline(ALine, 7, 4);
end;

procedure TForm1.Button4Click(Sender: TObject);
begin
xWorldMin := 0;
yWorldMin := 0;
xWorldMax := 100;
yWorldMax := 100;
AdjustScale;
PaintBox1.Refresh;
end;

end.

这里还同时奉送实数坐标到屏幕坐标的转换算法以及双线和粗线绘制的算法呢。
您作些调整可以完成您的要求。

B样条得到的曲线一般是不可导的, 也就是说比起可导曲线来说要"不光滑"得多.
可以用3次样条方法拟合, 如果偷懒些嘛, 用多项式拟合也可以凑合了. ^_^

用matlab试试

多谢指点！