Mega Code Archive
Convex Hull (2D)
<![CDATA[
Title: Convex Hull (2D)
Question: Given is a set of not ordered points in a plane.
How can I efficiently find the minimal ordered subset of points that build the convex boundary?
Answer:
{---------------------------------------------------------------------------
UCHULL Delphi Unit for V 5.x & 7.x
was former
UCHULL TURBO PASCAL Unit for TP V 6.0 and TPW 1.0, BP & BPW 7.x
---------------------------------------------------------------------------
Author: F.Honegger c/o LENO KBS CH-4031 BASEL
File : UCHULL.PAS
Associated File : UCHULL.TPU
Edit notes and History
1.st edit : 11.03.1992 creation
2.nd edit : 22.05.2000 adaption to D5
3.nd edit : 14.11.2007 compiler switches for D7
The routines are more or less self descriptive. For a intro-
duction to algorithms in computer graphics see
Wolfram Luther, Martin Ohsmann, Mathematischc Grundlagen der
Computergraphik, Vieweg Braunschweig 1988, ISBN 3-528-06302-5.
The routine chull calculates the convex hull of a set of
points. The algorithm implemented here is not subject of the
above reference. It has been developed by the author. The main
subject was to get a handy and flexible routine. The procedure
works as follows.
1. Retrieve points until a triangle is found, or input is
exhausted. If all the points processed lie on a straight
line then the extremal points are returned, and the
procedure finishes. If a triangle was found continue with
next step.
2. Order the points of the triangle mathematically positive
ie. counter clockwise.,
3. If points are exhausted we finish here, otherwise we
get a new point and go to the next step.
4. Verify if the point lies outside the current polygon. If
it does go to the next step, otherwise discard the point
and go to the previous step.
5. Find the left tangent from the point to the polygon.
6. Find the right tangent from the point to the polygon.
7. Build a new polygon, with the new point the tangent
points and the hidden points (points not view by the new
point). Repeat step 3 with the new polygon.
CHull returns a result structure (see type TCHull_Result below). The
dynamic x-field holds the x-coordinates, the dynamic y-field the
y-cordinates and the dynamic ix field the indexes of the returned points as
they are read. The first and last item of each field equals if and only
if the convex hull is a closed curve. CHull reads the point coordinates
using a user supplied input routine. Its address is passed by calling
CHull. If CHull fails, it returns nil arrays in its result record. The only
reasons of failure are heap overflow and structure overflow. If nupmax_idx
then a structure overflow has occurred. If nup has occurred.
}
{$IFNDEF win32}
The comiler switches and the code must be revised for version not having
win32 defined
{$ENDIF}
{$IFDEF ver130}
{$DEFINE CompileOk}
{$ENDIF}
{$IFDEF ver150}
{$DEFINE CompileOk}
{$ENDIF}
{$IFNDEF CompileOk}
The comiler switches and the code must be revised for version different to
13.0
{$ENDIF}
{$IFDEF CompileOk}
{$A-} {allign data or packed records- = byte }
{$B+} {Boolean complete evaluation - = on }
{$D+} {debug info - = off }
{$L+} {local symbols + = yes }
{$R-} {Range check - = off }
{$S-} {Stack checking on - = off }
{$I+} {I/O checking on + = on }
{$G-} {importdata references + = on }
{$X-} {extended sytax + = on }
{$O+} {Code optimisation + = on }
{$H+} {Long Strings + = on }
{$P+} {open string paramters if H- + = on }
{$V+} {varstring checks if short strings + = on }
{$C+} {Build code assertions + = on }
{$E-}
{$F-}
{$J+} {Writeable typed constants + = on }
{$K-}
{$M-}
{$N+}
{$Q-} {Overflow check + = on }
{$S-}
{$T-} {Typed addresse + = on }
{$U-} {Save pentium divide + = on }
{$W-} {Always make stackframes + = on }
{$Y+} {Symbol creation + = on }
{$Z1} {Minimum Size a Enum Type is stored (in conj. with C }
{$ENDIF}
unit Uchull;
interface
const
Max_idx=(Maxint SHR 3)-1;
type
TArrayOfDouble = array of double;
TArrayOfInteger = array of Integer;
TchullResult = record
Nup:Integer;
x:TArrayOfDouble;
y:TArrayOfDouble;
Ix:TArrayOfInteger;
end;
TOgetnp = function(var x,y:Double):Boolean of object;
Tgetnp = function(var x,y:Double):Boolean;
procedure Chull(OGetnp:TOgetnp;Getnp:Tgetnp; Var ResultRec:TchullResult);
implementation
Uses
Sysutils;
procedure SwapInteger(var a,b:Integer);
VAR
C:Integer;
{code is optimal when code optimisation switch is enabled}
BEGIN
c:=a;
a:=b;
b:=c;
END;
procedure SwapDouble(var a,b: double);assembler;
{c:=a; a:=b; b:=c; }
Asm
{load a on stop}
Fld Qword Ptr [eax]
{load b on stop}
Fld Qword Ptr [edx]
{save stop as a}
Fstp Qword Ptr [eax]
{save stop as b}
Fstp Qword Ptr [edx]
FWAIT
end;
procedure Chull(OGetnp:TOgetnp;Getnp:Tgetnp; Var ResultRec:TchullResult);
{Calculates iteratively the convex hull of a set of points, given by the
coordinate vectors x,y.
Input getnp address of a input routine of tye tgetnp
Output pointer to address record of the address fields.
}
label
End_pre,Chull_end,Error_exit;
var
{pointers to fields that hold coordinates of points}
x,y,X1,Y1:TArrayOfDouble;
{coordinates of point to be classified and processed}
Xp,Yp:Double;
{pointers to fileds that hold indexes of points}
Nn,Nn1:TArrayOfInteger;
{size of fields in bytes while classifying point}
Xysize,Nnsize:Word;
{size of fields in bytes after classifiying new point}
Nxysize,Nnnsize:Word;
{number of points of polygon befor/after processing new point}
Np,Npn:Integer;
{number of points processed}
Points_fetched:Integer;
{work variables}
Xi,Yi :Double;
Dx,Dy,Dx1,Dy1 :Double;
d,Di :Double;
Df :array[1..3] of double;
i,Ii,Ioi,SI,k,Dc :Integer;
{flags}
Iii :array[1..3] of Byte;
Ok,Out :Boolean;
begin
{we allocate space for three points, the extra field at zero offset is
used for memory management only}
Assert(Assigned(GetNp) xor Assigned(OGetNp),'Please pass one input routine only');
Np:=3;
Npn:=Np;
Xysize:=(Np+2) * SizeOf(Double);
Nnsize:=(Np+2) * SizeOf(Integer);
SetLength(Nn,Nnsize);
SetLength(x,Xysize);
SetLength(y,Xysize);
SetLength(Nn1,0);
SetLength(x1,0);
SetLength(y1,0);
Points_fetched:=0;
i:=1;
{try to read first three points}
ok:=false;
repeat
if Assigned(GetNp) then Ok:=Getnp(x[i],y[i]);
if Assigned(OGetNp) then Ok:=OGetnp(x[i],y[i]);
if Ok then
begin
Inc(i);
{increment pointcounter}
Inc(Points_fetched);
Nn[i]:=Points_fetched;
end;
until (not Ok) or (Points_fetched=3);
{ lets handle the trivial case (less then 3 points)}
if Points_fetched begin
if Points_fetched=0 then
begin
Resultrec.Nup:=0;
end else if Points_fetched=1 then
begin
Resultrec.Nup:=1;
end
else if Points_fetched=2 then
begin
if (x[1]=x[2]) and (y[1]=y[2]) then
begin
Resultrec.Nup:=1;
end else begin
Resultrec.Nup:=2;
end;
end;
end else begin
{then the non trivial case}
for i:=1 TO 4 do Nn[i]:=i;
{search for a triangle}
while TRUE do
begin
FillChar(Iii,SizeOf(Iii),0);
if ( x[1]=x[2] ) and ( y[1]=y[2] ) then Iii[1]:=1;
if ( x[1]=x[3] ) and ( y[1]=y[3] ) then Iii[2]:=1;
if ( x[2]=x[3] ) and ( y[2]=y[3] ) then Iii[3]:=1;
Ii:=0;
for i:=1 TO 3 do Ii:=Ii+Iii[i];
case Ii OF
3: begin
Ii:=2;
Iii[1]:=1;
Iii[2]:=0;
Iii[3]:=0;
end;
1: begin
if Iii[1]=1 then
begin
for i:=1 TO 2 do
begin
x[i]:=x[i+1];
y[i]:=y[i+1];
Nn[i]:=Nn[i+1];
end;
end;
for i:=1 TO 2 do Iii[i]:=1;
Iii[3]:=0;
end;
0: begin
{are points collinear ?}
d:=(y[2]-y[1])*(x[3]-x[1])-(y[3]-y[1])*(x[2]-x[1]);
{no we got a triangle}
if d0 then goto End_pre;
{yes we will extract the to extremal points, by comparing euclidean
distances}
Df[1]:=Sqr(x[1]-x[2])+Sqr(y[1]-y[2]);
Df[2]:=Sqr(x[1]-x[3])+Sqr(y[1]-y[3]);
Df[3]:=Sqr(x[3]-x[2])+Sqr(y[3]-y[2]);
d:=0;
for i:=1 TO 3 do
if Df[i]d then
begin
k:=i;
d:=Df[i];
end;
case k OF
2: begin
x[2]:=x[3];
y[2]:=y[3];
Nn[2]:=Nn[3];
Iii[1]:=1;
Iii[2]:=1;
end;
3: begin
for i:=1 TO 2 do
begin
x[i]:=x[i+1];
y[i]:=y[i+1];
Nn[i]:=Nn[i+1];
Iii[i]:=1;
end;
end;
1: begin
for i:=1 TO 2 do Iii[i]:=1;
end;
end;
Iii[3]:=0;
Ii:=1;
end;
end{case};
Ok:=TRUE;
{replace rejected points}
for i:=2 TO 3 do
if (Iii[i]=0) and Ok then
begin
if Assigned(GetNp) then Ok:=Getnp(Xp,Yp);
if Assigned(OGetNp) then Ok:=OGetnp(Xp,Yp);
if Ok then
begin
{increment pointcounter}
Inc(Points_fetched);
Nn[i]:=Points_fetched;
x[i]:=Xp;
y[i]:=Yp;
Dec(Ii);
end;
end;
{if the points are exhausted here the problem reduces to the trivial case}
if (not Ok) or (Ii0) then
begin
if Ii=2 then
begin
Resultrec.Nup:=1;
end else begin
Resultrec.Nup:=2;
end;
goto Chull_end;
end;
end;
End_pre:
{here we are with a triangle}
{lets run through it in the positive sense}
if d0 then
begin
SwapInteger(Nn[1],Nn[2]);
SwapDouble(x[1],x[2]);
SwapDouble(y[1],y[2]);
end;
if Assigned(GetNp) then Ok:=Getnp(Xp,Yp);
if Assigned(OGetNp) then Ok:=OGetnp(Xp,Yp);
{ for all new points repeat}
while Ok do
begin
{increment pointcounter}
Inc(Points_fetched);
{check if points lies inside or outside of the polygon}
i:=1; Out:=FALSE;
while (i begin
if i Xi:=x[i]; Yi:=y[i];
d:=(x[Ii]-Xi)*(Yp-Yi)-(y[Ii]-Yi)*(Xp-Xi);
if d end;
{ point lies outside the polygon. we have some work}
if Out then
begin
{ find the position in polygon where to insert the new point}
Ok:=FALSE;
Di:=0;
SI:=0;
while not Ok do
begin
{To enhance speed, we run clockwise or counter clockwise through the polygon
depending on the sign of the first d.}
if (SI=-1) or (SI=0) then
begin
if i1 then Dec(i) else i:=Np;
end else
begin
if i end;
if i
Xi:=x[i]; Yi:=y[i];
Dx1:=x[Ii]-Xi; Dy1:=y[Ii]-Yi; Dx:=Xp-Xi; Dy:=Yp-Yi;
d:=Dx*Dy1-Dy*Dx1;
if Abs(d)=0 then
begin
Out:=FALSE;
if Abs(Dx)=0 then
begin
if Dy*Dy10 then Ok:=TRUE
end else begin
if Dx*Dx10 then Ok:=TRUE
end;
SI:=1;
end else begin
if SI=0 then
begin
if d0 then
SI:=-1
else SI:=1;
Di:=-SI;
end;
if (SI*d0) and (SI*Di begin
Ok:=TRUE;
Out:=TRUE;
end;
end;
Di:=d;
end;
{if went clockwise we must adjust the position index}
if (SI=-1) and (d0) then
if i=Np then i:=1 else Inc(i);
Ioi:=i;
{figure out how many points after the newly inserted must be deleted}
if Out then
begin
{ preset delete counter}
Dc:=0;
if Ioi=Np then i:=1 else i:=Succ(Ioi);
end else begin
{ preset delete counter}
Dc:=1;
if Ioi=Np-1 then i:=1 else
if Ioi=Np then i:=2 else i:=Ioi+2;
end;
Ok:=TRUE;
while (iIoi) and Ok do
begin
if i=Np then Ii:=1 else Ii:=Succ(i);
Xi:=x[i]; Yi:=y[i];
d:=(Xi-Xp)*(y[Ii]-Yp)-(Yi-Yp)*(x[Ii]-Xp);
if d begin
i:=Ii;
Inc(Dc);
end else Ok:=FALSE;{i:=Ioi;}
end;
{adjust counter in the collinear case}
if Abs(d)=0 then Inc(Dc);
{allocate new dimensioned index/data buffers}
Npn:=(Np+1-Dc);
{structure overflow ?}
if NpnMax_idx then goto Error_exit;
Nxysize:=SizeOf(Double)*(Npn+2);
Nnnsize:=SizeOf(Integer)*(Npn+2);
Nn1:=Nn;
SetLength(Nn,Nnnsize);
X1:=x;
SetLength(x,Nxysize);
Y1:=y;
SetLength(y,Nxysize);
if Ioi=Np then i:=1 else i:=Succ(Ioi);
for k:=1 TO Dc do
if i=Np then i:=1 else Inc(i);
{ shuffle indexes}
k:=1;
x[k]:=X1[Ioi];
y[k]:=Y1[Ioi];
Nn[k]:=Nn1[Ioi];
Inc(k);
x[k]:=Xp;
y[k]:=Yp;
Nn[k]:=Points_fetched;
Inc(k);
while iIoi do
begin
x[k]:=X1[i];
y[k]:=Y1[i];
Nn[k]:=Nn1[i];
Inc(k);
if i=Np then i:=1 else Inc(i);
end;
Xysize:=Nxysize;
Nnsize:=Nnnsize;
Np:=Npn;
end;
if Assigned(GetNp) then Ok:=Getnp(Xp,Yp);
if Assigned(OGetNp) then Ok:=OGetnp(Xp,Yp);
end;
x[Np+1]:=x[1];
y[Np+1]:=y[1];
Nn[Np+1]:=Nn[1];
Resultrec.Nup:=Np+1;
end{else};
Chull_end:
Resultrec.x:=copy(x,1,Resultrec.Nup);
Resultrec.y:=copy(y,1,Resultrec.Nup);
Resultrec.Ix:=copy(Nn,1,Resultrec.Nup);
Exit;
Error_exit:
{reinit result record}
SetLength(Resultrec.Ix,0);
SetLength(Resultrec.x,0);
SetLength(Resultrec.y,0);
Resultrec.Nup:=Npn;
end{Chull};
end{Uchull}.
Attached is a zipped sample project that shows the unit in action!
Copy the text below the dashed line in to a text editor and save it as UCHullTest.b64. Check that the empty line after the MIME header is not removed. After you can use a tool like uudeView (http://www.miken.com/uud/) or the built in feature of TotalCommander
(http://www.ghisler.ch) do extract the zip.
-----------------------------------------------------------------
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