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{
From: ldeboer@cougar.multiline.com.au (Leon DeBoer)
{------------------------------------------------------------------------}
{ Catmull_Rom and BSpline Parametric Spline Program }
{ }
{ All source written and devised by Leon de Boer, (c)1994 }
{ E-Mail: ldeboer@cougar.multiline.com.au }
{ }
{ After many request and talk about spline techniques on the }
{ internet I decided to break out my favourite spline programs and }
{ donate to the discussion. }
{ }
{ Each of splines is produced using it's parametric basis matrix }
{ }
{ B-Spline: }
{ -1 3 -3 1 / }
{ 3 -6 3 0 / }
{ -3 0 3 0 / 6 }
{ 1 4 1 0 / }
{ }
{ CatMull-Rom: }
{ -1 3 -3 1 / }
{ 2 -5 4 -1 / }
{ -1 0 1 0 / 2 }
{ 0 2 0 0 / }
{ }
{ The basic differences between the splines: }
{ }
{ B-Splines only passes through the first and last point in the }
{ list of control points, the other points merely provide degrees of }
{ influence over parts of the curve (BSpline in green shows this). }
{ }
{ Catmull-Rom splines is one of a few splines that actually pass }
{ through each and every control point the tangent of the curve as }
{ it passes P1 is the tangent of the slope between P0 and P2 (The }
{ curve is shown in red) }
{ }
{ There is another spline type that passes through all the }
{ control points which was developed by Kochanek and Bartels and if }
{ anybody knows the basis matrix could they E-Mail to me ASAP. }
{ }
{ In the example shown the program produces 5 random points and }
{ displays the 2 spline as well as the control points. You can alter }
{ the number of points as well as the drawing resolution via the }
{ appropriate parameters. }
{------------------------------------------------------------------------}
PROGRAM Spline;
USES Graph;
TYPE
Point3D = Record
X, Y, Z: Real;
End;
VAR CtrlPt: Array [-1..80] Of Point3D;
PROCEDURE Spline_Calc (Ap, Bp, Cp, Dp: Point3D; T, D: Real; Var X, Y: Real);
VAR T2, T3: Real;
BEGIN
T2 := T * T; { Square of t }
T3 := T2 * T; { Cube of t }
X := ((Ap.X*T3) + (Bp.X*T2) + (Cp.X*T) + Dp.X)/D; { Calc x value }
Y := ((Ap.Y*T3) + (Bp.Y*T2) + (Cp.Y*T) + Dp.Y)/D; { Calc y value }
END;
PROCEDURE BSpline_ComputeCoeffs (N: Integer; Var Ap, Bp, Cp, Dp: Point3D);
BEGIN
Ap.X := -CtrlPt[N-1].X + 3*CtrlPt[N].X - 3*CtrlPt[N+1].X + CtrlPt[N+2].X;
Bp.X := 3*CtrlPt[N-1].X - 6*CtrlPt[N].X + 3*CtrlPt[N+1].X;
Cp.X := -3*CtrlPt[N-1].X + 3*CtrlPt[N+1].X;
Dp.X := CtrlPt[N-1].X + 4*CtrlPt[N].X + CtrlPt[N+1].X;
Ap.Y := -CtrlPt[N-1].Y + 3*CtrlPt[N].Y - 3*CtrlPt[N+1].Y + CtrlPt[N+2].Y;
Bp.Y := 3*CtrlPt[N-1].Y - 6*CtrlPt[N].Y + 3*CtrlPt[N+1].Y;
Cp.Y := -3*CtrlPt[N-1].Y + 3*CtrlPt[N+1].Y;
Dp.Y := CtrlPt[N-1].Y + 4*CtrlPt[N].Y + CtrlPt[N+1].Y;
END;
PROCEDURE Catmull_Rom_ComputeCoeffs (N: Integer; Var Ap, Bp, Cp, Dp: Point3D);
BEGIN
Ap.X := -CtrlPt[N-1].X + 3*CtrlPt[N].X - 3*CtrlPt[N+1].X + CtrlPt[N+2].X;
Bp.X := 2*CtrlPt[N-1].X - 5*CtrlPt[N].X + 4*CtrlPt[N+1].X - CtrlPt[N+2].X;
Cp.X := -CtrlPt[N-1].X + CtrlPt[N+1].X;
Dp.X := 2*CtrlPt[N].X;
Ap.Y := -CtrlPt[N-1].Y + 3*CtrlPt[N].Y - 3*CtrlPt[N+1].Y + CtrlPt[N+2].Y;
Bp.Y := 2*CtrlPt[N-1].Y - 5*CtrlPt[N].Y + 4*CtrlPt[N+1].Y - CtrlPt[N+2].Y;
Cp.Y := -CtrlPt[N-1].Y + CtrlPt[N+1].Y;
Dp.Y := 2*CtrlPt[N].Y;
END;
PROCEDURE BSpline (N, Resolution, Colour: Integer);
VAR I, J: Integer; X, Y, Lx, Ly: Real; Ap, Bp, Cp, Dp: Point3D;
BEGIN
SetColor(Colour);
CtrlPt[-1] := CtrlPt[1];
CtrlPt[0] := CtrlPt[1];
CtrlPt[N+1] := CtrlPt[N];
CtrlPt[N+2] := CtrlPt[N];
For I := 0 To N Do Begin
BSpline_ComputeCoeffs(I, Ap, Bp, Cp, Dp);
Spline_Calc(Ap, Bp, Cp, Dp, 0, 6, Lx, Ly);
For J := 1 To Resolution Do Begin
Spline_Calc(Ap, Bp, Cp, Dp, J/Resolution, 6, X, Y);
Line(Round(Lx), Round(Ly), Round(X), Round(Y));
Lx := X; Ly := Y;
End;
End;
END;
PROCEDURE Catmull_Rom_Spline (N, Resolution, Colour: Integer);
VAR I, J: Integer; X, Y, Lx, Ly: Real; Ap, Bp, Cp, Dp: Point3D;
BEGIN
SetColor(Colour);
CtrlPt[0] := CtrlPt[1];
CtrlPt[N+1] := CtrlPt[N];
For I := 1 To N-1 Do Begin
Catmull_Rom_ComputeCoeffs(I, Ap, Bp, Cp, Dp);
Spline_Calc(Ap, Bp, Cp, Dp, 0, 2, Lx, Ly);
For J := 1 To Resolution Do Begin
Spline_Calc(Ap, Bp, Cp, Dp, J/Resolution, 2, X, Y);
Line(Round(Lx), Round(Ly), Round(X), Round(Y));
Lx := X; Ly := Y;
End;
End;
END;
VAR I, J, Res, NumPts: Integer;
BEGIN
I := Detect;
InitGraph(I, J, 'e:\bp\bgi');
I := GetMaxX; J := GetMaxY;
Randomize;
CtrlPt[1].X := Random(I); CtrlPt[1].Y := Random(J);
CtrlPt[2].X := Random(I); CtrlPt[2].Y := Random(J);
CtrlPt[3].X := Random(I); CtrlPt[3].Y := Random(J);
CtrlPt[4].X := Random(I); CtrlPt[4].Y := Random(J);
CtrlPt[5].X := Random(I); CtrlPt[5].Y := Random(J);
Res := 20;
NumPts := 5;
BSpline(NumPts, Res, LightGreen);
CatMull_Rom_Spline(NumPts, Res, LightRed);
SetColor(Yellow);
For I := 1 To NumPts Do Begin
Line(Round(CtrlPt[I].X-3), Round(CtrlPt[I].Y),
Round(CtrlPt[I].X+3), Round(CtrlPt[I].Y));
Line(Round(CtrlPt[I].X), Round(CtrlPt[I].Y-3),
Round(CtrlPt[I].X), Round(CtrlPt[I].Y+3));
End;
ReadLn;
CloseGraph;
END.
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