Several Maths functions I

Determinar si 3 puntos están sobre la misma línea en 2D

function Collinear(x1, y1, x2, y2, x3, y3: Double): Boolean;

begin

Result := (((x2 - x1) * (y3 - y1) - (x3 - x1) * (y2 - y1)) = 0);

end;

(* End Of Collinear *)



// vérifier si 3 points appartiennent à une même droite bidimensionnelle

// c.a.d s'ils sont alignés.



...Determine if 3 points are collinear in 2D?



function Collinear(x1, y1, x2, y2, x3, y3: Double): Boolean;

begin

Result := (((x2 - x1) * (y3 - y1) - (x3 - x1) * (y2 - y1)) = 0);

end;

(* End Of Collinear *)



// vérifier si 3 points appartiennent à une même droite bidimensionnelle

// c.a.d s'ils sont alignés.





Determinar el punto en el que 2 segmentos 2D se tocan

procedure IntersectPoint(x1, y1, x2, y2, x3, y3, x4, y4: Double; var Nx, Ny: Double);

var

R: Double;

dx1, dx2, dx3: Double;

dy1, dy2, dy3: Double;

begin

dx1 := x2 - x1;

dx2 := x4 - x3;

dx3 := x1 - x3;



dy1 := y2 - y1;

dy2 := y1 - y3;

dy3 := y4 - y3;



R := dx1 * dy3 - dy1 * dx2;



if R <> 0 then

begin

R := (dy2 * (x4 - x3) - dx3 * dy3) / R;

Nx := x1 + R * dx1;

Ny := y1 + R * dy1;

end

else

begin

if Collinear(x1, y1, x2, y2, x3, y3) then

begin

Nx := x3;

Ny := y3;

end

else

begin

Nx := x4;

Ny := y4;

end;

end;

end;







function Collinear(x1, y1, x2, y2, x3, y3: Double): Boolean;

begin

Result := (((x2 - x1) * (y3 - y1) - (x3 - x1) * (y2 - y1)) = 0);

end;

(* End Of Collinear *)



// calcule le point d'intersection de 2 droites bidimensionnelles





Verifica si 2 segmentos 2D son paralelos

function SegmentsParallel(x1, y1, x2, y2, x3, y3, x4, y4: Double): Boolean;

begin

Result := (((y1 - y2) * (x1 - x2)) = ((y3 - y4) * (x3 - x4)));

end;

(* End Of SegmentsParallel *)







Verifica si 2 segmentos 3D son paralelos

function SegmentsParallel(x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4: Double): Boolean;

var

Dx1, Dx2: Double;

Dy1, Dy2: Double;

Dz1, Dz2: Double;

begin

{

Theory:

If the gradients in the following planes x-y, y-z, z-x are equal then it can be

said that the segments are parallel in 3D, However as of yet I haven't been able

to prove this "mathematically".



Worst case scenario: 6 floating point divisions and 9 floating point subtractions

}



Result := False;



{

There is a division-by-zero problem that needs attention.

My initial solution to the problem is to check divisor of the divisions.

}





Dx1 := x1 - x2;

Dx2 := x3 - x4;



//If (IsEqual(dx1,0.0) Or IsEqual(dx2,0.0)) And NotEqual(dx1,dx2) Then Exit;



Dy1 := y1 - y2;

Dy2 := y3 - y4;



//If (IsEqual(dy1,0.0) Or IsEqual(dy2,0.0)) And NotEqual(dy1,dy2) Then Exit;



Dz1 := z1 - z2;

Dz2 := z3 - z4;



//If (IsEqual(dy1,0.0) Or IsEqual(dy2,0.0)) And NotEqual(dy1,dy2) Then Exit;





if NotEqual(Dy1 / Dx1, Dy2 / Dx2) then Exit;

if NotEqual(Dz1 / Dy1, Dz2 / Dy2) then Exit;

if NotEqual(Dx1 / Dz1, Dx2 / Dz2) then Exit;



Result := True;

end;

(* End Of SegmentsParallel*)



const

Epsilon = 1.0E-12;



function IsEqual(Val1, Val2: Double): Boolean;

var

Delta: Double;

begin

Delta := Abs(Val1 - Val2);

Result := (Delta <= Epsilon); end; (* End Of Is Equal *) function NotEqual(Val1, Val2: Double): Boolean; var Delta: Double; begin Delta := Abs(Val1 - Val2); Result := (Delta > Epsilon);

end;

(* End Of Not Equal *)



Autor: Arash Partow

http://www.partow.net/