D2DConic2D

The package D2DConic2D provides functions for constructing various points, lines and line segments associated with conic curves.

Initialization

BeginPackage["D2DConic2D`", {"D2DCircle2D`", "D2DEllipse2D`", "D2DExpressions2D`", "D2DHyperbola2D`", "D2DLine2D`", "D2DParabola2D`", "D2DPoint2D`", "D2DSegment2D`", "D2DTransform2D`"}];

D2DConic2D::usage=
   "D2DConic2D is a package for constructing geometry associated with conic curves.";

Asymptotes2D::usage=
   "Asymptotes2D[hyperbola] constructs a list containing the two asymptote lines of a hyperbola.";

Directrices2D::usage=
   "Directrices2D[conic] constructs a list containing the directrix line(s) of a conic curve (one for a parabola, two for ellipses and hyperbolas).";

Eccentricity2D::usage=
   "Eccentricity2D[conic] computes the eccentricity of a conic curve (parabola, ellipse or hyperbola).";

FocalChords2D::usage=
   "FocalChords2D[conic] constructs a list containing the focal chords (line segments) of a conic curve (one for a parabola, two for ellipses and hyperbolas).";

Foci2D::usage=
   "Foci2D[conic] constructs a list containing the focus point(s) of a conic curve (one for a parabola, two for ellipses and hyperbolas).";

Vertices2D::usage=
   "Vertices2D[conic] contructs a list containing the vertex point(s) of a conic curve (one for a parabola, two for ellipses and hyperbolas).";

Begin["`Private`"];

Scalars

Eccentricity

Format: Eccentricity2D[ellipse]
Computes the eccentricity of an ellipse.

Eccentricity2D[Ellipse2D[{h_,k_},a_,b_,theta_]] := Sqrt[a^2-b^2]/a;

Format: Eccentricity2D[hyperbola]
Computes the eccentricity of a hyperbola.

Eccentricity2D[Hyperbola2D[{h_,k_},a_,b_,theta_]] := Sqrt[a^2+b^2]/a;

Format: Eccentricity2D[parabola]
Computes the eccentricity of a parabola (e=1).

Eccentricity2D[Parabola2D[{h_,k_},f_,theta_]] := 1;

Point Construction

Focus Points

Format: Foci2D[ellipse]
Constructs a list containing the two focus points of an ellipse.

Foci2D[E1:Ellipse2D[{h_,k_},a_,b_,theta_]] :=
   Module[{e=Eccentricity2D[E1]},
      {Point2D[Rotate2D[{h+a*e,k},theta,{h,k}]],
       Point2D[Rotate2D[{h-a*e,k},theta,{h,k}]]}];

Format: Foci2D[hyperbola]
Constructs a list containing the two focus points of a hyperbola.

Foci2D[H1:Hyperbola2D[{h_,k_},a_,b_,theta_]] :=
   Module[{e=Eccentricity2D[H1]},
      {Point2D[Rotate2D[{h+a*e,k},theta,{h,k}]],
       Point2D[Rotate2D[{h-a*e,k},theta,{h,k}]]}];

Format: Foci2D[parabola]
Constructs a list containing the single focus point of a parabola.

Foci2D[Parabola2D[{h_,k_},f_,theta_]] :=
   {Point2D[Rotate2D[{h+f,k},theta,{h,k}]]};

Vertex Points

Format: Vertices2D[ellipse]
Constructs a list containing the two vertex points of an ellipse.

Vertices2D[Ellipse2D[{h_,k_},a_,b_,theta_]] :=
   {Point2D[Rotate2D[{h+a,k},theta,{h,k}]],
    Point2D[Rotate2D[{h-a,k},theta,{h,k}]]};

Format: Vertices2D[hyperbola]
Constructs a list containing the two vertex points of a hyperbola.

Vertices2D[Hyperbola2D[{h_,k_},a_,b_,theta_]] :=
   {Point2D[Rotate2D[{h+a,k},theta,{h,k}]],
    Point2D[Rotate2D[{h-a,k},theta,{h,k}]]};

Format: Vertices2D[parabola]
Constructs a list containing the single vertex point of a parabola.

Vertices2D[Parabola2D[{h_,k_},f_,theta_]] := {Point2D[{h,k}]};

Line Construction

Asymptote Lines

Format: Asymptotes2D[hyperbola]
Constructs a list containing the two asymptote lines of a hyperbola.

Asymptotes2D[Hyperbola2D[{h_,k_},a_,b_,theta_]] :=
   {Rotate2D[Line2D[b, a,-a*k-b*h],theta,{h,k}],
    Rotate2D[Line2D[b,-a, a*k-b*h],theta,{h,k}]};

Directrix Lines

Format: Directrices2D[ellipse]
Constructs a list containing the two directrix lines of an ellipse.

Directrices2D::circular=
   "The ellipse `1` is circular; it has no (finite) directrix lines.";

Directrices2D[E1:Ellipse2D[{h_,k_},a_,b_,theta_]] :=
   Module[{e=Eccentricity2D[E1]},
      If[IsZero2D[e],
         Message[Directrices2D::circular,E1];{},
         {Rotate2D[Line2D[1,0,-(h+a/e)],theta,{h,k}],
          Rotate2D[Line2D[1,0,-(h-a/e)],theta,{h,k}]}] ];

Format: Directrices2D[hyperbola]
Constructs a list containing the two directrix lines of a hyperbola.

Directrices2D[H1:Hyperbola2D[{h_,k_},a_,b_,theta_]] :=
   Module[{e=Eccentricity2D[H1]},
      {Rotate2D[Line2D[1,0,-(h+a/e)],theta,{h,k}],
       Rotate2D[Line2D[1,0,-(h-a/e)],theta,{h,k}]} ];

Format: Directrices2D[parabola]
Constructs a list containing the single directrix line of a parabola.

Directrices2D[Parabola2D[{h_,k_},f_,theta_]] :=
   {Rotate2D[Line2D[1,0,-h+f],theta,{h,k}]};

Line Segment Construction

Focal Chords

Format: FocalChords2D[ellipse]
Constructs a list containing two line segments that are the focal chords of an ellipse.

FocalChords2D[E1:Ellipse2D[{h_,k_},a_,b_,theta_]] :=
   Module[{e,l1,l2},
      e=Eccentricity2D[E1];
      l1=Segment2D[{h+a*e,k+b^2/a},{h+a*e,k-b^2/a}];
      l2=Segment2D[{h-a*e,k+b^2/a},{h-a*e,k-b^2/a}];
      Map[Rotate2D[#,theta,{h,k}]&,{l1,l2}] ];

Format: FocalChords2D[hyperbola]
Constructs a list containing two line segments that are the focal chords of a hyperbola.

FocalChords2D[H1:Hyperbola2D[{h_,k_},a_,b_,theta_]] :=
   Module[{e,l1,l2},
      e=Eccentricity2D[H1];
      l1=Segment2D[{h+a*e,k+b^2/a},{h+a*e,k-b^2/a}];
      l2=Segment2D[{h-a*e,k+b^2/a},{h-a*e,k-b^2/a}];
      Map[Rotate2D[#,theta,{h,k}]&,{l1,l2}] ];

Format: FocalChords2D[parabola]
Constructs a list containing one line segment that is the single focal chord of the parabola.

FocalChords2D[Parabola2D[{h_,k_},f_,theta_]] :=
   {Rotate2D[Segment2D[{h+f,k+2*f},{h+f,k-2*f}],theta,{h,k}]};

Epilogue

End[ ]; (* end of "`Private" *)
EndPackage[ ]; (* end of "D2DConic2D`" *)