D2DParabola2D

The package D2DParabola2D implements the Parabola2D object.

Initialization

BeginPackage["D2DParabola2D`",{"D2DExpressions2D`", "D2DGeometry2D`", "D2DLine2D`", "D2DMaster2D`", "D2DNumbers2D`", "D2DPoint2D`", "D2DQuadratic2D`", "D2DSegment2D`", "D2DSketch2D`", "D2DTransform2D`"}];

D2DParabola2D::usage=
   "D2DParabola2D is a package that implements the Parabola2D object.";

FocalLength2D::usage=
   "FocalLength2D[parabola] returns the focal length of a parabola.";

Parabola2D::usage=
   "Parabola2D[{h,k},f,theta] is the standard form of a parabola that opens to the right (when theta=0); {x,y} is the vertex point of the parabola; 'f' is the distance from the vertex point to the focus; 'theta' is the counter-clockwise rotation (in radians) of the parabola about the vertex point.";

Begin["`Private`"];

Description

Representation

Format: Parabola2D[{h,k},f,θ]
Standard representation of a parabola in Descarta2D.  The first argument is a list of coordinates representing the position of the vertex of the parabola.  The second argument is a (positive) scalar representing the distance from the vertex to the focus (the focal length).  The third argument is the counter-clockwise rotation (in radians) of the parabola about the vertex point.

Equation

Format: Quadratic2D[parabola]
Constructs the quadratic representing the equation of a parabola.

Quadratic2D[Parabola2D[{h_,k_},f_,theta_]] :=
   Rotate2D[Quadratic2D[0,0,1,-4*f,-2*k,k^2+4*h*f],theta,{h,k}];

Evaluation

Format: Parabola2D[{h,k},f,θ][t]
Evaluates a parabola at a parameter t (-Infinity < t < +Infinity).  The end points of the latus rectum are at t=-1 and t=1.

Parabola2D[{h_,k_},f_,theta_][t_?IsScalar2D] :=
   Rotate2D[{h+f*t^2,k+2*f*t},theta,{h,k}];

Graphics

Provides graphics for a parabola by extending the Mathematica Display command. Executed when the package is loaded.

SetDisplay2D[
   Parabola2D[{h_,k_},f_,t_][{t1_?IsScalar2D,t2_?IsScalar2D}],
   MakePrimitives2D[Parabola2D[{h,k},f,t],{t1,t2}] ];

SetDisplay2D[
   Parabola2D[{h_,k_},f_,t_],
   MakePrimitives2D[Parabola2D[{h,k},f,t],
                    CurveLimits2D[{0,0},
                                  Parabola2D[{0,0},f,0]]] ];

Validation

Format: Parabola2D[{h,k},f,θ]
Detects a parabola with imaginary arguments and returns the $Failed symbol.  If the imaginary parts are insignificant, they are removed.

Parabola2D::imaginary=
   "An invalid parabola of the form 'Parabola2D[`1`, `2`, `3`]' has been detected; the arguments cannot be imaginary.";

Parabola2D[{h_,k_},f_,theta_] :=
   (Parabola2D @@ ChopImaginary2D[Parabola$2D[{h,k},f,theta]]) /;
(FreeQ[{h,k,f,theta},_Pattern] && IsTinyImaginary2D[{h,k,f,theta}]);

Parabola2D[{h_,k_},f_,theta_] :=
   (Message[Parabola2D::imaginary,{h,k},f,theta];$Failed) /;
(FreeQ[{h,k,f,theta},_Pattern] && IsComplex2D[{h,k,f,theta},0]);

Format: Parabola2D[{h,k},f,θ]
Detects a parabola with an invalid focal length.  If the focal length is negative, the parabola is rotated π radians to make it positive; if the focal length is zero, the $Failed symbol is returned.

Parabola2D::invalid=
   "An invalid parabola of the form 'Parabola2D[`1`, `2`, `3`]' has been detected; the focal length cannot be zero.";

Parabola2D[{h_,k_},f_,theta_] :=
   (Message[Parabola2D::invalid,{h,k},f,theta];$Failed) /;
(FreeQ[{h,k,f,theta},_Pattern] && IsZero2D[f,0]);

Parabola2D[{h_,k_},f_,theta_] :=
   Parabola2D[{h,k},-f,theta+Pi] /;
(FreeQ[{h,k,f,theta},_Pattern] && IsNegative2D[f,0]);

Format: Parabola2D[{h,k},f,θ]
Adjusts the rotation angle on a parabola to the range 0≤θ≤2π.

Parabola2D[{h_,k_},f_,theta_] :=
   Parabola2D[{h,k},f,PrimaryAngle2D[theta]] /;
(FreeQ[{h,k,f,theta},_Pattern] && (theta=!=PrimaryAngle2D[theta]));

Format: IsValid2D[parabola]
Verifies that a parabola is syntactically valid.

IsValid2D[Parabola2D[{h_?IsScalar2D,k_?IsScalar2D},
                     f_?IsScalar2D,
                     theta_?IsScalar2D]] := True;

Scalars

Angle of Rotation

Format: Angle2D[parabola]
Returns the rotation angle of a parabola.

Angle2D[Parabola2D[{h_,k_},f_,theta_]] := theta;

Focal Length

Format: FocalLength2D[parabola]
Returns the focal length of a parabola.

FocalLength2D[Parabola2D[{h_,k_},f_,theta_]] := f;

Transformations

Reflect

Format: Reflect2D[parabola,line]
Reflects a parabola in a line.

Reflect2D[Parabola2D[{h_,k_},f_,theta_],L2:Line2D[a2_,b2_,c2_]] :=
   Parabola2D[Reflect2D[{h,k},L2],f,ReflectAngle2D[theta,L2]];

Rotate

Format: Rotate2D[parabola,θ,coords]
Rotates a  parabola by an angle θ about a position specified by a coordinate list.  If the third argument is omitted, it defaults to the origin (see D2DTransform2D.html).

Rotate2D[Parabola2D[{h_,k_},f_,theta_],alpha_?IsScalar2D,
         {x0_?IsScalar2D,y0_?IsScalar2D}] :=
   Parabola2D[Rotate2D[{h,k},alpha,{x0,y0}],f,theta+alpha];

Scale

Format: Scale2D[parabola,s,coords]
Scales a parabola from a position given by coordinates.  If the third argument is omitted, it defaults to the origin (see D2DTransform2D.html).

Scale2D[Parabola2D[{h_,k_},f_,theta_],s_?IsScalar2D,
        {x0_?IsScalar2D,y0_?IsScalar2D}] :=
   Parabola2D[Scale2D[{h,k},s,{x0,y0}],s*f,theta] /;
Not[IsZeroOrNegative2D[s]];

Translate

Format: Translate2D[parabola,{u,v}]
Translates a parabola delta distance.

Translate2D[Parabola2D[{h_,k_},f_,theta_],{u_?IsScalar2D,v_?IsScalar2D}] :=
   Parabola2D[{h+u,k+v},f,theta];

Point Construction

Vertex Point

Format: Point2D[parabola]
Returns the vertex point of a parabola.

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

Pole Point

Format: Point2D[line,parabola]
Constructs the pole (point) of a polar (line) with respect to a parabola.  If the line is tangent to the parabola then the point is the point of tangency.

Point2D[L1:Line2D[a1_,b1_,c1_],P2:Parabola2D[{h_,k_},f_,theta_]] :=
   Point2D[L1,Quadratic2D[P2]];

Line Construction

Axis Line

Format: Line2D[parabola]
Constructs a line that contains the axis of a parabola.

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

Polar Line

Format: Line2D[point,parabola]
Constructs the polar (line) of a pole (point) with respect to a parabola.  If the point is on the parabola then the line is tangent to the parabola at the point.

Line2D[P1:Point2D[{x1_,y1_}],P2:Parabola2D[{h_,k_},f_,theta_]] :=
   Line2D[P1,Quadratic2D[P2]];

Parabola Construction

Parabola from Focus/Directrix

Format: Parabola2D[point,line]
Constructs a parabola from a focus point and a directrix line.

Parabola2D::invptln=
   "The focus `1` is on the directrix `2`; no valid parabola can be constructed.";

Parabola2D[P1:Point2D[{x1_,y1_}],L2:Line2D[a2_,b2_,c2_]] :=
   Module[{pt},
      If[IsOn2D[P1,L2],
         Message[Parabola2D::invptln,P1,L2];$Failed,
         pt=Point2D[P1,L2];
         Parabola2D[Coordinates2D[Point2D[P1,pt]],
                    Distance2D[P1,pt]/2,
                    ArcTan[x1-XCoordinate2D[pt],
                           y1-YCoordinate2D[pt]]]] ];

Epilogue

End[ ]; (* end of "`Private" *)
EndPackage[ ]; (* end of "D2DParabola2D`" *)