D2DEllipse2D

The package D2DEllipse2D implements the Ellipse2D object.

Initialization

BeginPackage["D2DEllipse2D`",{"D2DExpressions2D`", "D2DGeometry2D`", "D2DLine2D`", "D2DMaster2D`", "D2DNumbers2D`", "D2DPoint2D`", "D2DQuadratic2D`", "D2DSegment2D`", "D2DSketch2D`", "D2DTransform2D`"}];

D2DEllipse2D::usage=
   "D2DEllipse2D is a package that implements the Ellipse2D object.";

Ellipse2D::usage=
   "Ellipse2D[{h,k},a,b,theta] is the standard form of an ellipse centered at (h,k), semi-major axis length 'a', semi-minor axis length 'b' and rotation angle 'theta'.";

SemiMajorAxis2D::usage=
   "SemiMajorAxis2D[ellipse] returns the length of the semi-major axis of an ellipse.";

SemiMinorAxis2D::usage=
   "SemiMinorAxis2D[ellipse] returns the length of the semi-minor axis of an ellipse.";

Begin["`Private`"];

Description

Representation

Format: Ellipse2D[{h,k},a,b,θ]
Standard representation of an ellipse in Descarta2D.  The first argument is a list of coordinates representing the center of the ellipse.  The second and third arguments are (positive) scalars representing the lengths of the semi-major and semi-minor axes of the ellipse.  The fourth argument is the CCW rotation angle (in radians) of the ellipse about the center point.

Equation

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

Quadratic2D[Ellipse2D[{h_,k_},a_,b_,theta_]] :=
   Rotate2D[
      Quadratic2D[b^2,0,a^2,-2*b^2*h,-2*a^2*k,-a^2*b^2+b^2*h^2+a^2*k^2],
      theta,
      {h,k}];

Evaluation

Format: Ellipse2D[{h,k},a,b,θ][]
Evaluates a position on an ellipse at parameter and returns a list of coordinates {x,y}.  Parameters in the range cover the entire ellipse.

Ellipse2D[{h_,k_},a_,b_,theta_][t_?IsScalar2D] :=
   Rotate2D[{h+a*Cos[t],k+b*Sin[t]},theta,{h,k}];

Graphics

Provides the graphics primitives for an ellipse by extending the Mathematica Display command. Executed when the package is loaded.

SetDisplay2D[
   Ellipse2D[{h_,k_},a_,b_,t_][{t1_?IsScalar2D,t2_?IsScalar2D}],
   MakePrimitives2D[Ellipse2D[{h,k},a,b,t],
                    PrimaryAngleRange2D[{t1,t2}]] ];

SetDisplay2D[
   Ellipse2D[{h_,k_},a_,b_,t_],
   MakePrimitives2D[Ellipse2D[{h,k},a,b,t],{0,2Pi}] ];

Validation

Format: Ellipse2D[{h,k},a,b,θ]
Detects an ellipse with imaginary arguments and returns the $Failed symbol.  If the imaginary parts are insignificant, they are removed.

Ellipse2D::imaginary=
   "An invalid ellipse of the form 'Ellipse2D[`1`, `2`, `3`, `4`]' has been detected; the arguments of an ellipse cannot involve imaginary numbers.";

Ellipse2D[{h_,k_},a_,b_,theta_] :=
   (Ellipse2D @@ ChopImaginary2D[Ellipse$2D[{h,k},a,b,theta]]) /;
(FreeQ[{h,k,a,b,theta},_Pattern] && IsTinyImaginary2D[{h,k,a,b,theta}]);

Ellipse2D[{h_,k_},a_,b_,theta_] :=
   (Message[Ellipse2D::imaginary,{h,k},a,b,theta];$Failed) /;
(FreeQ[{h,k,a,b,theta},_Pattern] && IsComplex2D[{h,k,a,b,theta},0]);

Format: Ellipse2D[{h,k},a,b,θ]
Detects an ellipse with invalid arguments and returns the $Failed symbol.

Ellipse2D::invalid=
   "An invalid ellipse of the form 'Ellipse2D[`1`, `2`, `3`, `4`]' has been detected; the length of both the semi-major and semi-minor axes must be positive.";

Ellipse2D[{h_,k_},a_,b_,theta_] :=
   (Message[Ellipse2D::invalid,{h,k},a,b,theta];$Failed) /;
(FreeQ[{h,k,a,b,theta},_Pattern] && IsZeroOrNegative2D[{a,b},0]);

Format: Ellipse2D[{h,k},a,b,θ]
Detects a y-axis ellipse and rotates it π/2 radians.

Ellipse2D[{h_,k_},a_,b_,theta_] :=
   Ellipse2D[{h,k},b,a,theta+Pi/2] /;
(FreeQ[{h,k,a,b,theta},_Pattern] && IsNegative2D[a-b,0]);

Format: Ellipse2D[{h,k},a,b,θ]
Adjusts the rotation angle on all ellipses to the range 0≤θ<π.

Ellipse2D[{h_,k_},a_,b_,theta_] :=
   Ellipse2D[{h,k},a,b,PrimaryAngle2D[theta,Pi]] /;
(FreeQ[{h,k,a,b,theta},_Pattern] && (theta=!=PrimaryAngle2D[theta,Pi]));

Format: IsValid2D[ellipse]
Verifies that an ellipse is syntactically valid.

IsValid2D[Ellipse2D[{h_?IsScalar2D,k_?IsScalar2D},
                    a_?IsScalar2D,b_?IsScalar2D,
                    theta_?IsScalar2D]] := True;

Scalars

Angle of Rotation

Format: Angle2D[ellipse]
Returns the rotation angle of an ellipse.

Angle2D[Ellipse2D[{h_,k_},a_,b_,theta_]] := theta;

Semi-major Axis Length

Format:  SemiMajorAxis2D[ellipse]
Returns the length of the semi-major axis of an ellipse.

SemiMajorAxis2D[Ellipse2D[{h_,k_},a_,b_,theta_]] := a;

Semi-minor Axis Length

Format: SemiMinorAxis2D[ellipse]
Returns the length of the semi-major axis of an ellipse.

SemiMinorAxis2D[Ellipse2D[{h_,k_},a_,b_,theta_]] := b;

Transformations

Reflect

Format: Reflect2D[ellipse,line]
Reflects an ellipse in a line.

Reflect2D[Ellipse2D[{h_,k_},a_,b_,theta_],L:Line2D[p_,q_,r_]] :=
   Ellipse2D[Reflect2D[{h,k},L],a,b,ReflectAngle2D[theta,L]];

Rotate

Format: Rotate2D[ellipse,θ,coords]
Rotates an ellipse 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[Ellipse2D[{h_,k_},a_,b_,theta_],alpha_?IsScalar2D,
         {x0_?IsScalar2D,y0_?IsScalar2D}] :=
    Ellipse2D[Rotate2D[{h,k},alpha,{x0,y0}],a,b,alpha+theta];

Scale

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

Scale2D[Ellipse2D[{h_,k_},a_,b_,theta_],s_?IsScalar2D,
        {x0_?IsScalar2D,y0_?IsScalar2D}] :=
   Ellipse2D[Scale2D[{h,k},s,{x0,y0}],s*a,s*b,theta] /;
Not[IsZeroOrNegative2D[s]];

Translate

Format: Translate2D[ellipse,{u,v}]
Translates an ellipse delta distance.

Translate2D[Ellipse2D[{h_,k_},a_,b_,theta_],
            {u_?IsScalar2D,v_?IsScalar2D}] :=
    Ellipse2D[{h+u,k+v},a,b,theta];

Point Construction

Center Point

Format: Point2D[ellipse]
Constructs the center point of an ellipse.

Point2D[Ellipse2D[{h_,k_},a_,b_,theta_]] := Point2D[{h,k}];

Pole Point

Format: Point2D[line,ellipse]
Constructs the pole point of a line with respect to an ellipse.  If the line is tangent to the ellipse then the point is the point of tangency.

Point2D[L1:Line2D[a1_,b1_,c1_],E2:Ellipse2D[{h2_,k2_},a2_,b2_,theta2_]] :=
   Point2D[L1,Quadratic2D[E2]];

Line Construction

Axis Line

Format: Line2D[ellipse]
Constructs a line containing the major axis of an ellipse.

Line2D[Ellipse2D[{h_,k_},a_,b_,theta_]] :=
   Rotate2D[Line2D[0,1,-k],theta,{h,k}];

Polar Line

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

Line2D[P1:Point2D[{x1_,y1_}],E2:Ellipse2D[{h2_,k2_},a2_,b2_,theta2_]] :=
   Line2D[P1,Quadratic2D[E2]];

Ellipse Construction

Ellipse from Vertices and Eccentricity

Format: Ellipse2D[{point,point},e]
Constructs an ellipse from the vertices and eccentricity.

Ellipse2D::invdef=
   "The defining geometry or eccentricity is invalid; the eccentricity of an ellipse must be in the range 0<e<1, the foci and vertices cannot be coincident, and the focus cannot lie on the directrix.";

Ellipse2D[{P1:Point2D[{x1_,y1_}],P2:Point2D[{x2_,y2_}]},e_?IsScalar2D] :=
   Module[{a,b,h,k},
      If[IsZeroOrNegative2D[{e,1-e},Or] || IsCoincident2D[P1,P2],
         Message[Ellipse2D::invdef];$Failed,
         a=Distance2D[P1,P2]/2;
         b=a*Sqrt[1-e^2];
         {h,k}={(x1+x2)/2,(y1+y2)/2};
         Ellipse2D[{h,k},a,b,ArcTan[x2-x1,y2-y1]]] ];

Ellipse from Foci and Eccentricity

Format: Ellipse2D[point,point,e]
Constructs an ellipse from the foci and eccentricity.

Ellipse2D[P1:Point2D[{x1_,y1_}],P2:Point2D[{x2_,y2_}],e_?IsScalar2D] :=
   Module[{a,b,h,k},
      If[IsZeroOrNegative2D[{e,1-e},Or] || IsCoincident2D[P1,P2],
         Message[Ellipse2D::invdef];$Failed,
         a=Distance2D[P1,P2]/(2*e);
         b=a*Sqrt[1-e^2];
         {h,k}={(x1+x2)/2,(y1+y2)/2};
         Ellipse2D[{h,k},a,b,ArcTan[x2-x1,y2-y1]]] ];

Ellipse from Focus, Directrix and Eccentricity

Format: Ellipse2D[point,line,e]
Constructs an ellipse from a focus point, directrix line and eccentricity.

Ellipse2D[P1:Point2D[{x1_,y1_}],L2:Line2D[p_,q_,r_],e_?IsScalar2D] :=
   Module[{d,s,a,b,h,k},
      If[IsZeroOrNegative2D[{e,1-e},Or] || IsOn2D[P1,L2],
         Message[Ellipse2D::invdef];$Failed,
         d=Distance2D[P1,L2];
         s=(p*x1+q*y1+r)/(p^2+q^2);
         a=d*e/(1-e^2);
         b=a*Sqrt[1-e^2];
         {h,k}={x1,y1}+{p,q}*(a*s*e)/d;
         Ellipse2D[{h,k},a,b,ArcTan[p,q]]] ];

Epilogue

End[ ]; (* end of "`Private" *)
EndPackage[ ]; (* end of "D2DEllipse2D`" *)