Exploring Analyic Geometry with Mathematica®

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Parametric Equations of a Conic Arc

caparam.html

Exploration

Show that the parametric equations of a unit conic arc represent the same implicit quadratic equation as the one underlying the conic as derived from the control points "caparam_1.gif", "caparam_2.gif" and "caparam_3.gif" and ρ.

Approach

Create the unit conic arc. Eliminate t from the parametric equations and construct a quadratic from the result. Construct a quadratic directly from the conic arc. Verify that the two quadratics are identical.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create the unit conic arc.

Clear[a,b,p];
ca1=ConicArc2D[{0,0},{a,b},{1,0},p];

Eliminate t from the parametric equations.

Clear[xt,yt,t];
eq1=Eliminate[{xt==First[ca1[t]],yt==Last[ca1[t]]},{t}]

"caparam_4.gif"

Construct the quadratic represented by the parametric equations.

q1=Quadratic2D[eq1,{xt,yt}] //Simplify

"caparam_5.gif"

Construct the quadratic from the conic arc.

q2=Map[Simplify,Quadratic2D[ca1]] //Simplify

"caparam_6.gif"

Both quadratics are the same, ignoring the -1 factor.

IsCoincident2D[q1,q2]

"caparam_7.gif"


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