Exploring Analyic Geometry with Mathematica®

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Parameterization of a Quadratic

pquad.html

Exploration

Show that the quadratic "pquad_1.gif", that passes through the origin, can be parameterized by the equations

"pquad_2.gif" and "pquad_3.gif"

where -∞<t<+∞.

Approach

Let the parameter, t, be the slope of a line, L, passing through the origin. The coordinates of the point P(x(t),y(t)), which is the desired parameterization, is the intersection point of L with Q that is not coincident with the origin.

Initialize

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<<Descarta2D`

Solution

Define a function that returns parametric equations of a quadratic, given a point on the quadratic, in terms of a parameter, t.

Parameterize2D[Q:Quadratic2D[a_,b_,c_,d_,e_,f_],
               P:Point2D[{x0_,y0_}],
               t_Symbol]:=
Coordinates2D[First[Select[Points2D[Line2D[P,t],Q],
                    Not[IsCoincident2D[P,#]]&]]];

If the point on the quadratic is the origin, (0,0), then the equations are given by the following.

Clear[a,b,c,d,e,t];
Parameterize2D[Quadratic2D[a,b,c,d,e,0],Point2D[{0,0}],t] //Simplify

"pquad_4.gif"

Discussion

As an example, parameterize the quadratic "pquad_5.gif".

Clear[t];
Q=Quadratic2D[5,-3*Sqrt[3],4,-8,-14,0];
XtYt=Parameterize2D[Q,Point2D[{0,0}],t]

"pquad_6.gif"

Plot the quadratic using the parametric equations. Notice the gap in the graph as the parameter, t, approaches ±∞.

"pquad_7.gif"

"pquad_8.gif"

Graphics saved as "pquad01.eps".

Determine the locus of the quadratic in standard form.

crv=Loci2D[Q] //N

"pquad_9.gif"

An identical graph is produced from the equation in standard form. The gap is not present in this plot because the trigonometric parameterization of the ellipse, used to plot the standard form, avoids passing through infinity.

"pquad_10.gif"

"pquad_11.gif"

Graphics saved as "pquad02.eps".


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
www.Descarta2D.com