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Parameterization of a Quadratic

pquad.html

Exploration

Show that the quadratic , that passes through the origin, can be parameterized by the equations

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

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

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

Discussion

As an example, parameterize the quadratic .

Clear[t];

Q=Quadratic2D[5,-3*Sqrt[3],4,-8,-14,0];

XtYt=Parameterize2D[Q,Point2D[{0,0}],t]

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

Graphics saved as "pquad01.eps".

Determine the locus of the quadratic in standard form.

crv=Loci2D[Q] //N

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.

Graphics saved as "pquad02.eps".

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