GXWeb Pedal Curves: Hyperbola and Ellipse

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About Pedal Curves (from Wolfram MathWorld)

$a$
	0	3	10
$b$
	0	5	10
$\theta$
	0	1.56	6.283

App generated by GXWeb

The pedal of a curve with respect to a point O is the locus of the foot of the perpendicular from O to the tangent to the curve.

Two curves are formed here as the envelope curves of line AC, where point C then is tangent to the curve.

The ellipse and the hyperbola are closely related conics - can you see the values for a and b in this model for which it switches from hyperbola to ellipse and back again? Use the model to explore and study the cartesian equations below.

The Pedal Curve constructed here is a Lemniscate, the Lemniscate of Bernoulli for the hyperbola, and the Hippopede or Lemniscate of Booth for the ellipse. It is locus of point C where

$a = Length(OA)$

$b = Length(OB)$

and $angle(AOB) = \theta$ .

Pedal Hyperbola Cartesian equation:
Lemniscate of Bernoulli $X^4+2·X^2·Y^2+Y^4-X^2·a^2+Y^2·(-a^2+b^2)$

Hyperbolic Cartesian equation: $Y^2·a^2-a^4+a^2·b^2+X^2·(a^2-b^2)$

Pedal Ellipse Cartesian equation:
Lemniscate of Booth (Hippopede) $X^4+2·X^2·Y^2+Y^4-X^2·a^2-Y^2·b^2$

Elliptical Cartesian equation: $Y^2·a^2+X^2·b^2-a^2·b^2$

GXWeb Pedal Curves: Hyperbola and Ellipse

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WolframAlpha: CAS+

Pedal Hyperbola

Pedal Ellipse

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