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GXWeb Pedal Curves: Hyperbola and Ellipse
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GXWeb Curve Construction Collection (like this one!)
About Pedal Curves (from Wolfram MathWorld)
\(a\) 0 1 10 \(b\) 0 0 10 \(\theta\) 0 0 6.283
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: \[Y^2·a^2+X^2·b^2-a^2·b^2\]
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Pedal Hyperbola
Pedal Ellipse
Behind the Scenes
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