Using Geometer’s Sketchpad to Support Mathematical Thinking Envelope of a Lemniscate The lemniscate, also called the lemniscate of Bernoulli, is a polar curve whose most common form is the locus of points P , such that the product of the distances from to two fixed points and is the constant 2. The fixed P F 1 F 2 c points, or foci, are a distance 2 c apart. Expressed in terms of distances:€ € € € 2 € (Equation 1) PF ⋅ PF = c € 1 2 Using the coordinates P x,y , F c,0 and F −c,0 in Equation 1 yields the ( ) 1( ) 2 ( ) Cartesian equation: € 2 2 2 2 x − c + y − 0 ⋅ x + c + y − 0 = c2 ( ) ( ) ( ) ( ) € € € Now squaring both sides yields: 2 2 2 2 4 (Equation€ 2) (x − c) + y (x + c) + y = c [ ][ ] Expanding and simplifying then gives: 2 2 2 2 x − c x + c + x − c y2 + x + c y2 + y4 = c4 € ( ) ( ) ( ) ( ) 2 2 (x − c) (x + c) + x2 − 2xc + c2 y2 + x2 + 2xc + c2 y2 + y4 = c4 ( ) ( ) 2 2 (x − c) (x + c) + 2x2 y2 + 2c2 y2 + y4 = c4 € € Shelly Berman p. 1 of 3 Jo Ann Fricker € Lemniscate Envelope.doc Using Geometer’s Sketchpad to Support Mathematical Thinking x2 − 2xc + c2 x2 + 2xc + c2 + 2x2 y2 + 2c2 y2 + y4 = c4 ( )( ) 4 2 2 2 2 2 2 4 x − 2x c + 2x y + 2c y + y = 0 4 2 2 4 2 2 2 2 x + 2x y + y = 2x c − 2c y € 2 2 2 2 2 2 (Equation 3) x + y = 2c x − y . € ( ) ( ) € Translating to polar coordinates gives the equation: € 2 r 2 cos2 θ + r 2 sin2 θ = 2c2 r 2 cos2 θ − r 2 sin2 θ ( ) ( ) 2 r 4 cos2 θ + sin2 θ = 2c2r 2 cos2 θ − sin2 θ ( ) ( ) 4 2 2 € r = 2c r (cos 2θ) 2 2 (Equation€ 4) r = 2c cos(2θ) usually simplified to: € 2 2 (Equation 5) r = a cos(2θ). Adapted from Eric W. Weisstein.€ "Lemniscate." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Lemniscate.html € π Rotating the polar curve through an angle of would result in the curve: 4 2 2 π r = a cos 2 θ − . 4 Geometrically, the lemniscate can be€ generated as the envelope of circles centered on a rectangular hyperbola and passing through the center of the hyperbola (Wells 1991).€ The lemniscate pictured in the diagram is modeled by the equation: π r 2 = 80 cos 2θ − 2 € Shelly Berman p. 2 of 3 Jo Ann Fricker Lemniscate Envelope.doc Using Geometer’s Sketchpad to Support Mathematical Thinking Adapted from Exploring Precalculus with The Geometer’s Sketchpad 4.0 http://www.keypress.com/catalog/products/software/Prod_GSPModExpPrecalc.html Hide Cartesian Family of Lemniscates Hide Polar Hide Cartesian Family of Lemniscates Hide Polar 105° 90° 75° 105° 90° 75° + 120° 60° + 120° 60° 5 5 135° + 45° 135° + 45° 4 4 3 150° 30° 3 150° 30° 2 2 165° 15° 165° 15° 1 1 180° 0° 180° 0° -90° 90° 180° 270° 360° 450° -90° 90° 180° 270° 360° 450° -1 -1 -2 195° 345° -2 195° 345° -3 -3 210° 330° 210° 330° -4 -4 -5 225° - 315° -5 225° - 315° - 240° 300° - 240° 300° 255° 270° 285° 255° 270° 285° θ = 60° θ = 60° Edit the function below to try your own. Edit the function below to try your own. You can use parameters a and b in the You can use parameters a and b in the -180° -90° 30° 60°90° 180° 270° 450° 540° function you create. -180° -90° 30° 60°90° 180° 270° 450° 540° function you create. 0° 45° 360° 0° 45° 360° a = 4.00 a = 4.00 2 2 Animate f(θ) = a ⋅cos(2⋅θ-n⋅90) Animate f(θ) = a ⋅sin(2⋅θ-n⋅90) Exploring Precalculus with Sketchpad n = 0.00 Exploring Precalculus with Sketchpad n = 0.00 (C) 2005 by Key Curriculum Press (C) 2005 by Key Curriculum Press Hide Cartesian Family of Lemniscates Hide Polar Hide Cartesian Family of Lemniscates Hide Polar 105° 90° 75° 105° 90° 75° + 120° 60° + 120° 60° 5 5 135° + 45° 135° + 45° 4 4 3 150° 30° 3 150° 30° 2 2 165° 15° 165° 15° 1 1 180° 0° 180° 0° -90° 90° 180° 270° 360° 450° -90° 90° 180° 270° 360° 450° -1 -1 -2 195° 345° -2 195° 345° -3 -3 210° 330° 210° 330° -4 -4 -5 225° - 315° -5 225° - 315° - 240° 300° - 240° 300° 255° 270° 285° 255° 270° 285° θ = 60° θ = 60° Edit the function below to try your own. Edit the function below to try your own. You can use parameters a and b in the You can use parameters a and b in the -180° -90° 30° 60°90° 180° 270° 450° 540° function you create. -180° -90° 30° 60°90° 180° 270° 450° 540° function you create. 0° 45° 360° 0° 45° 360° a = 4.00 a = 4.00 2 2 Animate f(θ) = a ⋅cos(2⋅θ-n⋅90) Animate f(θ) = a ⋅sin(2⋅θ-n⋅90) Exploring Precalculus with Sketchpad n = 1.00 Exploring Precalculus with Sketchpad n = 1.00 (C) 2005 by Key Curriculum Press (C) 2005 by Key Curriculum Press Hide Cartesian Family of Lemniscates Hide Polar Hide Cartesian Family of Lemniscates Hide Polar 105° 90° 75° 105° 90° 75° + 120° 60° + 120° 60° 5 5 135° + 45° 135° + 45° 4 4 3 150° 30° 3 150° 30° 2 2 165° 15° 165° 15° 1 1 180° 0° 180° 0° -90° 90° 180° 270° 360° 450° -90° 90° 180° 270° 360° 450° -1 -1 -2 195° 345° -2 195° 345° -3 -3 210° 330° 210° 330° -4 -4 -5 225° - 315° -5 225° - 315° - 240° 300° - 240° 300° 255° 270° 285° 255° 270° 285° θ = 60° θ = 60° Edit the function below to try your own. Edit the function below to try your own. You can use parameters a and b in the You can use parameters a and b in the -180° -90° 30° 60°90° 180° 270° 450° 540° function you create. -180° -90° 30° 60°90° 180° 270° 450° 540° function you create. 0° 45° 360° 0° 45° 360° a = 4.00 a = 4.00 2 2 Animate f(θ) = a ⋅cos(2⋅θ-n⋅90) Animate f(θ) = a ⋅sin(2⋅θ-n⋅90) Exploring Precalculus with Sketchpad n = 2.00 Exploring Precalculus with Sketchpad n = 2.00 (C) 2005 by Key Curriculum Press (C) 2005 by Key Curriculum Press Shelly Berman p. 3 of 3 Jo Ann Fricker Lemniscate Envelope.doc .