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Some Well Known Curves Some Well Known Curves

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Some Well Known Curves Some Well Known Curves Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Some well known curves Circle: Cartesian equation: x2 y 2 a 2 Polar equation: ra Parametric equation: x acos t , y a sin t ,0 t 2 Pedal equation: pr Parabola: Cartesian equation: y2 4 ax Polar equation: l 1 cos (focus as pole) r Parametric equation: x at2 , y 2 at , t Pedal equation: p2 ar (focus as pole) Intrinsic equation: s alog cot cos ec a cot cos ec Parabola: Cartesian equation: x2 4 ay Polar equation: l 1 sin (focus as pole) r Parametric equation: x2 at , y at2 , t Pedal equation: (focus as pole) Intrinsic equation: s alog sce tan a tan sec Page 1 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Ellipse: xy22 Cartesian equation: 1 ab22 Polar equation: l 1e cos (focus as pole) r Parametric equation: x acos t , y b sin t ,0 t 2 ba2 2 Pedal equation: 1 pr2 Hyperbola: xy22 Cartesian equation: 1 ab22 l Polar equation: 1e cos r Parametric equation: x acosh t , y b sinh t , t ba2 2 Pedal equation: 1 pr2 Rectangular hyperbola: Cartesian equation: xy c2 Polar equation: rc22sin 2 2 Parametric equation: c x ct,, y t t Pedal equation: pr 2 c2 Page 2 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Cycloid: Parametric equation: x a t sin t , y a 1 cos t 02t Intrinsic equation: sa4 sin Inverted cycloid: Parametric equation: x a t sin t , y a 1 cos t 02t Intrinsic equation: sa 4 sin Asteroid: 2 2 2 Cartesian equation: x3 y 3 a 3 Parametric equation: x acos33 t , y a sin t 0 t 2 Pedal equation: r2 a 23 p 2 Intrinsic equation: 4sa 3 cos 2 0 Page 3 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Semi cubical parabola: Cartesian equation: x32 ay 00 a Parametric equation: x at23,, y at t Folium of Descartes: Cartesian equation: x33 y3 axy 3a sin cos Polar equation: r cos33 sin Parametric equation: 33at at 2 xy, 11tt33 Strophoid: Cartesian equation: x2 y 2 x ay 2 00 a Parametric equation: a11 t22 at t xy, 11tt22 Page 4 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Lemniscate of Bernoulli: Cartesian 2 equation:x2 y 2 a 2 x 2 y 2 Polar equation: ra22 cos 2 Pedal equation: r32 a p Cardiode: Cartesian equation: 2 x2 y 2 ax a x 2 y 2 Polar equation: ra1 cos Pedal equation: r32 2 ap Cardiode: Cartesian equation: 2 x2 y 2 ax a 2 x 2 y 2 Polar equation: ra1 cos Pedal equation: r32 2 ap Page 5 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Equiangular spiral: Polar equation: r aecot Pedal equation: pr sin Intrinsic equation: s a em 1 Reciprocal spiral: a Polar equation: r Pedal equation: p2 a 2 r 2 a 2 r 2 Three leaved rose: Polar equation: ra sin3 Pedal equation: pr sin 3 Page 6 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Four leaved rose: Polar equation: ra sin2 Pedal equation: pr sin 2 Eight leaved rose: Polar equation: ra sin4 Pedal equation: pr sin 4 Catenary: x Cartesian equation: yc cosh c Parametric equation: x clog sec tan , y sec Intrinsic equation: sc tan Page 7 of 7 .
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