Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves Some well known curves

Circle: Cartesian : x2 y 2 a 2 Polar equation: ra : x acos t , y  a sin t ,0  t  2 : 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: x2 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 : 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 : 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 02t  Intrinsic equation: sa4 sin

Inverted cycloid: Parametric equation: x a t sin t , y  a 1  cos t 02t  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, 11tt33

Strophoid: Cartesian equation: x2 y 2 x  ay 2 00 a   Parametric equation: a11 t22 at t  xy, 11tt22

Page 4 of 7 Dr. Sk Amanathulla Asst. Prof., Raghunathpur College Some Well Known Curves 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: ra1 cos  Pedal equation: r32 2 ap

Cardiode: Cartesian equation: 2 x2 y 2  ax  a 2 x 2  y 2  Polar equation: ra1 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 aecot 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