Parametrizations of curves on a plane (Sect. 11.1)

Transcription

1 Paramerizaions of curves on a plane (Sec..) Review: Curves on he plane. Parameric equaions of a curve. s of curves on he plane. The ccloid. Review: Curves on he plane Remarks: Curves on a plane can be described b he se of poins (, ) soluions of an equaion F (, ) = 0. A paricular case is he graph of a funcion = f (). In his case: F (, ) = f (). Circle cenered a P = (0, 0) radius r: = r 2. Circle cenered a P = ( 0, 0 ) radius r: ( 0 ) 2 + ( 0 ) 2 = r 2.

2 Review: Curves on he plane An ellipse cenered a P = (0, 0) wih radius a and b, 2 a b 2 =. A sphere is he paricular case a = b = r. A hperbola wih asmpoes = ±, 2 2 =. A hperbola wih asmpoes = ± b a, 2 a 2 2 b 2 =. Review: Curves on he plane A parabola wih minimum a (0, 0), = 2. A parabola wih minimum a (a, b), = c ( a) 2 + b, c > 0. A parabola wih maimum a (a, b), = c ( a) 2 + b, c > 0.

3 Paramerizaions of curves on a plane (Sec..) Review: Curves on he plane. Parameric equaions of a curve. s of curves on he plane. The ccloid. Parameric equaions of a curve Remarks: A curve on a plane can alwas be hough as he moion of a paricle as funcion of ime. Ever curve given b F (, ) = 0 can be described as he se of poins ( (), () ) raveled b a paricle for [a, b]. Definiion A curve on he plane is given in parameric form iff i is given b he se of poins ( (), () ), where he parameer I R. Remark: If he inerval I is closed, I = [a, b], hen ((a), (a)) and ((b), (b)) are called he iniial and erminal poins of he curve.

4 Paramerizaions of curves on a plane (Sec..) Review: Curves on he plane. Parameric equaions of a curve. s of curves on he plane. The ccloid. s of curves on he plane Describe he curve () = cos(), () = sin(), for [0, 2π]. The funcions and above saisf he equaion [()] 2 + [()] 2 = cos 2 () + sin 2 () =. cos () sin () This is a circle. This is he equaion of a circle radius r =, cenered a (0, 0). The circle is raversed in counerclockwise direcion, saring and ending a (, 0).

5 s of curves on he plane Describe he curve () = sin(), () = cos(), for [0, 2π]. The funcions and above saisf he equaion [()] 2 + [()] 2 = sin 2 () + cos 2 () =. cos () sin () This is a circle. This is he equaion of a circle radius r =, cenered a (0, 0). The circle is raversed in clockwise direcion, saring and ending a (0, ). s of curves on he plane Describe he curve () = 3 cos(), () = 3 sin(), for [0, π/2]. The funcions and above saisf he equaion [()] 2 + [()] 2 = 3 2 cos 2 () sin 2 () = cos () 3 sin () 3 This is a porion of a circle. This is he equaion of a /4 circle radius r = 3, cenered a (0, 0). The circle is raversed in counerclockwise direcion, saring a (3, 0) and ending a (0, 3).

6 s of curves on he plane Describe he curve () = 3 cos(2), () = 3 sin(2), for [0, π/2]. The funcions and above saisf he equaion [()] 2 + [()] 2 = 3 2 cos 2 (2) sin 2 (2) = 3 2. This is a porion of a circle. 3 cos () 3 sin () 3 This is he equaion of a /2 circle radius r = 3, cenered a (0, 0). The circle is raversed in counerclockwise direcion, saring a (3, 0) and ending a ( 3, 0). s of curves on he plane Describe he curve () = 3 cos(), () = sin(), for [0, 2π]. The funcions and above saisf he equaion [()] [()] 2 = cos 2 () + sin 2 () =. sin () 3 cos () 3 This is an ellipse. This is he equaion of an ellipse wih -radius 3 and -radius, cenered a (0, 0). The ellipse is raversed in counerclockwise direcion, saring and ending a (3, 0).

7 s of curves on he plane Describe he curve () = cosh(), () = sinh(), for [0, ). The funcions and above saisf he equaion () [()] 2 [()] 2 = () cosh 2 () sinh 2 () =. This is a porion of a hperbola. This is he equaion of a hperbola wih asmpoes = ±. The hperbola porion sars a (, 0). s of curves on he plane Describe he curve () = sec(), () = an(), for [0, π/2). Recall: an 2 () + = sec 2 (). Therefore, [()] 2 [()] 2 = sec 2 () an 2 () =. This is a porion of a hperbola. This is he equaion of a hperbola wih asmpoes = ±. The hperbola porion sars a (, 0). () ()

8 s of curves on he plane Describe he curve () = 2, () = +, for (, ). = ( ) 2 Since =, hen = ( ) 2. This is a parabola. This is he equaion of a parabola opening o he righ. Passing hrough (, 0) (for = ), hen (0, ) (for = 0), and hen (, 2) (for = ). Paramerizaions of curves on a plane (Sec..) Review: Curves on he plane. Parameric equaions of a curve. s of curves on he plane. The ccloid.

9 The ccloid Definiion A ccloid wih parameer a > 0 is he curve given b () = a( sin()), () = a( cos()), R. Remark: From he equaion of he ccloid we see ha () a = a sin(), () a = a cos(). Therefore, [() a] 2 + [() a] 2 = a 2. Remarks: This is no he equaion of a circle. The poin ((), ()) belongs o a moving circle. The ccloid plaed an imporan role in designing precise pendulum clocks, needed for navigaion in he 7h cenur.