Parametric Equations

Transcription

1 Parametric Equations Curves in the plane are often described b giving the -coordinates and -coordinates as functions of some parameter such as t. Eamples: Astroid: t) a cos t, t) a sin t Cissoid: t) a t + t, t) a t + t Conchoid: t) a tan t + sin t, t) a + cos t

2 Conic Sections: The parametric equations of a conic whose ais makes an angle α with the -ais are, where d is the semi-latus rectum and e is the eccentricit: t) d cos t + e cost α), t) d sin t + e cost α) Ccloid: t) at b sin t, t) a b cos t Hpoccloid: t) a b) cos t + b cos b a b t, t) a b) sin t b sin b a b t

3 Lemniscate: t) a cos t b cos t, t) a sin t b cos t Limacon: t) + c sin t)cos t, t) + c sin t)sin t Lissajous: t) a sin, t) b cos t Tractri: t) a sin t, t) cos t + log tan t Witch of Agnesi: t) a cot t, t) a sin t Those who wish to see much more comprehensive sets of curves should lookat: http : //www groups.dcs.st and.ac.uk/ histor /Java/ http : // ah/specialp lanecur ve_dir /specialp lanecur ves.html http : // o.vir ginia.edu/ eww6n/math/math.html

4 Slope and Concavit To find the slope and concavit of a curve in the plane described b parametric equations is eas: We have d d d d d d ) d d d d d d ẏ ẋ and ) d d d d ) d ) d d d ẋÿ ẍẏ ẋ d ) d d ẋ ẍ ẏ ÿ d d ) d d ) d d d ) d where we have used the dot notation for differentiation with respect to t. ẋ Eample: Astroid: t) a cos t, t) a sin t, ẋ a cos t sin t, ẍ a cos t sin t + cos t), ẏ a sin t cos t, ÿ a sin t cos t sin t) d d ẏ ẋ d ẋÿ ẍẏ d ẋ a sin t cos t a cos t sin t sin t cos t tan t a cos t sin t)a sin t cos t sin t)) a cos t sin t + cos t))a sin t cos t) a cos t sin t) 9a sin t cos t cos t sin t) + 9a sin t cos t sin t + cos t)) 7a cos 6 t sin 6 t cos t sin t) sin t + cos t) a cos t sin t a cos t sin t

5 Areas Under Parametric Curves We can also easil find the area between the ais and a parametric curve: A b a d tt tt t)ẋt) Eample: Find the area of the Astroid: t) a cos t, t) a sin t. Solution: Note that this is times the area of the astroid ling in the first quadrant, so we have a ) ) A d a sin t a cos t sin t t t a t sin t cos t a ) cos t + cos t t a cos t) cos t) + cos t) t ) a cos t) cos t t a t a t a t a sin t t a cos t a sin t cos t sin t t cos t a t t sin t 8 a ) cos t) sin t

6 Arc Length of Parametric Curves As mentioned previousl, the formula for the length of a parametric curve is tt L ẋ + ẏ tt We compute the length of the perimeter of the astroid: t L a cos t sin t) + a sin t cos t) t L a cos t sin t + sin t cos t t L a sin t + cos t)sin t cos t t L a sin t cos t t L 6a sin t a cos t t a cos ) cos ) a ) 6a Volumes of Solids of Revolution Similarl, we can compute volumes of solid of revolution of parametric curves: For eample, if we revolve a curve about the -ais we have: V b a d Let us do this for the astroid: V a t 6a asin t) a cos t sin t) 6a t cos t) cos t sin t letting u cos t ) u 6a u ) u u du) 6a u u sin t) cos t sin t u + u u 6) u du 6

7 u 6a u u + u 6 u 8 du 6a u u + u7 7 u9 9 u [ 6a + 7 [ 6a [ ] 7 6a a ] )] 6a [ 9 + u u ] Surface Areas As mentioned previousl, a formula for the surface area obtained b rotating a parametric curve about the -ais is tt S ẋ + ẏ tt We compute the surface area obtained b rotating the astroid about the -ais: S t a sin t) a cos t sin t) + a sin t cos t) t a sin ta cos t sin t + sin t cos t t a t a a sin t sin t sin t + cos t)sin t cos t t sin t sin t cos t a t a sin t cos t 7

8 Elimination of the Parameter It is sometimes possible to use the parametric equations of a curve to find an equation for the curve. If the equation can be seen to be that of a familiar curve, this gives us useful information. Eample : t) cos t, t) sin t clearl satisfies +, so the curve is just the unit circle. Eample : t) cos t, t) sin t: we have t) ) ) t) t) cos t + sin t +, or cos t and t) sin t, so + so the curve is an ellipse with centre, ), with horizontal semi-minor ais of length and vertical semi-major ais of length. Eample : t) + cos t, t) + sin t: we have t) ) ) t) t) + sin t, socos t + sin t +, or cos t and t) + ) + ) + so the curve is an ellipse with centre, ), with horizontal semi-minor ais of length and vertical semi-major ais of length. Eample : t) + t, t) t : we solve for t: t t) and t t) +,so ) t) t) + so the curve has the equation of a parabola: ) + or + ) 8