Parametric Equations Not all curves are functions. To deal with curves that are not of the form y = f (x) orx = g(y), we use parametric equations. Define both x and y in terms of a parameter t: x = x(t) y = y(t) It is typical to reuse x and y as their function names. Each value of t (time) givesapoint(x(t), y(t)) (position). Ranging over all possible values of t gives a curve, a parametric curve. Figure: Aparametriccurveanditsorientation As t increases, the curve gets an orientation. Math 267 (University of Calgary) Fall 2015, Winter 2016 1 / 12
Example Sketch the parametric curve with equations Describe the orientation of the curve. x = t 2 + t y =2t 1 1< t < 1 Solution: Method 1 Make a table of values. Plot and trace the movement of the point. t x y -2 2-5 -1 0-3 -1/2-1/4-2 0 0-1 1 2 1 Figure: Aparametriccurveanditsorientation Math 267 (University of Calgary) Fall 2015, Winter 2016 2 / 12
Solution: Method 2 Eliminate the parameter t. Start with y =2t 1, write t = y +1. 2 Substitute into x = t 2 + t: y +1 2 y +1 x = + 2 2 = 1 4 y 2 + y + 3 4 The curve is a parabola that opens right. Since y increases as t, thepointmovesinthedirectionofincreasingvalueofy. Math 267 (University of Calgary) Fall 2015, Winter 2016 3 / 12
Example Sketch and describe x =5sin(3t) y =5cos(3t) 0 apple t apple 2. Solution: Eliminate t. Instead of solving for t, weuse sin 2 +cos 2 =1 Using =3t: sin(3t) = x 5 and cos (3t) = y x 2 y 2 5 =) + =1 5 5 Multiply by 25: x 2 + y 2 =25. The curve is the circle centered at the origin with radius 5. As t ranges from 0 to 2, wetravelclockwisearoundthecircleexactlythreetimes. Figure: Aparametriccurvetracingthesamecircleclockwisethreetimes Math 267 (University of Calgary) Fall 2015, Winter 2016 4 / 12
Example Let a be a positive constant. The parametric equations represents an astroid. Eliminating t: x = a cos 3 t y = a sin 3 t x 2/3 + y 2/3 = a 2/3. Figure: Aplotofanastroid(a =1)withMaple18 The astroid can be viewed also as an example of a hypocycloid, acurvetracedoutbyafixedpointona smaller circle rolling on the inside of a bigger circle. See, for example, https://en.wikipedia.org/wiki/astroid for a demonstration. Math 267 (University of Calgary) Fall 2015, Winter 2016 5 / 12
Example Let a be a positive constant. The parametric equations x = a(t sin t) y = a(1 cos t) represents a cycloid. Figure: Aplotofacycloid(a =1)withMaple18 It is the curve traced out by a fixed point on a smaller rolling on the horizontal axis. See, for example, https://en.wikipedia.org/wiki/cycloid for a demonstration. Math 267 (University of Calgary) Fall 2015, Winter 2016 6 / 12
Let s trace the cycloid Let C be a circle of radius a. Place the centre of C at (0, a) andletp be the point on C which initially coincides with the origin. Let C roll along the positive x-axis, and we trace the movement of the point P. Figure: Finding x(t) andy(t) forthepointp that traces a cycloid After the circle has rotated through an angle of t radians, the arc from P to the point of contact of C with the x-axis has length s = at. C has moved a horizontal distance of at. The horizontal distance between P and the centre is a sin t. So, x(t) =at a sin t = a(t sin t). The vertical distance between P and the centre is a cos t. So, y(t) =a a cos t = a(1 cos t). Math 267 (University of Calgary) Fall 2015, Winter 2016 7 / 12
Further examples/exercises Plot su cient number of points and trace the curve. 1 x =3+5cost, y =2+5sint 2 x =2+t 2, y =3t + t 2 3 x =cost, y =1+cos 2 t Use WolframAlpha to plot the following parametric curves. Vary the domain to view di erent portions of the curve. 1 x =sin3t, y =cos5t 2 x = 0.5 +cost, y = 0.5tant +sint 3 x = t +sin2t, y = t 2 +cos5t 4 x = t 1+t 3, y = t2 1+t 3 Math 267 (University of Calgary) Fall 2015, Winter 2016 8 / 12
Calculus with parametric curves Derivatives: Suppose x = x(t), y = y(t) defineaparametriccurvesuchthaty varies with x in a di erentiable manner. Example By the chain rule, dy dt = dy dx dx dt. If dx dt dy 6=0,then dx = dy/dt dx/dt Find the tangent line to the parametric curve x = t 2, y = t 3 12t at the point (1, 11). Solution: The point (1, 11) occurs at t =1. dy dx = dy/dt dx/dt = 3t2 12 2t =) m = dy = 3t2 12 dx t=1 2t The tangent line has equation y ( 11) = 9 (x 1) 2 t=1 = 9 2. Figure: Tangent line to a parametric curve Math 267 (University of Calgary) Fall 2015, Winter 2016 9 / 12
Area with parametric curves Recall: If y = f (x) isanon-negativefunctionontheintervala apple x apple b, thentheareaunderthegraphoff and above the x-axis between x = a and x = b is given by A = Z b a f (x) dx If this function f has a parametric representation as x = x(t), y = y(t), where apple t apple y(t) =f (x(t)) and the above integral is equal to Z b Z Z A = f (x) dx = f (x(t)) x 0 (t) dt = y(t) x 0 (t) dt a,then Example Find the area under one arch of the cycloid x = a(t sin t) y = a(1 cos t). Solution. Thecycloidmeetsthex-axis exactly when y =0,i.e.,cost =1. In particular, one arch is formed on the interval 0 apple t apple 2. So, the area is A = Z 2 0 y(t)x 0 (t) dt = Z 2 Z 2 Z 2 = a 2 (1 cos t) 2 dt = a 2 0 = a 2 Z 2 0 1 2cost + 1+cos2t 2 0 a(1 cos t)a(1 cos t) dt 0 1 2cost +cos 2 t dt =... =3 a 2 Math 267 (University of Calgary) Fall 2015, Winter 2016 10 / 12 dt
Arclength with parametric curves Recall that the length of the curve y = f (x), a apple x apple b is given by where ds = L = Z x=b x=a ds, s q dy 2 (dx) 2 +(dy) 2 = 1+ dx. dx If this function f has a parametric representation as x = x(t), y = y(t), where apple t apple,then dx = x 0 (t) dt and dy = y 0 (t) dt. So, the arclength element ds is s q dx 2 dy 2 q ds = (dx) 2 +(dy) 2 = + dt = (x 0 (t)) 2 +(y 0 (t)) 2 dt dt dt The above integral is equal to A = Z t= t= Z ds = q (x 0 (t)) 2 +(y 0 (t)) 2 dt. Note that a polar curve r = g( ), apple apple,definestheparametriccurve x = r cos = g( )cos, y = r sin = g( )sin. One can prove that the arclength element can be simplified into s dx 2 s dy 2 dr 2 ds = + d = r 2 + d. d d d Math 267 (University of Calgary) Fall 2015, Winter 2016 11 / 12
Further Examples/Exercises 1 Find the length of one arch of a cycloid. 2 Find the area bounded inside the astroid x = a cos 3 t, y = a sin 3 t. 3 Find the total length of the astroid. 4 Find the length of the polar curve r =2cos. 5 Find the length of the polar curve r =, 0 apple apple 2. 6 Prove the arclength formula for a polar curve: s dr 2 ds = r 2 + d. d 7 Find a formula to compute the area of the surface obtained by revolving a polar curve about the polar axis. 8 Let 0 < b < a. Showthatthetotallengthoftheellipse x 2 a + y 2 2 b =1 2 is given by Z /2 p L =4a 1 e2 sin 2 d, 0 where e is the eccentricity of the ellipse p a 2 b 2 e =. a Math 267 (University of Calgary) Fall 2015, Winter 2016 12 / 12