82 14 Vector Calculus 14 1 Parametric Equations of Curves In this section we will see how Maple V can be used to plot interesting curves using parametric representation First of all consider the circle centered at the origin with radius 2 The equation for this curve is x2 y2 4 If you wish plot this curve then you soon become aware that the set of points represented by the above equation cannot satisfy a relationship of the form y f x where f is a single valued function the vertical line test fails The graph of the function f 1 x 4 x2 is the upper semi circle of radius 2 P1 plot sqrt 4 x 2 x 2 2 Whereas the following plot gives the bottom half P2 plot sqrt 4 x 2 x 2 2 We can now use display to put both plots together See Figure 70 with plots display P1 P2 scaling constrained We can obtain a parameterization for the entire circle Since the following identity is true for all real values of t 2 20 1 2 1 00 10 1 x 3 2 2 1 00 1 2 3 10 1 20 2 Figure 71 Parabola x t y t3 2t Figure 70 Circle of Radius 2 sin t 2 cos t 2 1 it follows that x 2 sin t y 2 cos t satisfy the equation x2 y2 4 for all values of t Moreover the image of the mapping defined by the parametric equations x 2 cos t y 2 sin t generate the entire circle which is shown in Figure 70 as t varies for 0 to 2 with one plot statement using the parametric plot syntax plot 2 cos t 2 sin t t 0 2 Pi scaling constrained 14 VECTOR CALCULUS 83 If a curve is given as an explicit function of x or y then it is easy to write a parameterization if a curve is described by the relation y f x for a x b then a parameterization for this curve is x t y f t for a t b and similarly if a curveis described as x g y then the parameterization can be of the form x g t y t For example suppose y x3 2 x is given Then either of the following plot commands will give the Figure 71 for any range say 3 x 3 plot x 3 2 x x 3 3 plot t t 3 2 t t 3 3 Similarly if x y2 y is the function you wish to plot then the following command does the trick See Figure 72 plot t 2 t t t 2 2 2 8 6 1 4 00 1 2 3 5 4 6 2 1 1 2 00 1 2 3 2 Figure 72 Parabola x t2 t y t Figure 73 Line x 1 2t y 3 5t Straight lines are very easy to parameterize For example if y mx b is the equation of the line then x t y mt b gives a parameterization But you can sometimes write the parametric equations directly from their geometric description For example a portion of the straight line through the point 1 3 parallel to the vector 2 5 has parametric equations x 1 2t y 3 5t is shown in Figure 73 and can be plotted by the command plot 1 2 t 3 5 t t 1 1 On occasion it possible to plot curves that would be very complicated to express in even implicit algebraic terms For example the following curve is known as a Lissajous Figure and is shown in Figure 74 plot cos 3 t sin 5 t t 0 2 Pi scaling constrained Solutions of differential equations are usually represented in parametric form For example x cos 2t y 2 sin 2t is a solution of the initial value problem d x t x t dt d y t 4 x t dt x 0 1 y 0 0 as can easily be verified A plot of this curve is given by plot cos 2 t 2 sin 2 t t 0 Pi scaling constrained 14 VECTOR CALCULUS 1 0 5 84 1 2 0 5 1 00 0 5 1 1 0 5 00 0 5 1 1 2 Figure 74 Lissajous Figure x cos 3t y sin 5t 0 5 1 Figure 75 Ellipse x cos 2t y 2sin 2t See Figure 75 When plotting curves in three dimensions using Maple V we need to use the command spacecurve which is part of the plots package Consider the curve written parametrically as x cos t y sin t z t It s plot Figure 76 over the range 0 2 is given by spacecurve cos t sin t t t 0 2 Pi Figure 76 Cylindrical Helix x cos t y sin t z t Figure 77 A Closed Curve in Three Space The complicated curve shown in Figure 77 is actually a simple closed curve and can be obtained as follows spacecurve 4 sin 3 t cos 2 t 4 sin 3 t sin 2 t cos 3 t t 0 2 Pi Exercises 14 1 1 Make a Maple V plot of the conical helix given by the parametric equations x t cos 3t y t sin 3t z t t 14 VECTOR CALCULUS 85 2 Write parametric equations for first octant portion of the curve of intersection of the sphere and cylinder x2 y2 z2 64 x2 y 4 2 16 Make a Maple V plot of this curve 3 Show that x sin 2t cos t y sin 2t sin t 0 t 2 is a parametric representation for the curve given implicitly by the equation x2 y2 3 4x2 y2 Plot the graph of this curve using Maple V using the parametric representation and also using the explicit repesentation Is it worthwhile to obtain the parametric representation 14 VECTOR CALCULUS 14 2 86 Parametric Equations of Surfaces As with the implicit representation of a circle the implicit representation of a sphere may require a piece meal approach when making plots In order to plot the graph of the sphere x2 y2 z2 4 we could solve for z and obtain a parameterization for the lower half with p z f 1 x y 4 x2 y2 0 x2 y2 4 and obtain a plot with plot3d P1 plot3d sqrt 4 x 2 y 2 x 2 2 y sqrt 4 x 2 sqrt 4 x 2 style patchnogrid numpoints 2500 In order to get the top half we type in the following P2 plot3d sqrt 4 x 2 y 2 x 2 2 y sqrt 4 x 2 sqrt 4 x 2 style patchnogrid numpoints 2500 Finally to put both plots as shown in Figure 78 at the same time we write with plots display P1 P2 axes normal scaling constrained with plots display P1 P2 scaling constrained 2 2 2 1 1 1 0 00 1 1 1 2 2 2 Figure 78 Sphere of Radius 2 Figure 79 Cone Plot with …