Section 8.4 Plane Curves and Parametric Equations Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called parametric equations). Each value of t determines a point (x,y), which we can plot in a coordinate plane. As t varies, the point (x,y) = (f(t),g(t)) varies and traces out a curve C, which we call a parametric curve. EXAMPLE: Sketch and identify the curve defined by the parametric equations x = cost, y = sint 0 t π Solution: If we plot points, it appears that the curve is a circle (see the figure below and page 6). We can confirm this impression by eliminating t. If fact, we have x +y = cos t+sin t = Thus the point (x,y) moves on a unit circle x +y =. EXAMPLE: Sketch and identify the curve defined by the parametric equations x = sint, y = cost 0 t π
EXAMPLE: Sketch and identify the curve defined by the parametric equations x = sint, y = cost 0 t π Solution: If we plot points, it appears that the curve is a circle (see the Figure below). We can confirm this impression by eliminating t. If fact, we have x +y = sin t+cos t = Thus the point (x,y) moves on a unit circle x +y =. EXAMPLE: Sketch and identify the curve defined by the parametric equations x = cost, y = sint 0 t π Solution: If we plot points, it appears that the curve is an ellipse (see page 8). We can confirm this impression by eliminating t. If fact, we have x ( + y x ( y ) 4 ) = + = cos t+sin t = Thus the point (x,y) moves on an ellipse x + y 4 =. x cos t, y sin t, 0 t Pi 6 4 EXAMPLE: Sketch and identify the curve defined by the parametric equations x = tcost, y = tsint t > 0
EXAMPLE: Sketch and identify the curve defined by the parametric equations x = tcost, y = tsint t > 0 Solution: Ifweplotpoints, itappearsthatthecurveisaspiral(seepage9). Wecanconfirmthisimpression by the following algebraic manipulations: x +y = (tcost) +(tsint) = t sin t+t cos t = t (sin t+cos t) = t = x +y = t To eliminate t completely, we observe that y x = tsint tcost = sint ( y ) cost = tant = t = arctan x ( y ) Substituting this into x +y = t, we get x +y = arctan. x x t cos t, y t sin t, 0 t Pi 0 0 0 0 EXAMPLE: Sketch and identify the curve defined by the parametric equations x = t +t, y = t
EXAMPLE: Sketch and identify the curve defined by the parametric equations x = t +t, y = t Solution: If we plot points, it appears that the curve is a parabola (see page 0). We can confirm this impression by eliminating t. If fact, we have y = t = t = y + ( ) y + = x = t +t = + y + = 4 y +y + 4 Thus the point (x,y) moves on a parabola x = 4 y +y + 4. x t^ t, y t, t.0..0. EXAMPLE: Sketch and identify the curve defined by the parametric equations x = sin t, y = cost Solution: If we plot points, it appears that the curve is a restricted parabola (see page ). We can confirm this impression by eliminating t. If fact, we have y = cost = y = 4cos t = 4x+y = 4sin t+4cos t = 4 = x = y 4 We also note that 0 x and y. Thus the point (x,y) moves on the restricted parabola x = y 4. x sin^ t, y cos t, 0 t Pi 6 0.40.60.8.0. 4
The Cycloid EXAMPLE: The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid (see the Figure below). If the circle has radius r and rolls along the x-axis and if one position of P is the origin, find parametric equations for the cycloid. Solution: We choose as parameter the angle of rotation θ of the circle (θ = 0 when P is at the origin). Suppose the circle has rotated through θ radians. Because the circle has been in contact with the line, we see from the Figure below that the distance it has rolled from the origin is OT = arc PT = rθ Therefore, the center of the circle is C(rθ,r). Let the coordinates of P be (x,y). Then from the Figure above we see that x = OT PQ = rθ rsinθ = r(θ sinθ) Therefore, parametric equations of the cycloid are y = TC QC = r rcosθ = r( cosθ) x = r(θ sinθ), y = r( cosθ) θ R () One arch of the cycloid comes from one rotation of the circle and so is described by 0 θ π. Although Equations were derived from the Figure above, which illustrates the case where 0 < θ < π/, it can be seen that these equations are still valid for other values of θ. Although it is possible to eliminate the parameter θ from Equations, the resulting Cartesian equation in x and y is very complicated and not as convenient to work with as the parametric equations: x r +π x πr ( = y ) cos y ( y ) r r r
Unit Circle where t = kπ/6, k = 0,...,. x = cost, y = sint x cos t, y sin t, 0 t Pi 6.0 x cos t, y sin t, 0 t Pi 6.0 x cos t, y sin t, 0 tpi 6.0.0.0.0.0.0.0.0.0.0 x cos t, y sin t, 0 tpi 6.0 x cos t, y sin t, 0 tpi 6.0 x cos t, y sin t, 0 t 6Pi 6.0.0.0.0.0.0.0.0.0.0 x cos t, y sin t, 0 t 7Pi 6.0 x cos t, y sin t, 0 t 8Pi 6.0 x cos t, y sin t, 0 t 9Pi 6.0.0.0.0.0.0.0.0.0.0 x cos t, y sin t, 0 t 0Pi 6.0 x cos t, y sin t, 0 t Pi 6.0 x cos t, y sin t, 0 t Pi 6.0.0.0.0.0.0.0.0.0.0 6
Unit Circle where t = kπ/6, k = 0,...,. x sin t, y cos t, 0 t Pi 6.0 x = sint, y = cost x sin t, y cos t, 0 t Pi 6.0 x sin t, y cos t, 0 tpi 6.0.0.0.0.0.0.0.0.0.0 x sin t, y cos t, 0 tpi 6.0 x sin t, y cos t, 0 tpi 6.0 x sin t, y cos t, 0 t 6Pi 6.0.0.0.0.0.0.0.0.0.0 x sin t, y cos t, 0 t 7Pi 6.0 x sin t, y cos t, 0 t 8Pi 6.0 x sin t, y cos t, 0 t 9Pi 6.0.0.0.0.0.0.0.0.0.0 x sin t, y cos t, 0 t 0Pi 6.0 x sin t, y cos t, 0 t Pi 6.0 x sin t, y cos t, 0 t Pi 6.0.0.0.0.0.0.0.0.0.0 7
Ellipse where t = kπ/6, k = 0,...,. x = cost, y = sint x cos t, y sin t, 0 t Pi 6 x cos t, y sin t, 0 t Pi 6 x cos t, y sin t, 0 tpi 6 4 4 4 x cos t, y sin t, 0 tpi 6 x cos t, y sin t, 0 tpi 6 x cos t, y sin t, 0 t 6Pi 6 4 4 4 x cos t, y sin t, 0 t 7Pi 6 x cos t, y sin t, 0 t 8Pi 6 x cos t, y sin t, 0 t 9Pi 6 4 4 4 x cos t, y sin t, 0 t 0Pi 6 x cos t, y sin t, 0 t Pi 6 x cos t, y sin t, 0 t Pi 6 4 4 4 8
Spiral where t = kπ/4, k = 0,...,. x t cos t, y t sin t, 0 t Pi..0 x = tcost, y = tsint x t cos t, y t sin t, 0 t Pi x t cos t, y t sin t, 0 tpi..0.0..0. x t cos t, y t sin t, 0 tpi 4 x t cos t, y t sin t, 0 tpi 4 x t cos t, y t sin t, 0 t 6Pi 6 4 4 4 6 4 6 6 x t cos t, y t sin t, 0 t 7Pi 6 4 x t cos t, y t sin t, 0 t 8Pi x t cos t, y t sin t, 0 t 9Pi 6 4 6 6 x t cos t, y t sin t, 0 t 0Pi 0 x t cos t, y t sin t, 0 t Pi 0 x t cos t, y t sin t, 0 t Pi 0 0 0 0 0 0 0 0 0 0 9
Parabola where t = +k/4, k = 0,...,. x t^ t, y t, t x = t +t, y = t x t^ t, y t, t x t^ t, y t, t.0..0..0..0..0..0. x t^ t, y t, t x t^ t, y t, t x t^ t, y t, t 6.0..0..0..0..0..0. x t^ t, y t, t 7 x t^ t, y t, t 8 x t^ t, y t, t 9.0..0..0..0..0..0. x t^ t, y t, t 0 x t^ t, y t, t x t^ t, y t, t.0..0..0..0..0..0. 0
Restricted Parabola where t = kπ/6, k = 0,...,. x = sin t, y = cost x sin^ t, y cos t, 0 t Pi 6 x sin^ t, y cos t, 0 t Pi 6 x sin^ t, y cos t, 0 tpi 6 0.40.60.8.0. 0.40.60.8.0. 0.40.60.8.0. x sin^ t, y cos t, 0 tpi 6 x sin^ t, y cos t, 0 tpi 6 x sin^ t, y cos t, 0 t 6Pi 6 0.40.60.8.0. 0.40.60.8.0. 0.40.60.8.0. x sin^ t, y cos t, 0 t 7Pi 6 x sin^ t, y cos t, 0 t 8Pi 6 x sin^ t, y cos t, 0 t 9Pi 6 0.40.60.8.0. 0.40.60.8.0. 0.40.60.8.0. x sin^ t, y cos t, 0 t 0Pi 6 x sin^ t, y cos t, 0 t Pi 6 x sin^ t, y cos t, 0 t Pi 6 0.40.60.8.0. 0.40.60.8.0. 0.40.60.8.0.
Butterfly Curve [ ( )] [ ( )] t t x = sint e cost cos(4t)+sin, y = cost e cost cos(4t)+sin where t = kπ/6, k = 0,...,.
Butterfly Curve [ ( )] [ ( )] t t x = sint e cost cos(4t)+sin, y = cost e cost cos(4t)+sin where t = kπ/, k = 0,...,.
Parametric Curve x = t+sint, y = t+cost where t = π +kπ/, k = 0,...,. x t sin t,y t cost, Pi t Pi Pi x t sin t,y t cost, Pi t Pi Pi x t sin t,y t cost, Pi t PiPi 6 4 6 6 4 6 6 4 6 x t sin t,y t cost, Pi t PiPi x t sin t,y t cost, Pi t PiPi x t sin t,y t cost, Pi t Pi 6Pi 6 4 6 6 4 6 6 4 6 x t sin t,y t cost, Pi t Pi 7Pi x t sin t,y t cost, Pi t Pi 8Pi x t sin t,y t cost, Pi t Pi 9Pi 6 4 6 6 4 6 6 4 6 x t sin t,y t cost, Pi t Pi 0Pi x t sin t,y t cost, Pi t Pi Pi x t sin t,y t cost, Pi t Pi Pi 6 4 6 6 4 6 6 4 6 4
Parametric Curves where k =,...,0. x =.cost coskt, y =.sint sinkt x.cos t cos t, y.sin t sin t x.cos t cos t, y.sin t sin t x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cos 6t, y.sin t sin 6t x.cos t cos 7t, y.sin t sin 7t x.cos t cos 8t, y.sin t sin 8t x.cos t cos 9t, y.sin t sin 9t x.cos t cos 0t, y.sin t sin 0t x.cos t cos t, y.sin t sin t x.cos t cos t, y.sin t sin t x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cos 6t, y.sin t sin 6t x.cos t cos 7t, y.sin t sin 7t x.cos t cos 8t, y.sin t sin 8t x.cos t cos 9t, y.sin t sin 9t x.cos t cos 0t, y.sin t sin 0t
Parametric Curves where k =,...,0. x =.cost cost, y =.sint sinkt x.cos t cost, y.sin t sin t x.cos t cost, y.sin t sin t x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cost, y.sin t sin 6t x.cos t cost, y.sin t sin 7t x.cos t cost, y.sin t sin 8t x.cos t cost, y.sin t sin 9t x.cos t cost, y.sin t sin 0t x.cos t cost, y.sin t sin t x.cos t cost, y.sin t sin t x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cost, y.sin t sint x.cos t cost, y.sin t sin 6t x.cos t cost, y.sin t sin 7t x.cos t cost, y.sin t sin 8t x.cos t cost, y.sin t sin 9t x.cos t cost, y.sin t sin 0t 6