Parametric Curves. EXAMPLE: Sketch and identify the curve defined by the parametric equations

Transcription

1 Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves 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. x = cost, y = sin t 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 =. x = sin t, y = cos t 0 t π

2 Section 9. Parametric Curves 00 Kiryl Tsishchanka x = sin t, y = cos t 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 =. x = cost, y = sin t 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 x = t cost, y = t sin t t > 0

3 Section 9. Parametric Curves 00 Kiryl Tsishchanka x = t cost, y = t sin t t > 0 Solution: If we plot points, it appears that the curve is a spiral (see page 9). We can confirm this impression by the following algebraic manipulations: x + y = (t cost) + (t sin t) = t sin t + t cos t = t (sin t + cos t) = t = x + y = t To eliminate t completely, we observe that y x = t sin t t cost = sin t ( y ) cost = tan t = 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 x = t + t, y = t

4 Section 9. Parametric Curves 00 Kiryl Tsishchanka 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 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 = 4 cos t = 4x + y = 4 sin t + 4 cos 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

5 Section 9. Parametric Curves 00 Kiryl Tsishchanka 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 r sin θ = r(θ sin θ) Therefore, parametric equations of the cycloid are y = TC QC = r r cosθ = r( cosθ) x = r(θ sin θ), 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 ( = cos y ) y ( y ) r r r

6 Section 9. Parametric Curves 00 Kiryl Tsishchanka Unit Circle where t = kπ/6, k = 0,...,. x = cost, y = sin t 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 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 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 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

7 Section 9. Parametric Curves 00 Kiryl Tsishchanka Unit Circle where t = kπ/6, k = 0,...,. x = sin t, y = cos t x sin t, y cos t, 0 t Pi 6.0 x sin t, y cos t, 0 t Pi 6.0 x sin t, y cos t, 0 tpi 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 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 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

8 Section 9. Parametric Curves 00 Kiryl Tsishchanka Ellipse where t = kπ/6, k = 0,...,. x = cost, y = sin t 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 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 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 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

9 Section 9. Parametric Curves 00 Kiryl Tsishchanka Spiral where t = kπ/4, k = 0,...,. x = t cost, y = t sin t x t cos t, y t sin t, 0 t Pi..0 x t cos t, y t sin t, 0 t Pi x t cos t, y t sin t, 0 tpi 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 x t cos t, y t sin t, 0 t 7Pi x t cos t, y t sin t, 0 t 8Pi x t cos t, y t sin t, 0 t 9Pi 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

10 Section 9. Parametric Curves 00 Kiryl Tsishchanka 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 x t^ t, y t, t x t^ t, y t, t x t^ t, y t, t x t^ t, y t, t 7 x t^ t, y t, t 8 x t^ t, y t, t x t^ t, y t, t 0 x t^ t, y t, t x t^ t, y t, t

11 Section 9. Parametric Curves 00 Kiryl Tsishchanka 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 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 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 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

12 Section 9. Parametric Curves 00 Kiryl Tsishchanka Butterfly Curve [ ( )] [ ( )] t t x = sin t e cos t cos(4t) + sin, y = cost e cos t cos(4t) + sin where t = kπ/6, k = 0,...,.

13 Section 9. Parametric Curves 00 Kiryl Tsishchanka Butterfly Curve [ ( )] [ ( )] t t x = sin t e cos t cos(4t) + sin, y = cost e cos t cos(4t) + sin where t = kπ/, k = 0,...,.

14 Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curve where t = π + kπ/, k = 0,...,. x = t + sin t, y = t + cost 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 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 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 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

15 Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves x =. cost coskt, y =. sint sin kt where k =,..., 0. 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

16 Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves x =. cost cos t, y =. sin t sin kt where k =,..., 0. 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