Curves Given Parametrically

1 Parametrized curve 1.1 Parametrized curve Parametrized curve Parametrized curve A parametrized Curve is a path in the xy-plane traced out by the point (x(t), y(t)) as the parameter t ranges over an interval I.  C = (x(t), y(t)) : t ∈ I Examples 1. • The graph of a function y = f (x), x ∈ I, is a curve C that is parametrized by x(t) = t, y(t) = f (t), t ∈ I. • The graph of a polar equation r = ρ(θ), θ ∈ I, is a curve C that is parametrized by the functions x(t) = r cos t = ρ(t) cos t, y(t) = r sin t = ρ(t) sin t, 1 t ∈ I. 1.2 Examples Example: Line Segment Line Segment: y = 2x, x ∈ [1, 3] • Set x(t) = t, then y(t) = 2t, t ∈ [1, 3] • Set x(t) = t + 1, then y(t) = 2t + 2, t ∈ [0, 2] • Set x(t) = 3 − t, then y(t) = 6 − 2t, t ∈ [0, 2] • Set x(t) = 3 − 4t, then y(t) = 6 − 8t, t ∈ [0, 1/2] 2 • Set x(t) = 2 − cos t, then y(t) = 4 − 2 cos t, t ∈ [0, 4π] We parametrize the line segment in different ways and interpret each parametrization as the motion of a particle with the parameter t being time. Example: Parabola Parabola Arc: x = 1 − y 2 , −1 ≤ y ≤ 1 • Set y(t) = t, then x(t) = 1 − t2 , t ∈ [−1, 1] ⇒ changing the domain to all real t gives us the whole parabola. • Set y(t) = cos t, then x(t) = 1 − cos2 t, t ∈ [0, π] ⇒ changing the domain to all real t does not give us any more of the parabola. Example: Spiral of Archimedes Spiral of Archimedes: r = θ, θ ≥ 0 • The curve is a nonending spiral. Here it is shown in detail from θ = 0 to θ = 2π. • The parametric representation is x(t) = t cos t, y(t) = t sin t, 3 t ≥ 0. Example: Limaçons Limaçons (Snails): r = a + b cos θ The parametric representation is

x(t) = a + b cos t cos t, y(t) = a + b cos t sin t, t ∈ [0, 2π]. Example: Petal Curves Petal Curves (Flowers): r = a cos nθ, r = a sin nθ The parametric representations are

x(t) = a cos(nt) cos t, y(t) = a cos(nt) sin t,
x(t) = a sin(nt) cos t, 2 2.1
y(t) = a sin(nt) sin t, Locus Circles Circles: C = {P : d(P, O) = |a|} 4 t ∈ [0, 2π]. t ∈ [0, 2π]. Center O at (0, 0) ⇒ x2 + y 2 = a2 ⇒ r = a ⇒ t ∈ [0, 2π], x(t) = a cos t, y(t) = a sin t Center O at (0, a) ⇒ x2 + (y(− a)2 = a2 ⇒ r = 2a sin θ x(t) = 2a sin t cos t = a sin 2t, ⇒ t ∈ [0, π], y(t) = 2a sin t sin t = a(1 − cos 2t). Another parametric representation is by translation ⇒ t ∈ [0, 2π], x(t) = a cos t, y(t) = a sin t + a Center O at (a, 0) ⇒ (x − a)2(+ y 2 = a2 ⇒ r = 2a cos θ x(t) = 2a cos t cos t = a(1 + cos 2t), π 3π ⇒ t∈[ , ], 2 2 y(t) = 2a cos t sin t = a sin 2t. Another parametric representation is by translation ⇒ t ∈ [0, 2π], x(t) = a cos t + a, y(t) = a sin t. 2.2 Ellipses Ellipses A ellipse is the set of points P in a plane that the sum of whose distances from two fixed points (the foci F1 and F2 ) separated by a distance 2c is a given positive constant 2a.  E = P : |d(P, F1 ) + d(P, F2 )| = 2a √ With F1 at (−c, 0) and F2 at (c, 0) and setting b = a2 − c2 ,   x2 y2 E = (x, y) : 2 + 2 = 1 a b Ellipses: Cosine and Sine The ellipse can also be given by a simple parametric form analogous to that of a circle, but with the x and y coordinates having different scalings, x = a cos t, y = b sin t, t ∈ (0, 2π). Note that cos2 t + sin2 t = 1. 5 2.3 Hyperbolas Hyperbolas A hyperbola is the set of points P in a plane that the difference of whose distances from two fixed points (the foci F1 and F2 ) separated by a distance 2c is a given positive constant 2a.  H = P : |d(P, F1 ) − d(P, F2 )| = 2a √ With F1 at (−c, 0) and F2 at (c, 0) and setting b = c2 − a2 , we have   y2 x2 H = (x, y) : 2 − 2 = 1 a b Hyperbolas: Hyperbolic Cosine and Hyperbolic Sine The right branch of a hyperbola can be parametrized by x = a cosh t, y = b sinh t, t ∈ (−∞, ∞). The left branch can be parametrized by x = −a cosh t, y = b sinh t, t ∈ (−∞, ∞).

Note that cosh t = 21 et + e−t , sinh t = 21 et − e−t and cosh2 t − sinh2 t = 1. Hyperbolas: Other Parametric Representation 6 Another parametric representation for the right branch of the hyperbola is
x = a sec t, y = b tan t, t ∈ −π/2, π/2 . Parametric equations for the left branch is x = −a sec t, 2.4 y = b tan t, Lemniscates Lemniscates (Ribbons): r2 = a2 cos 2θ 7
t ∈ −π/2, π/2 . A lemniscate is the set of points P in a plane that the product of whose distances from two fixed points (the foci F1 and F2 ) a distance 2c away is the constant  c2 . R = P : d(P, F1 ) · d(P, F2 )| = c2 With F1 at (−c, 0) and F2 at (c, 0),

2 x2 + y 2 = 2c2 x2 − y 2 Switching to polar coordinates gives π π
3π 5π
r2 = 2c2 cos 2θ, θ ∈ − , ∪ , 4 4 4 4 The parametric equations for the lemniscate with a2 = 2c2 is a sin t cos t a cos t , y= , t ∈ (0, 2π). x= 1 + sin2 t 1 + sin2 t Outline Contents 1 Parametrized curve 1.1 Parametrized curve . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 Locus 2.1 Circles . . . 2.2 Ellipses . . 2.3 Hyperbolas 2.4 Lemniscates 4 4 5 6 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .