offer a powerful way to represent curves using two functions of a . They can describe complex shapes like circles and cycloids that regular equations can't handle. This flexibility makes them useful in physics and engineering.
Mastering parametric equations involves understanding their construction, behavior, and applications. From projectile motion to planetary orbits, these equations model real-world phenomena. They also connect to calculus, allowing for differentiation and integration of complex curves.
Parametric Equations
Parametric vs rectangular equations
Parametric equations represent a curve using two equations, x(t) and y(t), where t is an independent variable called the parameter
Rectangular equations represent a curve using a single equation, y=f(x), where y is a function of x
Parametric equations can represent curves that fail the vertical line test and cannot be represented by a single (x2+y2=1, a circle)
Some curves are easier to represent using parametric equations (cycloids, trochoids)
Construction of parametric equations
Circles centered at the origin: x(t)=rcos(t) and y(t)=rsin(t), where r is the radius and 0≤t≤2π
Ellipses centered at the origin: x(t)=acos(t) and y(t)=bsin(t), where a and b are the horizontal and vertical radii, and 0≤t≤2π
Line segments between points (x1,y1) and (x2,y2): x(t)=x1+(x2−x1)t and y(t)=y1+(y2−y1)t, where 0≤t≤1
Cycloids generated by a circle of radius r rolling along the x-axis: x(t)=r(t−sin(t)) and y(t)=r(1−cos(t))
Lissajous curves: x(t)=Asin(at+δ) and y(t)=Bsin(bt), where A, B, a, b, and δ are constants that determine the shape of the curve
Behavior of parametric curves
The parameter t often represents time, and as t increases, the point (x(t),y(t)) traces out the curve
The direction of the curve depends on how x(t) and y(t) change with respect to t
If both x(t) and y(t) increase with t, the curve moves up and to the right
If x(t) increases and y(t) decreases, the curve moves down and to the right
The speed of the point along the curve is determined by the rate of change of x(t) and y(t) with respect to t
If dtdx and dtdy are both large, the point moves quickly along the curve
If dtdx and dtdy are both small, the point moves slowly along the curve
Parametric curves can self-intersect or have cusps and loops (folium of Descartes, x(t)=1+t33at and y(t)=1+t33at2)
Curve tracing techniques can be used to visualize and analyze the behavior of parametric curves
Applications of parametric equations
Projectile motion: x(t)=v0cos(θ)t and y(t)=v0sin(θ)t−21gt2, where v0 is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity
Determine the maximum height: ymax=2gv02sin2(θ)
Calculate the range: R=gv02sin(2θ)
Find the time of flight: tflight=g2v0sin(θ)
Planetary motion: Kepler's laws can be modeled using parametric equations for elliptical orbits: x(t)=a(cos(t)−e) and y(t)=b1−e2sin(t), where a and b are the semi-major and semi-minor axes, and e is the eccentricity
Harmonic motion: x(t)=Acos(ωt+ϕ), where A is the amplitude, ω is the angular frequency, and φ is the phase shift
Model the motion of a spring-mass system or a pendulum
Analyze the relationship between position, velocity, and acceleration
Vector-valued functions can be used to represent parametric curves in higher dimensions
Calculus with parametric equations
Parametric differentiation: Find the derivative of a parametric curve using the chain rule
dxdy=dx/dtdy/dt, where dtdy and dtdx are the derivatives of y(t) and x(t) with respect to t
Parametric integration: Calculate the arc length or area under a parametric curve
Arc length: L=∫t1t2(dtdx)2+(dtdy)2dt
Area: A=∫t1t2y(t)dtdxdt
Phase plane analysis: Study the behavior of dynamical systems by plotting parametric curves in the phase plane, where each axis represents a different variable of the system
Key Terms to Review (5)
Conic sections: Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These curves include circles, ellipses, parabolas, and hyperbolas.
Curvilinear path: A curvilinear path is the trajectory of a point that moves along a curve in a plane. It is often described using parametric equations in algebra and trigonometry.
Parameter: A parameter is a variable that defines a family of objects or functions, often serving as an independent variable in parametric equations. It allows for the representation of curves and surfaces using coordinates that depend on this variable.
Parametric equations: Parametric equations represent a set of related quantities as explicit functions of one or more independent variables called parameters. They are often used to describe the motion of objects and to graph complex curves.
Rectangular equation: A rectangular equation is an equation that expresses a relationship between $x$ and $y$ coordinates in the Cartesian coordinate system. These equations are typically written in the form $y = f(x)$ or $g(x, y) = 0$.