Parametric Equations and Polar Coordinates



Parametric Equations

Parametric equations are a set of equations in which the coordinates (e.g., [latex]x[/latex] and [latex]y[/latex]) are expressed in terms of a single third parameter.

Learning Objectives

Express two variables in terms of a third variable using parametric equations

Key Takeaways

Key Points

  • Parametric equations are useful for drawing curves, as the equation can be integrated and differentiated term-wise.
  • A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter.
  • Equations can be converted between parametric equations and a single equation.

Key Terms

  • coordinate: a number representing the position of a point along a line, arc, or similar one-dimensional figure

In mathematics, a parametric equation of a curve is a representation of the curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter. For example,

[latex]x = \cos(t) \\ y = \sin(t)[/latex]

is a parametric equation for the unit circle, where [latex]t[/latex] is the parameter. A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter. The notion of parametric equation has been generalized to surfaces of higher dimension with a number of parameters equal to the dimension of the manifold (dimension one and one parameter for curves, dimension two and two parameters for surfaces, etc.)

For example, the simplest equation for a parabola [latex]y=x^2[/latex] can be parametrized by using a free parameter [latex]t[/latex], and setting [latex]x=t[/latex] and [latex]y = t^2[/latex].

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise. Thus, one can describe the velocity of a particle following such a parametrized path as follows:

[latex]v(t) = (x'(t), y'(t))[/latex]

This is a function of the derivatives of [latex]x[/latex] and [latex]y[/latex] with respect to the parameter [latex]t[/latex].

Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations. If one of these equations can be solved for [latex]t[/latex], the expression obtained can be substituted into the other equation to obtain an equation involving [latex]x[/latex] and [latex]y[/latex] only. If there are rational functions, then the techniques of the theory of equations such as resultants can be used to eliminate [latex]t[/latex]. In some cases there is no single equation in closed form that is equivalent to the parametric equations.

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Parametric Example: One example of a sketch defined by parametric equations. Note that it is graphed on parametric axes.

Calculus with Parametric Curves

Calculus can be applied to parametric equations as well.

Learning Objectives

Use differentiation to describe the vertical and horizontal rates of change in terms of [latex]t[/latex]

Key Takeaways

Key Points

  • Parametric equations are equations that depend on a single parameter.
  • A common example comes from physics. The trajectory of an object is well represented by parametric equations.
  • Writing the horizontal and vertical displacement in terms of the time parameter makes finding the velocity a simple matter of differentiating by the parameter time. Parameterizing makes this kind of analysis straight-forward.

Key Terms

  • acceleration: the change of velocity with respect to time (can include deceleration or changing direction)
  • displacement: a vector quantity which denotes distance with a directional component
  • trajectory: the path of a body as it travels through space

Parametric equations are equations which depend on a single parameter. You can rewrite [latex]y=x[/latex] such that [latex]x=t[/latex] and [latex]y=t[/latex] where [latex]t[/latex] is the parameter.

A common example occurs in physics, where it is necessary to follow the trajectory of a moving object. The position of the object is given by [latex]x[/latex] and [latex]y[/latex], signifying horizontal and vertical displacement, respectively. As time goes on the object flies through its path and [latex]x[/latex] and [latex]y[/latex] change. Therefore, we can say that both [latex]x[/latex] and [latex]y[/latex] depend on a parameter [latex]t[/latex], which is time.

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Trajectories: A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time.

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise. Thus, one can describe the velocity of a particle following such a parametrized path as:

[latex]v(t)=r'(t) \\ \, \quad =(x'(t),y'(t),z'(t)) \\ \,\quad =(-a\sin(t),a \cos(t),b)[/latex]

where [latex]v[/latex] is the velocity, [latex]r[/latex] is the distance, and [latex]x[/latex], [latex]y[/latex], and [latex]z[/latex] are the coordinates. The apostrophe represents the derivative with respect to the parameter.

The acceleration can be written as follows with the double apostrophe signifying the second derivative:

[latex]a(t)=r''(t) \\ \, \quad =(x''(t),y''(t),z''(t)) \\ \, \quad =(-a\cos(t),-a \sin(t),b)[/latex]

Writing these equations in parametric form gives a common parameter for both equations to depend on. This makes integration and differentiation easier to carry out as they rely on the same variable. Writing [latex]x[/latex] and [latex]y[/latex] explicitly in terms of [latex]t[/latex] enables one to differentiate and integrate with respect to [latex]t[/latex]. The horizontal velocity is the time rate of change of the [latex]x[/latex] value, and the vertical velocity is the time rate of change of the [latex]y[/latex] value. Writing in parametric form makes this easier to do.

Polar Coordinates

Polar coordinates define the location of an object in a plane by using a distance and an angle from a reference point and axis.

Learning Objectives

Use a polar coordinate to define a point with [latex]r[/latex] (distance from pole), and [latex]\theta[/latex](angle between axis and ray)

Key Takeaways

Key Points

  • Polar coordinates use a distance from a central point called a radial distance, usually specified as [latex]r[/latex].
  • Polar coordinates use an angle measurement from a polar axis, which is usually positioned as horizontal and pointing to the right. Counterclockwise is usually positive.
  • To convert from Polar coordinates to Cartesian coordinates, draw a triangle from the horizontal axis to the point. The [latex]r[/latex] coordinate is [latex]r \cos \theta[/latex] and the y coordinate is [latex]r \sin \theta[/latex].

Key Terms

  • polar: of a coordinate system, specifying the location of a point in a plane by using a radius and an angle

We use coordinate systems every day, even if we don’t realize it. For example, if you walk 20 meters to the right of the parking lot to find the car, you are using a coordinate system. Coordinate systems are a way of determining the location of a point or object of interest in relation to something else. The coordinate system you are most likely familiar with is the [latex]xy[/latex]-coordinate system, where locations are described as horizontal ([latex]x[/latex]) and vertical ([latex]y[/latex]) distances from an arbitrary point. This is called the Cartesian coordinate system.

The [latex]xy[/latex] or Cartesian coordinate system is not always the easiest system to use for every problem. In certain problems, like those involving circles, it is easier to define the location of a point in terms of a distance and an angle. Such definitions are called polar coordinates.

In polar coordinates, each point on a plane is defined by a distance from a fixed point and an angle from a fixed direction.

The distance is known as the radial distance and is usually denoted as [latex]r[/latex].

The angle is known as the polar angle, or radial angle, and is usually given as [latex]\theta[/latex].

A positive angle is usually measured counterclockwise from the polar axis, and a positive radius is in the same direction as the angle. A negative radius would be opposite the direction of the angle and a negative angle would be measured clockwise from the polar axis. The polar axis is usually drawn horizontal and pointing to the right.

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Polar Coordinates: A set of polar coordinates. Note the polar angle increases as you go counterclockwise around the circle with 0 degrees pointing horizontally to the right.

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Relation between Cartesian and Polar Coordinates: The [latex]x[/latex] Cartesian coordinate is given by [latex]r \cos \theta[/latex] and the [latex]y[/latex] Cartesian coordinate is given by [latex]r \sin \theta[/latex].

Polar coordinates in [latex]r[/latex] and [latex]\theta[/latex] can be converted to Cartesian coordinates [latex]x[/latex] and [latex]y[/latex]. This can be done by noting that the radial distance [latex]r[/latex] and the polar angle [latex]\theta[/latex] can define a triangle with a horizontal length [latex]x[/latex] and vertical length [latex]y[/latex]. Thus, using trigonometry, it can be shown that the [latex]x[/latex] coordinate is [latex]r \cos \theta[/latex] and the [latex]y[/latex] coordinate is [latex]r \sin \theta[/latex].

Area and Arc Length in Polar Coordinates

Area and arc length are calculated in polar coordinates by means of integration.

Learning Objectives

Evaluate arc segment area and arc length using polar coordinates and integration

Key Takeaways

Key Points

  • Arc length is the linear length of the curve if it were straightened out.
  • The area is the size of the region defined by the curve radius and the angle and length of the connection lines enclosing the area.
  • To calculate these dimensions, use integration over the angle.

Key Terms

  • polar: of a coordinate system, specifying the location of a point in a plane by using a radius and an angle

Arc Length

If you were to straighten a curved line out, the measured length would be the arc length. Since it can be very difficult to measure the length of an arc linearly, the solution is to use polar coordinates. Using polar coordinates allows us to integrate along the length of the arc in order to compute its length.

The arc length of the curve defined by a polar function is found by the integration over the curve [latex]r(\theta)[/latex]. Let [latex]L[/latex] denote this length along the curve starting from points [latex]A[/latex] through to point [latex]B[/latex], where these points correspond to [latex]\theta = a[/latex] and [latex]\theta = b[/latex] such that [latex]0 < b-a < 2 \pi[/latex]. The length of [latex]L[/latex] is given by the following integral:

[latex]\displaystyle{L = \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta}[/latex]

Solving this integral will give the length of the arc.

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Arc Length: The curved lines bounding the region [latex]R[/latex] are arcs. Their length can be calculated with calculus. The area of the region [latex]R[/latex] can also be calculated by integration.

Arc Segment Area

To find the area enclosed by the arcs and the radius and polar angles, you again use integration.

Let [latex]R[/latex] denote the region enclosed by a curve [latex]r(\theta)[/latex] and the rays [latex]\theta = a[/latex] and [latex]\theta = b[/latex], where [latex]0 < b-a < 2 \pi[/latex]. Then, the area of [latex]R[/latex] is:

[latex]\displaystyle{\frac{1}{2} \int_a^b r^2 d\theta}[/latex]

This result can be found as follows.

First, the interval [latex][a, b][/latex] is divided into [latex]n[/latex] subintervals, where [latex]n[/latex] is an arbitrary positive integer. Thus [latex]\Delta \theta[/latex], the length of each subinterval, is equal to [latex]b-a[/latex] (the total length of the interval), divided by [latex]n[/latex], the number of subintervals. For each subinterval [latex]i = 1, 2, \cdots, n[/latex], let [latex]\theta_i[/latex] be the midpoint of the subinterval, and construct a sector with the center at the pole, radius [latex]r(\theta_i)[/latex], central angle [latex]\Delta \theta[/latex] and arc length [latex]r(\theta_i)\Delta\theta[/latex]. The area of each constructed sector is therefore equal to

[latex]\displaystyle{\frac{1}{2} r^2 \Delta\theta}[/latex]

And the total area is the sum of these sectors. An infinite sum of these sectors is the same as integration.

Conic Sections

Conic sections are defined by intersections of cones with planes.

Learning Objectives

Identify conic sections as curves obtained from the intersection of a cone with a plane

Key Takeaways

Key Points

  • Conic sections are curves obtained from an intersection of a cone with a plane.
  • Graphs of quadratic equations in two variables are conic sections.
  • Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton’s law of universal gravitation are conic sections if their common center of mass is considered to be at rest.

Key Terms

  • cone: a surface of revolution formed by rotating a segment of a line around another line that intersects the first line

In mathematics, a conic section (or just “conic”) is a curve obtained from the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible, one of the most useful being that a conic consists of those points whose distances to some other point (called a focus) and some other line (called a directrix) are in a fixed ratio, called the eccentricity.

Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of such interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity—those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic, the circle is a limiting case with eccentricity 0. In modern geometry, certain degenerate cases—such as the union of two lines—are included as conics as well.

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section—though it may be degenerate—and all conic sections arise in this way.

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton’s law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations.

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Types of Conic Sections: There are three types of conic sections: 1.Parabola; 2. Ellipse; 3. Hyperbola.

Arc Length and Speed

Arc length and speed in parametric equations can be calculated using integration and the Pythagorean theorem.

Learning Objectives

Calculate arc length by integrating the speed of a moving object with respect to time

Key Takeaways

Key Points

  • Arc length is the length of a curve. To calculate it in parametric equations, employ the Pythagorean Theorem.
  • Arc lengths can be calculated by adding up a series of infinitesimal lengths along the arc. To do this, set up an integral over the parameter.
  • Speed is the rate of change of the arc length with respect to time. The derivatives of [latex]x[/latex] and [latex]y[/latex] with respect to time are plugged into the Pythagorean Theorem to give the speed of an object traveling in an arc.

Key Terms

  • Pythagorean Theorem: A theorem stating that the hypotenuse of a right triangle is equal to the square root of the sum of the square of the other two sides
  • curve: a simple figure containing no straight portions and no angles

The length of a curve can be difficult to measure. A curve may be thought of as an infinite number of infinitesimal straight line segments, each pointing in a slightly different direction to make up the curve. Adding up all these lengths together would be equivalent to stretching the curve out straight and measuring its length. The length of the curve is called the arc length.

In order to calculate the arc length, we use integration because it is an efficient way to add up a series of infinitesimal lengths.

Arc lengths can be used to find the distance traveled by an object with an arcing path. Consider a case in which an object movies along a path in the Cartesian plane (the [latex]xy[/latex]-plane). Its position horizontally is given by [latex]x=f(t)[/latex] and its position vertically is given by [latex]y=g(t)[/latex], where [latex]f[/latex] and [latex]g[/latex] are functions which depend on a parameter, [latex]t[/latex]. Since there are two functions for position, and they both depend on a single parameter—time—we call these equations parametric equations.

The distance, or arc length, the object travels through its motion is given by the equation:

[latex]\displaystyle{D = \int_{t_1}^{t_2} \sqrt{ \left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2} dt}[/latex]

This equation is obtained using the Pythagorean Theorem. The arc length is calculated by laying out an infinite number of infinitesimal right triangles along the curve. Each of these triangles has a width [latex]dx[/latex] and a height [latex]dy[/latex], standing for an infinitesimal increase in [latex]x[/latex] and [latex]y[/latex]. By the Pythagorean Theorem, each hypotenuse will have length [latex]\sqrt{dx^2 + dy^2}[/latex]. Adding up each tiny hypotenuse yields the arc length.

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Approximating Arc Length with Hypotenuses: The length of a curve can be approximated by using a series of right triangles with the hypotenuses lying along the curve. The smaller the triangles one uses, the closer the approximation will be.

However, since [latex]x[/latex] and [latex]y[/latex] depend on the parameter [latex]t[/latex], we will want to integrate over [latex]t[/latex], not over [latex]x[/latex] and [latex]y[/latex]. Taking the derivative of [latex]x[/latex] and [latex]y[/latex] with respect to [latex]t[/latex], we find the rate of change of the distance with time. This is also known as the speed. As shown previously using the Pythagorean Theorem, it is given by:

[latex]\displaystyle{\sqrt{ \left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2}}[/latex]

where the rate of change of the hypotenuse length depends on the rate of change of [latex]x[/latex] and [latex]y[/latex]. Integrating the speed with respect to time gives the distance as shown above.

Conic Sections in Polar Coordinates

Conic sections are sections of cones and can be represented by polar coordinates.

Learning Objectives

Identify types of conic sections using polar coordinates

Key Takeaways

Key Points

  • Conic sections are the intersections of cones with a plane.
  • The three types of conic sections are the hyperbola, parabola, and ellipse.
  • Polar coordinates offer us a useful way of representing conic sections.

Key Terms

  • cone: a surface of revolution formed by rotating a segment of a line around another line that intersects the first line
  • hyperbola: a conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone

In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. There are a number of other geometric definitions possible. One of the most useful definitions, in that it involves only the plane, is that a conic consists of those points whose distances to some point—called a focus—and some line—called a directrix—are in a fixed ratio, called the eccentricity.

Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of such sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic, the circle is a limiting case with eccentricity 0. In modern geometry, certain degenerate cases, such as the union of two lines, are included as conics as well.

In polar coordinates, a conic section with one focus at the origin is given by the following equation:

[latex]\displaystyle{r = \frac{l}{1+ecos(\theta)}}[/latex]

where e is the eccentricity and l is half the latus rectum. As in the figure, for [latex]e = 0[/latex], we have a circle, for [latex]0 < e < 1[/latex] we obtain an ellipse, for [latex]e = 1[/latex] a parabola, and for [latex]e > 1[/latex] a hyperbola.