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3: Parametric Equations and Polar Coordinates

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    Parametric equations define a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization. The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth.

    The chambered nautilus is a fascinating creature. This animal feeds on hermit crabs, fish, and other crustaceans. It has a hard outer shell with many chambers connected in a spiral fashion, and it can retract into its shell to avoid predators. When part of the shell is cut away, a perfect spiral is revealed, with chambers inside that are somewhat similar to growth rings in a tree.

    A photo of a cross section of a seashell that spirals from big chambers to smaller and smaller ones.
    Figure 1: The chambered nautilus is a marine animal that lives in the tropical Pacific Ocean. Scientists think they have existed mostly unchanged for about 500 million years. (Credit:CC-BY-NC-SA modification of work by Jitze Couperus, Flickr)

    The mathematical function that describes a spiral can be expressed using rectangular (or Cartesian) coordinates. However, if we change our coordinate system to something that works a bit better with circular patterns, the function becomes much simpler to describe. The polar coordinate system is well suited for describing curves of this type. How can we use this coordinate system to describe spirals and other radial figures?

    In this chapter we also study parametric equations, which give us a convenient way to describe curves, or to study the position of a particle or object in two dimensions as a function of time. We will use parametric equations and polar coordinates for describing many topics later in this text.

    • 3.1: Parametric Equations
      In this section we examine parametric equations and their graphs. In the two-dimensional coordinate system, parametric equations are useful for describing curves that are not necessarily functions. The parameter is an independent variable that both x and y depend on, and as the parameter increases, the values of x and y trace out a path along a plane curve.
    • 3.2: Calculus of Parametric Curves
      Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?
    • 3.3: Polar Coordinates
      The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs. The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates.
    • 3.4: Area and Arc Length in Polar Coordinates
      In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. In particular, if we have a function y=f(x) defined from x=a to x=b where f(x)>0 on this interval, the area between the curve and the x-axis is given by A=∫f(x)dx. This fact, along with the formula for evaluating this integral, is summarized in the Fundamental Theorem of Calculus. In this section, we study analogous formulas for area and arc length in the polar coordinate system.
    • 3.5: Chapter 3 Review Exercises

    Thumbnail: A polar coordinate system graph with the pole at the center and radial lines extending outward at 30-degree intervals. The radial lines are labeled with their corresponding angles in degrees. (CC BY NC SA 4.0; OpenStax)

    Contributors and Attributions

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

    This page titled 3: Parametric Equations and Polar Coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, Vinh Kha Nguyen and Jyothsna Viswanadha via source content that was edited to the style and standards of the LibreTexts platform.