# 11: Parametric Equations and Polar Coordinates

- Page ID
- 2579

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

- 11.0: Prelude to Parametric Equations and Polar Coordinates
- 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.

- 11.1: 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.

- 11.2: 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?

- 11.3: 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.

- 11.4: 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.

- 11.5: Conic Sections
- Conic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the y-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle.

- 11.E: Parametric Equations and Polar Coordinates (Exercises)
- These are homework exercises to accompany OpenStax's "Calculus" Textmap.

## 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.