Learning Objectives

- Distinguish between and understand the difference between a rectangular coordinate system and a polar coordinate system.
- Plot points with polar coordinates on a polar plane.

Over (12) kilometers from port, a sailboat encounters rough weather and is blown off course by a (16)-knot wind (see Figure (PageIndex{1})). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.

## Plotting Points Using Polar Coordinates

When we think about plotting points in the plane, we usually think of *rectangular coordinates* ((x,y)) in the *Cartesian coordinate plane*. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled ((r, heta)) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

The polar grid is scaled as the unit circle with the positive (x)*-*axis now viewed as the polar axis and the origin as the pole. The first coordinate (r) is the radius or length of the directed line segment from the pole. The angle ( heta), measured in radians, indicates the direction of (r). We move counterclockwise from the polar axis by an angle of ( heta),and measure a directed line segment the length of (r) in the direction of ( heta). Even though we measure ( heta) first and then (r), the polar point is written with the (r)-coordinate first. For example, to plot the point (left(2,dfrac{pi}{4}
ight)), we would move (dfrac{pi}{4}) units in the counterclockwise direction and then a length of (2) from the pole. This point is plotted on the grid in Figure (PageIndex{2}).

Example (PageIndex{1}): Plotting a Point on the Polar Grid

Plot the point (left(3,dfrac{pi}{2} ight)) on the polar grid.

**Solution**

The angle (dfrac{pi}{2}) is found by sweeping in a counterclockwise direction (90°) from the polar axis. The point is located at a length of (3) units from the pole in the (dfrac{pi}{2}) direction, as shown in Figure (PageIndex{3}).

Exercise (PageIndex{1})

Plot the point (left(2, dfrac{pi}{3} ight)) in the polar grid.

- Answer

Example (PageIndex{2}): Plotting a Point in the Polar Coordinate System with a Negative Component

Plot the point (left(−2, dfrac{pi}{6} ight)) on the polar grid.

**Solution**

We know that (dfrac{pi}{6}) is located in the first quadrant. However, (r=−2). We can approach plotting a point with a negative (r) in two ways:

- Plot the point (left(2,dfrac{pi}{6} ight)) by moving (dfrac{pi}{6}) in the counterclockwise direction and extending a directed line segment (2) units into the first quadrant. Then retrace the directed line segment back through the pole, and continue (2) units into the third quadrant;
- Move (dfrac{pi}{6}) in the counterclockwise direction, and draw the directed line segment from the pole (2) units in the negative direction, into the third quadrant.

See Figure (PageIndex{5a}). Compare this to the graph of the polar coordinate (left(2,dfrac{pi}{6} ight)) shown in Figure (PageIndex{5b}).

Exercise (PageIndex{2})

Plot the points (left(3,−dfrac{pi}{6} ight)) and (left(2,dfrac{9pi}{4} ight)) on the same polar grid.

- Answer

## Converting from Polar Coordinates to Rectangular Coordinates

When given a set of **polar coordinates**, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables (x), (y), (r), and ( heta).

(cos heta=dfrac{x}{r} ightarrow x=r cos heta)

(sin heta=dfrac{y}{r} ightarrow y=r sin heta)

Dropping a perpendicular from the point in the plane to the *x-*axis forms a right triangle, as illustrated in Figure (PageIndex{7}). An easy way to remember the equations above is to think of (cos heta) as the adjacent side over the hypotenuse and (sin heta) as the opposite side over the hypotenuse.

CONVERTING FROM POLAR COORDINATES TO RECTANGULAR COORDINATES

To convert polar coordinates ((r, heta)) to rectangular coordinates ((x, y)), let

[cos heta=dfrac{x}{r} ightarrow x=r cos heta]

[sin heta=dfrac{y}{r} ightarrow y=r sin heta]

How to: Given polar coordinates, convert to rectangular coordinates.

- Given the polar coordinate ((r, heta)), write (x=r cos heta) and (y=r sin heta).
- Evaluate (cos heta) and (sin heta).
- Multiply (cos heta) by (r) to find the (x)
*-*coordinate of the rectangular form. - Multiply (sin heta) by (r) to find the (y)
*-*coordinate of the rectangular form.

Example (PageIndex{3A}): Writing Polar Coordinates as Rectangular Coordinates

Write the polar coordinates (left(3,dfrac{pi}{2} ight)) as rectangular coordinates.

**Solution**

Use the equivalent relationships.

[egin{align*} x&= r cos heta x&= 3 cos dfrac{pi}{2} &= 0 y&= r sin heta y&= 3 sin dfrac{pi}{2} &= 3 end{align*}]

The rectangular coordinates are ((0,3)). See Figure (PageIndex{8}).

Example (PageIndex{3B}): Writing Polar Coordinates as Rectangular Coordinates

Write the polar coordinates ((−2,0)) as rectangular coordinates.

**Solution**

See Figure (PageIndex{9}). Writing the polar coordinates as rectangular, we have

[egin{align*} x&= r cos heta x&= -2 cos(0) &= -2 y&= r sin heta y&= -2 sin(0) &= 0 end{align*}]

The rectangular coordinates are also ((−2,0)).

Exercise (PageIndex{3})

Write the polar coordinates (left(−1,dfrac{2pi}{3} ight)) as rectangular coordinates.

- Answer
((x,y)=left(dfrac{1}{2},−dfrac{sqrt{3}}{2} ight))

## Converting from Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

CONVERTING FROM RECTANGULAR COORDINATES TO POLAR COORDINATES

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in Figure (PageIndex{10}).

(cos heta=dfrac{x}{r}) or (x=r cos heta)

(sin heta=dfrac{y}{r}) or (y=r sin heta)

(r^2=x^2+y^2)

( an heta=dfrac{y}{x})

Example (PageIndex{4}): Writing Rectangular Coordinates as Polar Coordinates

Convert the rectangular coordinates ((3,3)) to polar coordinates.

**Solution**

We see that the original point ((3,3)) is in the first quadrant. To find ( heta), use the formula ( an heta=dfrac{y}{x}). This gives

[egin{align*} an heta&= dfrac{3}{3} an heta&= 1 { an}^{-1}(1)&= dfrac{pi}{4} end{align*}]

To find (r), we substitute the values for (x) and (y) into the formula (r=sqrt{x^2+y^2}). We know that (r) must be positive, as (dfrac{pi}{4}) is in the first quadrant. Thus

[egin{align*} r&= sqrt{3^2+3^2} r&= sqrt{9+9} r&= sqrt{18} &= 3sqrt{2} end{align*}]

So, (r=3sqrt{2}) and ( heta=dfrac{pi}{4}), giving us the polar point ((3sqrt{2},dfrac{pi}{4})). See Figure (PageIndex{11}).

**Analysis**

There are other sets of polar coordinates that will be the same as our first solution. For example, the points (left(−3sqrt{2}, dfrac{5pi}{4} ight)) and (left(3sqrt{2},−dfrac{7pi}{4} ight)) will coincide with the original solution of (left(3sqrt{2}, dfrac{pi}{4} ight)). The point (left(−3sqrt{2}, dfrac{5pi}{4} ight)) indicates a move further counterclockwise by (pi), which is directly opposite (dfrac{pi}{4}). The radius is expressed as (−3sqrt{2}). However, the angle (dfrac{5pi}{4}) is located in the third quadrant and, as (r) is negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point as (left(3sqrt{2}, dfrac{pi}{4} ight)). The point (left(3sqrt{2}, −dfrac{7pi}{4} ight)) is a move further clockwise by (−dfrac{7pi}{4}), from (dfrac{pi}{4}). The radius, (3sqrt{2}), is the same.

## Extra Practice

- Plot the point with polar coordinates (left(3, frac{pi}{6} ight)).
- Plot the point with polar coordinates (left(5,-frac{2 pi}{3} ight))
- Convert (left(6,-frac{3 pi}{4} ight)) to rectangular coordinates.
- Convert (left(-2, frac{3 pi}{2} ight)) to rectangular coordinates.
- Convert (7,-2) to polar coordinates.
- Convert (-9,-4) to polar coordinates.

## Key Equations

Conversion formulas | (cos heta=dfrac{x}{r} ightarrow x=r cos heta) (sin heta=dfrac{y}{r} ightarrow y=r sin heta) (r^2=x^2+y^2) ( an heta=dfrac{y}{x}) |

## Key Concepts

- The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
- To plot a point in the form ((r, heta)), ( heta>0), move in a counterclockwise direction from the polar axis by an angle of ( heta), and then extend a directed line segment from the pole the length of (r) in the direction of ( heta). If ( heta) is negative, move in a clockwise direction, and extend a directed line segment the length of (r) in the direction of ( heta). See Example (PageIndex{1}).
- If (r) is negative, extend the directed line segment in the opposite direction of ( heta). See Example (PageIndex{2}).
- To convert from polar coordinates to rectangular coordinates, use the formulas (x=r cos heta) and (y=r sin heta). See Example (PageIndex{3}) and Example (PageIndex{4}).
- To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: (cos heta=dfrac{x}{r}), (sin heta=dfrac{y}{r}), ( an heta=dfrac{y}{x}), and (r=sqrt{x^2+y^2}). See Example (PageIndex{5}).

## Polar Coordinates

In a plane, suppose you have a point O called the origin, and an axis through that point &ndash say the x -axis &ndash called the polar axis.

Then the polar coordinates ( r , &theta ) describe the point lying a distance of r units away from the origin, at an angle of &theta to the x -axis. The value of &theta may be given in degrees or radians .

To convert from polar coordinates to Cartesian coordinates, you can use:

To convert from Cartesian coordinates to polar coordinates:

So, the Cartesian ordered pair ( x , y ) converts to the Polar ordered pair ( r , &theta ) = ( x 2 + y 2 , tan &minus 1 ( y x ) ) .

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Geodetic coordinates (sometimes called geographic coordinates ) are angular coordinates (longitude and latitude), closely related to spherical polar coordinates, and are defined relative to a particular Earth geodetic datum (described in Geodetic Datum). For more information about geodetic coordinate support, see Geodetic Coordinate Support.

Projected coordinates are planar Cartesian coordinates that result from performing a mathematical mapping from a point on the Earth's surface to a plane. There are many such mathematical mappings, each used for a particular purpose.

## 6.2: General Planar Motion in Polar Coordinates

- Contributed by Timon Idema
- Associate Professor (Bionanoscience) at Delft University of Technology
- Sourced from TU Delft Open

Although in principle all planar motion can be described in Cartesian coordinates, they are not always the easiest choice. Take, for example, a central force field (a force field whose magnitude only depends on the distance to the origin, and points in the radial direction), as we&rsquoll study in the next section. For such a force field polar coordinates are a more natural choice than Cartesians. However, polar coordinates do carry a few subtleties not present in the Cartesian system, because the direction of the axes depends on position. We will therefore first derive the relevant expressions for the position, velocity and acceleration vector, as well as the components of the force vector, in polar coordinates for the general case.

As we already know (see Appendix A.2), the position vector (vec

For the velocity and acceleration vectors we then find:

Note that Equations 5.1.3 and 5.1.6 are the special cases of Equations
ef

The two terms in (F_r) are readily identified as the radial acceleration (ddot**Coriolis force**

and is associated with a velocity in both the radial and the angular direction. It is fairly weak on everyday length scales, but gets large on global length scales. In particular, if you move over the surface of the Earth (necessarily with a nonzero angular component of your velocity), it tends to deflect you from a straight path. On the Northern hemisphere, if you move horizontally, it tends to push you to the right it also pushes you west when going up, and east when going down. Coriolis forces are responsible for the rotational movement of air around high and low pressure zones, causing respectively clockwise and counterclockwise currents around them on the Northern hemisphere (Figure (PageIndex<1>)). We&rsquoll encounter the Coriolis force again in the more general three-dimensional setting in Section 7.2.

## 6.1: Polar Coordinates

A plane is a flat 2-dimensional region such as the surface of a page.

To specify locations on the plane we can lay down two number lines or axes perpendicularly on the plane as shown to the right. The horizontal axis is usually called the x axis and the vertical axis is usually called the y axis. The point where the axes cross is usually chosen to be where x = 0 and where y = 0 (this point is called the origin ). The origin is indicated by a circle in the center of the picture. The axes divide the plane into four regions called quadrants , which are numbered counterclockwise as shown. This construction, a plane plus two perpendicular axes, is called a cartesian plane (named after René Decartes, 1596 - 1650).

If we draw points on a cartesian plane or if we draw a curve representing a function on a cartesian plane then we say that we are graphing or drawing a graph of the points or of the function.

The study of geometric shapes such as lines, circles, triangles, etc. can be done on the ordinary, blank plane or on the cartesian plane. When done on the ordinary plane, their study is called plane geometry and when done on the cartesian plane, it is called analytical geometry . Trigonometry, the study of triangles, is done partly using plane geometry and partly using analytical geometry.

Any point on the cartesian plane can be located by giving its horizontal distance to the right of the origin (this is called its x coordinate ) and its vertical distance above the origin (this is called its y coordinate ). For example the red dot in the cartesian plane shown to the right has an x coordinate of 2 and a y coordinate of 1. Sometimes we say that this point is at x = 2 and y = 1 but most often we use the ordered pair notation (2, 1) to describe this point. (Note that the number before the comma is always the x coordinate and the number after the comma is always the y coordinate.)

This method of using an x value and a y value to locate a point in the plane is called the rectangular coordinate system (notice the dotted rectangle shown in the picture). Another coordinate system in common use, for example for complex numbers, is the polar coordinate system it uses a distance from the origin and a direction to locate a point.

### Identifying points on graphs

We saw above that giving an x coordinate and a y coordinate locates a point in the cartesian plane. But we can turn the logic around: we could say that a point in the cartesian plane represents the simultaneous values of two quantities x and y . This interpretation is very important in science and technology where x and y can represent almost any quantities.

The following examples show how to identify points on progressively more complicated graphs.

- In this graph the horizontal axis is called p and the vertical axis q . Thus points on this graph represent values of p and q .

(Actually the fact that p is midway between 2 and 3 means that p = 2.5 only because the horizontal scale is linear , which means that a movement of, say 1 centimeter, in the horizontal direction represents the same change in the horizontal quantity anywhere on the graph. There are other types of graphs such as logarithmic graphs where this is not true.)

Example 2: The labels on the axes now have units. The label t (ms) means that t is measured in milliseconds and the label d (cm) means that d is measured in centimeters. Thus point B is at t = 250 ms and d = 7.5 cm. Graphs with units appear often in scientific or technical applications.

Example 3: The quantities plotted horizontally and vertically are now algebraic expressions . Thus at point C , the quantity mg has a value of 3 Newtons and the quantity r 2 has a value of 12.5 square meters.

Graphs with expressions on the axes appear often in scientific or technical applications. Their use makes it possible to transform a curve into a straight line, which is much easier to analyze.

If you found this page in a web search you won&rsquot see the **Table of Contents** in the frame on the left.

Click here to display it.

## Determining Pairs of Polar Coordinates

Determine four pairs of polar coordinates that represent the following point (P(r, heta )) such that (&minus360^

Figure (PageIndex<6>)

#### Plotting Polar Coordinates

Plot the following coordinates in polar form and give their description in polar terms: ((1,0)), ((0,1)), ((-1,0)), ((-1,1)).

Figure (PageIndex<7>)

The points plotted are shown above. Since each point is 1 unit away from the origin, we know that the radius of each point in polar form will be equal to 1.

The first point lies on the positive 'x' axis, so the angle in polar coordinates is (0^

Earlier, you were asked to plot the positions of your darts using a polar coordinate system.

Since you have the positions of the darts on the board with both the distance from the origin and the angle they make with the horizontal, you can describe them using polar coordinates.

Figure (PageIndex<8>)

As you can see, the positions of the darts are:

Plot the point (Mleft(2.5, 210^

Figure (PageIndex<9>)

Plot the point (Sleft(&minus3.5,dfrac<5pi ><6> ight)).

Figure (PageIndex<10>)

Plot the point (Aleft(1, dfrac<3pi ><4> ight)).

Figure (PageIndex<11>)

### Review

Plot the following points on a polar coordinate grid.

- ((3,150^
)) - ((2,90^
)) - ((5,60^
)) - ((4,120^
)) - ((3,210^
)) - ((&minus2,120^
)) - ((4,&minus90^
)) - ((&minus5,&minus30^
)) - ((2,&minus150^
)) - ((&minus3,300^
))

Give three alternate sets of coordinates for the given point within the range (&minus360^

## 10.3 Polar Coordinates (# 2)

**Introduction:** In this lesson we will learn how to graph points using polar coordinates. We will also graph a variety of polar curves. We will also learn how we can generalize the idea of the derivative to find the slope of a polar curve. This will also allow us to determine when the tangent line is vertical or horizontal.

**Objectives:** After this lesson you should be able to:

- Understand the polar coordinate system.
- Rewrite rectangular coordinates and equations in polar form and vice versa.
- Sketch the graph of an equation given in polar form.
- Find the slope of a tangent line to a polar graph.
- Identify several types of polar graphs.

**Video & Notes:** Fill out the note sheet for this lesson (10-3-Polar-Coordinates-2) as you watch the video. If you prefer, you could read section 10.3 and work out the problems on the notes on your own as practice. Remember, notes must be uploaded to Blackboard weekly for a grade! If for some reason the video below does not load you can access it on YouTube here.

**Homework:** Go to Web Assign and complete the .3 Polar Coordinates and Polar Graphs” assignment.

## Switching Coordinates

We've really enjoyed our slice of pie, and we've even gone back for seconds while nobody was watching. We need to pack the rest into a box and take it with us to share, though. We can only find square-shaped boxes, so how do we know if the last 2 ⁄_{3} of the pie is going to fit into a box?

We would need to have a way to translate from polar to rectangular coordinates and vice versa. This is no problem, since we can describe where a point is using either polar or rectangular coordinates, and we only need a few tools to switch between coordinate systems.

We'll start with points in the first quadrant. Any point in the first quadrant can be described by rectangular coordinates

(*x*, *y*) where *x*, *y* ≥ 0 or by polar coordinates

(*r*, *θ*) where *r* ≥ 0 and

To translate between coordinate systems, draw a right triangle whose hypotenuse connects the origin and the point. One leg of the triangle should be on the *x*-axis.

The rectangular coordinates tell us the lengths of the legs of the triangle:

The polar coordinates tell us the hypotenuse and one angle of the triangle (since it's a right triangle, we know all the angles):

Each set of coordinates is telling us different information about this triangle. We can use one set of coordinates and a small selection of our tools for working with triangles to find the other set of coordinates.

Going from rectangular to polar coordinates is the same thing as finding the magnitude and direction of a vector.

To go from polar to rectangular coordinates, remember the definitions of the sine and cosine functions.

It's a little trickier to translate between coordinates when the points are in other quadrants.

The first thing we do, no matter which way we're translating, is recognize which quadrant the point is in. After translating between coordinates, the point should still be in the same quadrant. This will help us know if our answer is reasonable.

To go from rectangular to polar coordinates is still the same thing as finding the magnitude and direction of a vector. To find the direction we may need to use a reference angle.

To go from polar to rectangular coordinates, we don't need to worry about reference angles. All we need to do is use the definitions of sine and cosine.

Translating points between rectangular and polar coordinates may be a bit tedious. Going from rectangular to polar coordinates is the same thing as

finding the magnitude and direction of a vector. Going from polar to rectangular coordinates means plugging values into the formulas

Also, if the point in question lies on the *x*- or *y*-axis, there's no need for fancy formulas. We can translate visually.

## Polarplot

is a real row vector of size nc. The style to use for curve i is defined by style(i) . The default style is 1:nc (1 for the first curve, 2 for the second, etc.).

if style(i) is negative, the curve is plotted using the mark with id abs(style(i))+1 . See polyline properties to see the mark ids.

if style(i) is strictly positive, a plain line with color id style(i) or a dashed line with dash id style(i) is used. See polyline properties to see the line style ids.

When only one curve is drawn, style can be the row vector of size 2 [sty,pos] where sty is used to specify the style and pos is an integer ranging from 1 to 6 which specifies a position to use for the caption. This can be useful when a user wants to draw multiple curves on a plot by calling the function plot2d several times and wants to give a caption for each curve.

is a string of length 3 "xy0" .

controls the display of captions,

captions are displayed. They are given by the optional argument leg .

controls the computation of the frame. same as frameflag

the current boundaries (set by a previous call to another high level plotting function) are used. Useful when superposing multiple plots.

the optional argument rect is used to specify the boundaries of the plot.

the boundaries of the plot are computed using min and max values of x and y .

like y=1 but produces isoview scaling.

like y=2 but produces isoview scaling.

like y=1 but plot2d can change the boundaries of the plot and the ticks of the axes to produce pretty graduations. When the zoom button is activated, this mode is used.

like y=2 but plot2d can change the boundaries of the plot and the ticks of the axes to produce pretty graduations. When the zoom button is activated, this mode is used.

like y=5 but the scale of the new plot is merged with the current scale.

like y=6 but the scale of the new plot is merged with the current scale.

a string. It is used when the first character x of argument strf is 1. leg has the form &#[email protected]@. " where leg1 , leg2 , etc. are respectively the captions of the first curve, of the second curve, etc. The default is "" .

This argument is used when the second character y of argument strf is 1, 3 or 5. It is a row vector of size 4 and gives the dimension of the frame: rect=[xmin,ymin,xmax,ymax] .

### Description

polarplot creates a polar coordinate plot of the angle theta versus the radius rho. theta is the angle from the x-axis to the radius vector specified in radians rho is the length of the radius vector specified in dataspace units. Note that negative rho values cause the corresponding curve points to be reflected across the origin.

## The University of Sydney - School of Mathematics and Statistics

By two dimensional space we mean a plane surface extending infinitely in all directions. A table top, for example, is a subset of two dimensional space. To fix a point Q in two dimensional space requires two numbers. First select any point, call it the origin and mark it as O . All measurements will from now on originate from this point O . Next place two mutually perpendicular axes OX , OY through O . Within this reference frame we look for the given point Q , shown in the diagram below.

Drop perpendiculars from Q to OX and OY .

Every point on the OX axis corresponds to a real number: by convention, positive numbers to the east of O and negative numbers to the west of O . Similarly, every point on the OY axis also corresponds to a real number: positive numbers to the north of O and negative numbers to the south of O . Thus in the diagram above, both R and S correspond to positive numbers, say x and y respectively. We call the pair of numbers ( x,y ) the Cartesian coordinates of the point Q . Notice that the order in which the numbers are written is important: (1 , 2) and (2 , 1) are the Cartesian coordinates of different points.

All points in the first quadrant have Cartesian coordinates ( x,y ) in which x and y are both positive. Points in the second quadrant have coordinates ( x,y ) in which x is negative and y is positive, points in the third quadrant have coordinates ( x,y ) in which both x and y are negative, and points in the fourth quadrant have coordinates ( x,y ) in which x is positive and y is negative.

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