Joshua Dover's Calculus 3 Project: Point in Cartesian, Polar, Cylindrical and Spherical coordinate systems
Overview
This Project has been completed as part of a standard Calculus 3 synchronous online course during Spring 2021 Semester at MassBay Community College, Wellesley Hills, MA.
Introduction
Point in Cartesian, Polar, Cylindrical and Spherical Coordinate Systems
Joshua Dover
Introduction
If you were asked to describe an objects location, especially in space, it would be really hard without a system of references. Consider if you were telling a friend how to familiar location, like the grocery store, you would probably try and direct them from a known reference point. You might say something like "drive to the post office and take the first left after you pass it." This method helps you relay the information using a frame of reference. In mathematics, a point in is best described as a location or exact position, and it can exist on a plane, surface, or in space. It is considered to be one of the fundemental objects of Euclidean Geometry and has no size; therefore it lacks dimensions like width, height, and depth. In the 17th century, Descarte invented a coordinate system that united algebra and geometry, called Cartesian coordinates. Using this method geometric shapes, including lines and curves, could be drawn. This was fundemental for the development of calculus and was used to generalize vector spaces. Cartesian coordinates can even be used in graphs and three-dimensional images. From here several other methods were for desingating a points location were developed, they include: Polar, Cylindrical, and Spherical coordinate systems.
Cartesian Coordinates
Cartesian Coordinates, also known as rectangular coordinates, describe the location of a point in the plane, or in three-dimensional space. A plane is two dimensional flat surface and is usually divided by two axes, typically, x and y. Traditionally, the x-axis is the horizontal axis and the y-axis is the vertical one. Each axis operates with it's own number line that extends infinitely in either direction so that any real number can be expressed as a coordinate. Where the axes intersect is know as the origin and is desgingated by the coordinates (0, 0). On the x-axis, positions to the left of the origin have negative values while positions to the right are expressed wth positive values. The same applies to the vertical axis. As you move up from the origin the numbers are positive and as you move down from the origin the values are negative. A Cartesian coordinate in the two dimensional plane describes how far you move left or right in the horizontal direction and then how far up or down you move in the vertical direction to arrive at the designated point. They are expressed as a set of ordered pairs such as (-3, 2) where 3 refers to the distance along the x-axis from the origin in the negative direction and 2 refers to the distance vertically from the origin in the positive direction (see figure 1 below) [9].
Figure 1[9]
User: n/a - Added: 12/14/09
In three-dimensional space the Cartesian coordinates are expressed as a triplet of ordered numbers and is based on three mutually perpendicular axes. The additional axis is usually referred to as the z-axis. Where the three axes meet is called the origin and it is desgniated by the points (0, 0, 0). Picturing three perpendicular axes can be difficult but a good way to picture it would a corner in a room. In this scenario the corner represents the origin and the xy plane would be the floor. One axis would extend horizontally out from the corner in each direction. This only allows you to picture one quardrant but if you imagine the lines continuing out past the wall you would have the entire plane. The seam of the corner would be the z axis and adds a vertical element to the xy plane. If that seam was extended past the floor you would have be able to plot negative values for z [9]. Three-dimensional Cartesian coordinates allow us to specify points in space and are usually expressed in the format (x, y, z). As in the two-dimensional version, values correspond to the distance from the origin along the respective axis.
Polar Coordinates
The polar coordinate system provides an alternative method of mapping points to ordered pairs. To find the coordinates of a point in the polar coordinate system, consider Figure 2. The point P has Cartesian coordinates (x, y). The line segment connecting the origin to the point P measures the distance from the origin to P and has length r. The angle between the positive x -axis and the line segment has measure θ. This observation suggests a natural correspondence between the coordinate pair (x, y) and the values r and θ. This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values associated with it. In the polar coordinate system, each point also has two values associated with it: r and θ [1, section 1.3].
Figure 2 An arbitrary point in the Cartesian plane [1, section 1.3].
Using right-triangle trigonometry, the following equations are true for the point P:
\(\cos \theta=\dfrac{x}{r} ~~\text{so} ~~~x=r\cos\theta\)
\(\sin \theta = \dfrac{y}{r}~~\text{so}~~~y=r\sin \theta\) .
Furthermore,
\(r^2=x^2+y^2 \)
\(\tan \theta =\dfrac{y}{x}\)
Each point (x, y) in the Cartesian coordinate system can therefore be represented as an ordered pair (r, θ) in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate.
In order to find a unique solution where the polar plot is in the same quadrant as the original point (x, y) it is necessary to restric the solutions of the equation tan θ = y/x. Normally, this equation has an infinite number of solutions, however, if we restrict the solutions to values between 0 and 2π then the solution will be in the quadrant relative to the original point. Also, the corresponding value of r is positive, so r2 = x2 + y2 [1, section 1.3].
In order to convert points between coordinate systems use the following equations:
Given a point P in the plane with Cartesian coordinates (x, y) and polar coordinates (r, θ), the following conversion formulas hold true:
x = r cos θ
y = r sin θ
\(r^2=x^2+y^2 \\ \tan \theta =\dfrac{y}{x}\)
These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates. Below are worked examples and tutorial videos.
1. How to convert the rectangular coordinates (3, 6) to polar coordinates:
First, we will find r
\(r^2=x^2+y^2 \\ r^2=3^2+6^2\\ r^2=45\\ r=\sqrt {45} ~~or ~3\sqrt{5}\)
Next, lets find θ
\(\tan{\theta}= \frac{y}{x}\\ \tan{\theta}= \frac{6}{3}\\ \theta = \tan^{-1}(2)\\ \theta \approx1.107\)
Therefore, the polar coordinates are \((3\sqrt{5}, 1.107)\) .
User: n/a - Added: 4/1/16
2. How to convert the polar coordinates \((8,\frac{2\pi}{3})\) to Cartesian coordinates
First, lets find the x coordinate
\(x=8 \cos(\frac{2\pi}{3}) \\ x=8(-.05)\\ x=-4\)
Next, lets find the y coordinate
\(y=8 \sin(\frac{2\pi}{3}) \\ x=8(\frac{2\pi}{3})\\ x=4\sqrt{3}\)
Therfore, the Cartesian coordinate is \((-4, 4\sqrt{3}).\)
User: n/a - Added: 6/7/11
Cylindrical Coordinates and Spherical Coordinates
The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces can be difficult to model with equations based on the Cartesian system. Cylindrical and Spherical coordinates, based on extensions of polar coordinates, are used to describe a location on these difficult surfaces. Cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures [1, section 2.7].
Cylindrical Coordinates
By adding an axis (z) to the traditional Cartesian coordinate system (x,y), a three dimensional point can be plotted which is represented as a point with (x, y, z) components. The same technique can be applied to polar coordinates. By incorporating the z axis into the polar coordinate system, we have a new method to plot three dimensional points. This new system, known as the Cylindrical coordinate system, provides an extension of polar coordinates to three dimensions. In this new system a point in space is represented by the ordered triple (r, θ, z). The first two components
(r, θ) are the polar coordinates of the point in the xy-plane, and z is the usual z-coordinate found in the Cartesian coordinate system (see Figure 3) [1, section 2.7].
In the xy-plane, the right triangle shown in Figure 3 [1, section 2.7] provides the key to converting cylindrical and Cartesian coordinates.
In order to convert between Cylindrical (r, θ, z) and Cartesian (x, y, z) Coordinates use the following equations:
From cylindrical coordinates to rectangular use:
x = r cos θ
y = r sin θ
z = z coordinates
From rectangular coordinates to cylindrical use:
r2 = x2 + y2
\(\tan \theta =\dfrac{y}{x}\)
z = z coordinates
Once again, the equation tan θ = y/x has an infinite number of solutions, but restricting θ to values between 0 and 2π will provide a unique solution based on the quadrant in which the original cartesian point is located [1, section 2.7].
Below are worked examples of conversions from Cartesian coordinates to cylindrical and cylindrical to Cartesian with tutorial videos.
1. How to convert the Cartesian coordinates (1, 5, 5) to cylindrical coordinates
\(r^2=x^2+y^2 \\ r^2=1^2+5^2\\ r^2=26\\ r=\sqrt{26}\)
\(\tan \theta =\dfrac{y}{x} \\ \tan \theta = \dfrac{5}{1} \\ \theta = \tan^{-1}(5)\\ \theta \approx 1.37\)
Therefore, the cylindrical coordinates are
\((\sqrt{26},1.37,5)\).
User: n/a - Added: 7/22/14
2. How to convert the cylindrical coordinates \((4, \frac{11 \pi}{6},4)\) to Cartesian coordinates
\(x=r\cos \theta \\ x =4 \cos \Big(\dfrac{11 \pi}{6}\Big)\\ x=4 \Big(\dfrac{\sqrt{3}}{2}\Big) \\ x=2\sqrt{3}\)
\(y=r\sin \theta \\ x =4 \sin \Big(\dfrac{11 \pi}{6}\Big)\\ x=4 \Big(\dfrac{-1}{2}\Big) \\ x=-2\)
Therefore, the rectangular coordinates are \( (2\sqrt{3},-2,4).\)
User: n/a - Added: 7/22/14
Sperical Coordinates
In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the spherical coordinate system, the triple describes one distance and two angles. Cylidrical coordinates make it easier to describe a point on a cylinder and, in the same context, spherical coordinates make it simple to describe a point in a sphere. Grid lines for spherical coordinates are based on angle measurements, smiliar to the angle measurment used in polar coordinates [1, section 2.7].
In the spherical coordinate system, a point P in space (Figure 4) is represented by the ordered triple (ρ, θ, φ) where:
• ρ (the Greek letter rho) is the distance between P and the origin (ρ ≠ 0);
• θ is the same angle used to describe the location in cylindrical and polar coordinates;
• φ (the Greek letter phi) is the angle formed by the positive z-axis and line segment \({OP}\) where O is the origin and 0 ≤ φ ≤ π.
Figure 4 [1, section 2.7], Comparing spherical, rectangular, and cylindrical coordinates.
The origin is represented as (0, 0, 0) in spherical coordinates.
To convert spherical, cylindrical, and rectangular Coordinates use the following information:
Rectangular coordinates and spherical coordinates of a point are related as follows:
These equations are used to convert from spherical coordinates (ρ, θ, φ) to rectangular coordinates (x, y, z)
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ.
These equations are used to convert from rectangular coordinates (x, y, z) to spherical coordinates (ρ, θ, φ)
ρ2 = x2 + y2 + z2
\(\tan \theta =\dfrac{y}{x} \\ \phi=\arccos \Big( \dfrac{z}{\sqrt{x^2+y^2+z^2}}\Big)\)
These equations are used to convert from spherical coordinates (ρ, θ, φ) to cylindrical coordinates (r, θ, z)
r = ρ sin φ
θ = θ
z = ρ cos φ
The following equations are used to convert from cylindrical coordinates (r, θ, z) to spherical coordinates (ρ, θ, φ)
ρ = r2 + z2
θ = θ
\(\phi=\arccos \Big( \dfrac{z}{\sqrt{r^2+z^2}}\Big)\)
As before, we must be careful when using the formula tan θ = y/x to choose the correct value of θ.
Below are worked examples on how to convert Cartesian coordinates to spherical coordinates and spherical coordinates to Cartesian coordinates.
How to convert the Cartesian coordinates \((-5,-2,5)\) spherical coordinates
\(\rho^2=x^2+y^2+z^2\\ \rho^2=(-5)^2+(-2)^2+5^2\\ \rho^2=25+4+25\\ \rho^2=54\\ \rho^2=\sqrt{54} ~~~\text{or}~~~3\sqrt{6} \)
\(\tan \theta =\dfrac{y}{x} \\ \tan \theta = \dfrac{-2}{-5} \\ \theta = \tan^{-1}(\dfrac{2}{5}) \\ \theta \approx 0.3805~~~\text{ but this is in the first quadrant. To get this in the right quadrant we should had}~\pi~~\text{so}\\ \theta \approx 3.52\)
\(\)
\(\cos \phi= \dfrac{z}{\sqrt{x^2+y^2+z^2}}\\ \cos \phi= \dfrac{5}{3\sqrt{6}}\\ \phi = \cos^{-1}\Big(\dfrac{5}{3\sqrt{6}}\Big)\\ \phi \approx 0,822\)
User: n/a - Added: 7/24/14
How to convert the spherical coordinates \((3, \dfrac{\pi}{2},\dfrac{\pi}{4})\) to Cartesian coordinates.
\(x=\rho \sin \phi \cos \theta \\ x=3 \Big(\dfrac{\sqrt{2}}{2} \Big) (0)\\ x=0\)
User: n/a - Added: 7/24/14
Conclusion
The development of the Cartesian coordinate system united algebra and geometry by making it possible visualizing curves and functions. This made the concepts of both disciplines less abstract, and helped pave the way for the elements found in calculus. Additionally, there are many business, engineering, and mathematical applications for the graphs these points can be used to create. Through the addition of the polar, cylindrical, and spherical coordinate system, it became easier express the location of point in a circle, a cylinder, and a sphere repesctively. These additional coordinate systems are help engineers, scientist, and mathematicians increase our understanding and improve the world around us.
References
[1] Herman, E., & Strang, G. (77005), Calculus Volume 3, Houston, TX: OpenStax, 2018.
[2] Mathispower4u, "Plotting Points on the Coordinate Plane," https://www.youtube.com/watch?v=s7NKLWXkEEE, 2009.
[3] Mathispower4u, "Ex: Convert Cartesian Coordinates to Polar Coordinates," https://www.youtube.com/watch?v=byfFX7FMhzQ, 2016.
[4] Mathispower4u, "Example: Convert a Point in Polar Coordinates to Rectangular Coordinates," https://www.youtube.com/watch?v=Txx9rvLnuTA, 2011
[5] Mathispower4u, "Ex 1: Convert Cartesian Coordinates to Cylindrical Coordinates," https://www.youtube.com/watch?v=3eA1UowemQs, 2014.
[6] Mathispower4u, "Ex: Convert Cylindrical Coordinates to Cartesian Coordinates," https://www.youtube.com/watch?v=jNaPT_vcrNQ, 2014.
[7] Mathispower4u, "Ex1: Convert Cartesian Coordinates to Spherical Coordinates," https://www.youtube.com/watch?v=ZI0f426X-rA, 2014.
[8] Mathispower4u, "Ex1: Convert Spherical Coordinates to Cartesian Coordinates," https://www.youtube.com/watch?v=8J4B83Y-KHQ, 2014.
[9] Nykamp DQ, “Cartesian coordinates,” From Math Insight, http://mathinsight.org/cartesian_coordinate
[10] Udacity, "Cartesian Coordinates - Interactive 3D Graphic," https://www.youtube.com/watch?v=N4o3s5t0n9g, 2015.