SUNY Geneseo Department of Mathematics

Cylindrical and Spherical Coordinates

Thursday, January 30

Math 223 01
Spring 2020
Prof. Doug Baldwin

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Questions?

Cylindrical Coordinates

First part of section 11.7.

Key Points

“Like polar coordinates with a z axis”

There are equations for converting to/from cylindrical/rectangular (based on trigonometry and Pythagorean Theorem - the important things to know are that the equations exist, and maybe enough about where they come from that you can quickly understand them when you see them - memorizing them isn’t particularly valuable considering how quickly you can look them up if you need them).

Example

The computer animated snake has coiled itself around a perfectly cylindrical tree of radius 1, aligned with the universe’s positive z axis, in such a manner that the snake’s nose is at ( -√2/2, √2/2, 5 ). Where is the nose in the cylindrical coordinate system defined by the tree?

Snake wrapped around a cylinder, with coordinates for its nose.

You can do some of this problem intuitively, e.g., knowing the radius of the tree tells you that the r coordinate must be 1, you can visually judge what angle would give x = -√2/2 and y = √2/2. But you can also do it more rigorously with the formulas for rectangular to cylindrical conversion:

Snake on tree, r is square root of x squared plus y squared, theta is arctangent of y over x, z is z

The tip of the snake’s tail is 1 unit from the tree’s center, π/6 radians counterclockwise around the tree from the x axis, and 2 units up the tree. Where is the tail in rectangular coordinates?

Use the formulas for cylindrical-to-rectangular conversion to solve this:

Snake on tree with cylindrical coordinates for tail, x equals r cosine theta, y equals r sine theta, z equals z

Spherical Coordinates

Second part of section 11.7.

Key Points

Spherical coordinates are an extension of polar coordinates with 2 angles (Θ counterclockwise from the x axis, Φ down from the z axis) and one distance (from origin).

Coordinate system and line to a point with angles from z axis to that line and x axis to line's projection

Beware of different notations in different disciplines.

There are equations to convert between spherical and rectangular.

Example

Geneseo is at latitude 43° N and longitude 78° W. Earth’s radius is about 4000 miles. What are Geneseo’s rectangular coordinates in a coordinate system whose origin is at the center of the Earth, whose x axis runs through the Prime Meridian, and whose z axis runs through the north pole?

There are 2 steps here. First, realize that latitude and longitude aren’t quite standard spherical coordinates. In particular, the angles are defined differently: latitude is an angle up from the equator, not down from the pole, and longitude may be measured either counterclockwise, if east, or clockwise, if west. So start by converting the problem description into standard spherical coordinates:

Earth with Geneseo marked, theta is 90 minus latitude, phi is 360 minus longitude

Then use spherical-to-rectangular conversion formulas:

x is rho times sine phi times cosine theta, y is rho times sine phi times sine theta, z is rho times cosine phi

Next

Problem set, see handout for details.

Vectors in 3 dimensions.

Read “Working with Vectors in R3” in section 11.2.

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