YourName-HW03

your name
the date you completed the first version;  date of most recent revision
This template was revised February 28, 2021.

Introduction and Statement of Problem

The purpose of this notebook is to gain some experience finding the angular and circumferential distance between two points on a spherical Earth.

Statement of Problem.  Imagine yourself standing at Neenach in southern California (34°44’24”N, 118°37’24”W), and you want to know the direction of the shortest straight-line path to Pinnacles in west-central California (36°29’13”N, 121°10’01”W).  The azimuth of a bearing is measured in a clockwise direction relative to true north.  In a general sort of way, you could start by looking toward true north (90°N, 0°E) and slowly rotating clockwise while keeping track of your total rotation angle until you are looking directly toward Pinnacles (assuming you could see that far and over the curvature of Earth).  If you could perform that angular measurement accurately, that total rotation angle would be the azimuth from Neenach to Pinnacles.  
     Judging from the geographic coordinates of Neenach and Pinnacles, would you have to rotate clockwise less than or more than 180° from true north to be looking along the shortest great-circle path to Pinnacles?  You might want to sketch a map of the situation to form your judgement.     
     Write a Mathematica notebook that analyzes the input data to complete the following tasks.
     (a) Convert the given geographic coordinates of Neenach and Pinnacles to decimal geographic coordinates (see section 2.2.3).
     (b) Convert the decimal geographic coordinates to unit location vectors(locVectNeenach and locVectPinnacles), recalling that south latitudes and west longitudes are negative numbers.  For the location vector to the North Pole, define locVectNorthPole = {0,0,1}
     (c) Determine the dihedral angle between (1) the plane defined by the location vector to Neenach and the location vector to the North Pole, and (2) the plane defined by the location vector to Neenach and the location vector to Pinnacles  (see sections 2.4.1 and 2.4.2).
     (d) The dihedral angle you just determined is related in some quantitative way to the azimuth of the great-circle path from Neenach to Pinnacles.  Conjure-up some spatial reasoning to determine that azimuth, and write a mathematical expression to indicate the relationship between the dihedral angle you computed and the azimuth in this case.  For example, 
•  azimuth = dihedral angle
•  azimuth = 360° – (dihedral angle)
•  azimuth = 180° + (dihedral angle)
et cetera.

Input Data

The geographic coordinates of the North Pole are 90°N, 0°E.

The geographic coordinates of a given point within Pinnacles National Monument are latitude 36°29’13”N, longitude 121°10’01”W.

The geographic coordinates of a given point within the Neenach volcanic field are latitude 34°44’24”N, longitude 118°37’24”W.

User-defined Function

The user-defined function makeLocationVector acts on input geographic coordinates (decimal latitude and decimal longitude, both expressed in degrees) and returns the Cartesian-geographic coordinates of the unit location vector to that point.  This user-defined function is copied from the document “Kinematics-Chapter-2.nb.pdf” by Vince Cronin.

makeLocationVector[lat_, long_]:={Cos[lat Degree] Cos[long Degree],Cos[lat Degree] Sin[long Degree],Sin[lat Degree]};

Computation

Convert the given geographic coordinates to decimal geographic coordinates (Kinematics chapter 2, sec. 2.2.3)

The variables decimalLatPinnacles and decimalLongPinnacles are the decimal geographic coordinates associated with a point within Pinnacles National Monument in California (36°29’13”N, 121°10’01”W).  Remember that west longitudes are negative.

The variables decimalLatNeenach and decimalLongNeenach are the decimal geographic coordinates associated with a point within the Neenach volcanic field in California (34°44’24”N, 118°37’24”W).  Remember that west longitudes are negative.

Convert the decimal geographic coordinates to cartesian geographic coordinates (Kinematics chapter 2, sec. 2.2.3)

The Code Fairy gave us a present by telling us the unit location vector locVectNorthPole.

locVectNorthPole={0,0,1};

The user-defined function makeLocationVector[] acts on the arguments decimalLatPinnacles and decimalLongPinnacles to yield the unit location vector locVectPinnacles.

The user-defined function makeLocationVector[] acts on the arguments decimalLatNeenach and decimalLongNenach to yield the unit location vector locVectNeenach.

Determine the dihedral angle (in degrees) between Pinnacles and Neenach (Kinematics chapter 2, sec. 2.4.2)

The built-in Mathematica function Cross[] acts on the arguments locVectNorthPole,  locVectPinnacles and locVectNeenach to yield two vectors, and the built-in function VectorAngle[] and the conversion factor (180/π) degrees per radian complete the computation to provide a value for the variable dihedralAngDegrees.

Output Data

Decimal geographic coordinates of the input points

decimalLatPinnacles

decimalLongPinnacles

decimalLatNeenach

decimalLongNeenach

Cartesian geographic coordinates of the input points

locVectPinnacles

locVectNeenach

Dihedral angle between the Neenach-North Pole great circle and the Neenach-Pinnacles great circle (degrees)

dihedralAngDegrees

Analysis

Given the value of variable dihedralAngDegrees, what is the azimuth of the shortest line from Neenach to Pinnacles?

(Replace this text with your analytical statement)

References

Cronin, V.S., 2021, Kinematics-Chapter-2.nb:  unpublished draft, http://croninprojects.org/Vince/PlateKinematics/KinematicsCourse/Kinematics-Chapter-2.nb.pdf

Cronin, V.S., 2017, A brief primer on plate kinematics, with Mathematica:  unpublished draft, http://croninprojects.org/Vince/PlateKinematics/KinematicsPrimer/KinematicsCh2.nb