Math 214 Spring 01

Assignment 5

 

This assignment is based upon the material found on page 474.  It will be due

on Thursday, April 12.  It will be worth 20 points.

 

Neatly present each problem on engineering paper (not the lined side!).

 

1a) Read the problem.  In the second paragraph, it is stated that for curved

lines to appear as straight lines on a flat map that latitude lines must be stretched horizontally by a factor of secf, where f is the angle of the latitude line.  For the map to preserve the angles between latitude and longitude lines, the lengths of longitude lines

are also stretched by a scaling factor of secf at latitude f.

 

Show that secf is indeed the scaling factor needed to transform the curved lines

to a flat map.

 

Hint:                             c = 2pr

                                                            When you take the lines from the globe

                                     C = 2pR         and flatten them you want c = C, where R

                                                            is the radius of the globe.

 

 

 


    r                               Set up a right triangle to show that R = (r)(secf)

   R              f              Now, you want c = C, let x be the scaling factor.

                                    That is (x)2pr = 2pR.  It should be easy to show

                                    that x = secf.  This is the horizontal scaling factor.

 

 

 


For the vertical factor use this diagram:

 


                                               

                                           L                           Show that L = (x)(l) where x = secf

                                                                        Use similar triangles.

                                                            l

                f

 

 

 


Answer the four questions at the end of the problem.

 

Do not hesitate to corner me with questions.  I view this as a learning assignment

and not just a problem to test your skills.