Geometry Applications: Lattitude and Longitude

Geometry Applications: Lattitude and Longitude

[Music]

[Music]

IN 1884, THE CITY OF GREENWICH ENGLAND

BECAME THE LOCATION OF THE PRIME MERIDIAN.

BY AGREEMENT AMONG A GROUP OF COUNTRIES,

GREENWICH WAS GIVEN LONGITUDE 0 DEGREES.

EVEN TODAY'S GPS SATELLITES USE GREENWICH

AS THE PRIME MERIDIAN IN THEIR DETERMINATIONS

OF LONGITUDE AND LATITUDE.

THE DESIGNATION OF GREENWICH AS THE LONGITUDINAL CENTER

OF THE WORLD ENDED A CENTURIES LONG ENDEAVOR

TO DEVELOP AN ACCURATE SYSTEM OF NAVIGATING SHIPS.

INVENTIONS THAT WE TAKE FOR GRANTED TODAY

WERE CRUCIAL IN THE DEVELOPMENT OF AN ACCURATE

SYSTEM OF LONGITUDE AND LATITUDE AND YET LATITUDE

WAS ALWAYS AVAILABLE AS AN ACCURATE MEASURE.

WHY IS LATITUDE SO MUCH EASIER TO MEASURE?

LETS TAKE A LOOK.

LONGITUDE AND LATITUDE ARE USED

AS A COORDINATE SYSTEM ON A SPHERE.

THE SPHERE REPRESENTS THE EARTH.

AS THE EARTH ROTATES, ITS AXIS OF ROTATION

CREATES A NATURAL ALIGNMENT

WITH STARS COLLINEAR WITH THE AXIS.

THIS IS HOW THE NORTH STAR CAME TO BE USED

AND NAMED AS A MEANS OF MEASURING LATITUDE.

LATITUDE IS A SET OF CONCENTRIC CIRCLES

ALONG THE SPHERE THAT ARE PARALLEL TO EACH OTHER.

DEPENDING ON YOUR POSITION ON THE EARTH'S SURFACE,

THE ANGLE TO THE NORTH STAR CHANGES.

LATITUDE IS AN ANGLE MEASURE

AND LATITUDE IS MEASURED RELATIVE TO THE EQUATOR

WHICH CORRESPONDS TO 0 DEGREES.

THE MAXIMUM VALUE FOR LATITUDE AT EACH POLE

IS 90 DEGREES.

SUBDIVISIONS OF LATITUDE ARE MINUTES AND SECONDS.

60 MINUTES EQUALS ONE DEGREE

AND 60 SECONDS EQUALS ONE MINUTE.

YOU WILL SOMETIMES SEE LATITUDE MEASURES

EXPRESSED IN DEGREES, MINUTES AND SECONDS

DEPENDING ON HOW PRECISE THE LOCATION.

LATITUDE CAN BE MEASURED ON LAND AND ON THE SEA

USING A NUMBER OF MEASUREMENT DEVICES

COMMON TO THE TIME.

SO LATITUDE BECAME AN ACCURATE WAY

OF MEASURING THE LOCATION OF A SHIP

NORTH OR SOUTH OF THE EQUATOR.

BUT THE CONVENIENCE OF LATITUDE MEASUREMENTS

COMES AT THE EXPENSE OF LONGITUDE

OR EAST-WEST MEASUREMENTS.

BECAUSE THE ROTATION OF THE EARTH IS ALONG

THE EAST-WEST DIRECTION THERE IS NO FIXED POINT

COMPARABLE TO THE NORTH STAR.

YET SAILORS NEEDED TO KNOW BOTH LONGITUDE

AND LATITUDE TO NAVIGATE THE SEAS.

THERE ARE NO LANDMARKS ON THE OCEAN.

AND ALTHOUGH MEASURING LATITUDE WAS POSSIBLE,

WITHOUT A LONGITUDE MEASUREMENT

A SAILOR WOULD BE UNSURE WHETHER THE SHIP

WAS MOVING IN AN EAST-WEST DIRECTION.

SHIPS LOST AT SEA WERE A COMMON OCCURRENCE

BEFORE A WAY OF MEASURING LONGITUDE WAS FOUND.

THE SOLUTION CAME WITH THE DEVELOPMENT

OF ACCURATE CLOCKS.

HERE'S WHY.

THE LONGITUDE COORDINATE SYSTEM

IS MADE UP OF A SERIES OF GREAT CIRCLES

THAT INTERSECT THE NORTH AND SOUTH POLES.

THESE GREAT CIRCLES ARE KNOWN AS MERIDIANS.

BECAUSE THE ROTATION OF THE EARTH'S AXIS IS A

360 DEGREE TURN, THEN LONGITUDE IS MEASURED

FROM 0 TO 180 DEGREES TO THE EAST OR WEST

OF THE MERIDIAN THAT CORRESPONDS TO 0 DEGREES,

ALSO KNOWN AS THE PRIME MERIDIAN.

GREENWICH ENGLAND IS ALONG THE PRIME MERIDIAN.

AS WITH LATITUDE, LONGITUDE CAN ALSO BE

EXPRESSED IN DEGREES, MINUTES AND SECONDS.

THE EARTH ROTATES 360 DEGREES IN A 24-HOUR PERIOD.

IF YOU DIVIDE 360 BY 24

YOU GET 15 DEGREES PER HOUR.

EVERY 15 DEGREE CHANGE IN YOUR LONGITUDE

MEANS THAT YOU ARE ONE HOUR AHEAD

OR BEHIND YOUR PREVIOUS LOCATION.

THIS IS HOW A LONGITUDE MEASUREMENT WORKS:

LET'S SAY A SHIP IS LEAVING GREENWICH ENGLAND.

THE SHIP'S CLOCKS ARE SET TO GREENWICH MEAN TIME.

AT NOON GREENWICH MEAN TIME

THE SUN IS DIRECTLY OVERHEAD.

BUT AFTER SEVERAL DAYS OF SAILING

YOU FIND THAT THE SUN IS OVERHEAD TO YOUR LOCATION

WHEN IT IS 2:00 PM IN GREENWICH.

THIS MEANS THAT YOU ARE TWO HOURS BEHIND GREENWICH

AND THEREFORE 30 DEGREES WEST OF GREENWICH.

AS YOU CAN SEE, LONGITUDE IS REALLY A TIME MEASUREMENT

AND HAVING ACCURATE CLOCKS IS ESSENTIAL.

PRIOR TO THE DEVELOPMENT OF

ACCURATE MARINE CHRONOMETERS

TIME MEASUREMENTS RELIED ON

PENDULUM CLOCKS AND SUNDIALS

BUT THESE TOOLS DON'T WORK AS WELL ON A SHIP.

THE MOVEMENT OF THE WATER AND THE SHIP

INTRODUCES A LOT OF INACCURACY

TO THE TIME MEASUREMENTS

AND GIVEN THAT THE MARGIN OF ERROR WAS ONE HOUR

IT WOULD HAVE BEEN EASY TO BE OFF BY MANY DEGREES

AND MANY MILES.

THE DEVELOPMENT OF ACCURATE CLOCKS

ALLOWED FOR THE ACCURATE MEASUREMENT OF LONGITUDE

AND CREATED A REVOLUTION IN SAILING AND NAVIGATION.

THE BRITISH WERE INSTRUMENTAL

IN THE DEVELOPMENT OF THE TECHNOLOGY

OF THESE INSTRUMENTS.

IT SHOULD NOT BE SURPRISING THAT, AS A RESULT,

THE BRITISH NAVY RULED THE SEAS FOR CENTURIES.

LONGITUDE AND LATITUDE MAKE UP A THREE-DIMENSIONAL

COORDINATE SYSTEM THAT IS VERY DIFFERENT

FROM THE CARTESIAN COORDINATE SYSTEM.

THE CARTESIAN SYSTEM IS TWO-DIMENSIONAL

AND RECTANGULAR WHILE THE LONGITUDE-LATITUDE

SYSTEM IS SPHERICAL. AND YET IT IS NECESSARY

TO USE INFORMATION FROM ONE SYSTEM IN ANOTHER.

WHILE LONGITUDE AND LATITUDE COORDINATES

ARE EXTREMELY USEFUL FOR SAILING AND NAVIGATION,

FINDING THE DISTANCE BETWEEN TWO POINTS

IS MUCH MORE SUITABLE TO THE RECTANGULAR SYSTEM

WHERE YOU CAN USE THE DISTANCE FORMULA.

BUT HOW DO YOU TRANSLATE THREE-DIMENSIONAL DATA

INTO TWO-DIMENSIONAL COORDINATES?

ONE WAY IS TO USE A TWO-DIMENSIONAL

PROJECTION OF A THREE-DIMENSIONAL OBJECT.

A SIMPLE WAY TO THINK OF THIS IS TO PICTURE

A PERSON'S SHADOW ON A SUNNY DAY.

THE PERSON IS THREE-DIMENSIONAL

BUT THEIR SHADOW IS TWO-DIMENSIONAL.

WITH THIS IN MIND TAKE THE THREE-DIMENSIONAL SPHERE

AND PROJECT IT TO A TWO-DIMENSIONAL

RECTANGULAR GRID...

THE RESULT IS WHAT IS KNOWN AS A MERCATOR PROJECTION.

THERE IS ACTUALLY AN INTERMEDIATE STEP

WHERE THE SPHERE IS PROJECTED TO A CYLINDER

AND THE CYLINDER IS FLATTENED TO A RECTANGLE.

THE CYLINDER IS A MUCH EASIER THREE-DIMENSIONAL

FIGURE TO FLATTEN INTO A TWO-DIMENSIONAL FIGURE

THAN A SPHERE.

THE EASE OF VIEWING A TWO-DIMENSIONAL

WORLD MAP COMES AT A PRICE.

THE PARTS OF THE MAP CLOSER TO THE POLLS ARE DISTORTED

COMPARED TO THE PARTS OF THE MAP NEAR THE EQUATOR.

FOR EXAMPLE, NOTICE HOW GREENLAND

IS NEARLY THE SAME SIZE AS THE CONTINENT OF AFRICA

WHEN IN FACT AFRICA IS MUCH LARGER.

TO SEE AN EXAMPLE OF CONVERTING FROM

ONE COORDINATE SYSTEM TO ANOTHER

LET'S USE THE TI-NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

PRESS CONTROL G TO HIDE THE FUNCTION ENTRY LINE.

YOU'LL BE CONSTRUCTING A CIRCLE

AND LOOKING AT ITS COORDINATES.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER ABOVE THE ORIGIN AND PRESS ENTER.

PRESS AND HOLD THE RIGHT ARROW

TO MOVE THE POINTER ALONG THE X AXIS.

STOP AT X EQUALS FIVE AND PRESS ENTER.

YOU HAVE CONSTRUCTED A CIRCLE OF RADIUS FIVE.

NOW PLACE A POINT ON THE CIRCLE IN QUADRANT ONE.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER TO A POINT ON THE CIRCLE.

YOU SHOULD SEE THE ONSCREEN MESSAGE "POINT".

PRESS ENTER.

TO SEE THE CARTESIAN COORDINATES OF THIS POINT

PRESS MENU AND UNDER ACTIONS

SELECT COORDINATES AND EQUATIONS.

MAKE SURE THE POINTER IS ABOVE THE POINT.

PRESS ENTER TWICE AND THEN PRESS ESCAPE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU NOW HAVE THE xy RECTANGULAR COORDINATES

FOR THIS POINT ON THE CIRCLE.

AS YOU MOVE THE POINT THE COORDINATES CHANGE

TO OTHER RECTANGULAR COORDINATES.

ANOTHER TYPE OF COORDINATES

THAT YOU COULD USE ARE POLAR COORDINATES.

WITH POLAR COORDINATES YOU ARE INTERESTED IN

THE DISTANCE FROM THE ORIGIN

AND THE ANGLE RELATIVE TO THE X AXIS.

A POLAR COORDINATE IS SIMILAR TO

A LATITUDE COORDINATE.

BEFORE MEASURING THE POLAR COORDINATES

CONSTRUCT A RADIUS FOR THE CIRCLE

THROUGH THE POINT YOU CREATED.

PRESS MENU AND UNDER POINTS AND LINES SELECT SEGMENT.

MOVE THE POINTER ABOVE THE ORIGIN AND PRESS ENTER.

NEXT, MOVE THE POINTER ABOVE THE POINT

ON THE CIRCLE AND PRESS ENTER AGAIN.

YOU SHOULD NOW HAVE A RADIUS.

WE CAN NOW CREATE A CORRESPONDING PAIR OF

POLAR COORDINATES FOR THE POINT ON THE CIRCLE.

USE THE MEASUREMENT TOOL.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ALONG THE RADIUS

YOU JUST CONSTRUCTED AND PRESS ENTER.

YOU WILL SEE A RADIUS MEASURE.

NOW MOVE THE POINTER TO THE LOWER RIGHTHAND

PART OF THE SCREEN AND PRESS ENTER AGAIN

TO PLACE THE RADIUS MEASUREMENT ONSCREEN.

NEXT, MEASURE THE ANGLE THE RADIUS MAKES

WITH THE X AXIS.

MAKE SURE THAT THE ANGLE MEASURE DISPLAYED

IS IN ANGLES NOT RADIANS.

GO TO THE DOCUMENTS SETTINGS WINDOW

AND CHANGE FROM RADIANS TO ANGLES

UNDER THE GRAPHING ANGLE OPTION.

NOW MEASURE THE ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

MOVE THE POINTER ABOVE THE POINT ON THE CIRCLE

AND PRESS ENTER.

MOVE THE POINTER TO THE ORIGIN

AND PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER ABOVE POINT 5,0 ON THE X AXIS

AND PRESS ENTER ONCE MORE.

YOU SHOULD NOW SEE AN ANGLE MEASURE APPEAR.

PRESS ESCAPE.

MOVE THE POINTER ABOVE THE ANGLE MEASURE.

PRESS AND HOLD THE CLICK BUTTON

TO SELECT THE ANGLE MEASURE.

MOVE IT SO THAT IT IS TO THE RIGHT OF THE RADIUS MEASURE

SO THAT YOU GET WHAT LOOKS LIKE TWO COORDINATES

WHICH ARE THE POLAR COORDINATES.

YOU CAN ALSO MOVE THE xy COORDINATES

ABOVE THE POLAR COORDINATES

TO MAKE FOR AN EASY COMPARISON OF THE VALUES.

AS YOU MOVE THE POINT ON THE CIRCLE

YOU CAN SEE HOW BOTH SETS OF COORDINATES UPDATE.

DIFFERENT SITUATIONS MAKE EACH SET OF COORDINATES

THE BETTER CHOICE.

IF YOU NEED TO KNOW THE PRECISE LOCATION

OF THE POINT ALONG THE x AND y AXES THEN THE

RECTANGULAR COORDINATES ARE THE ONES TO USE.

IF, AS IN THE CASE OF A RADAR SCREEN,

YOU NEED TO TRACK THE DISTANCE AND ANGLE,

THEN POLAR COORDINATES ARE THE BETTER CHOICE.

AS YOU SAW WITH THE MERCATOR PROJECTION

THERE IS A RELATIONSHIP BETWEEN

THE TWO SETS OF COORDINATES.

IN THE CASE OF THE POLAR COORDINATES

TO CHANGE FROM CARTESIAN TO POLAR COORDINATES

WE USE THESE EQUATIONS.