TransformationExpanded--FullProgram.mp4

Geometry Applications: Transformations

[Music]

Title: Geometry Applications: Transformations

Title: Geometry Applications: Translations and Rotations

[Music]

THE TEXAS TITAN IS ONE OF THE LARGEST

ROLLER COASTERS IN THE COUNTRY.

IT RISES 255 FEET AND REACHES SPEEDS

OF NEARLY 90 MPH.

THE JITTERY FEELING YOU GET AFTER RIDING A FAST ROLLER

COASTER MAY FEEL LIKE A TRANSFORMATIVE EXPERIENCE.

PERHAPS PART OF THIS HAS TO DO WITH THE FACT

THAT YOU ARE EXPERIENCING

A SERIES OF GEOMETRIC TRANSFORMATIONS.

WHENEVER AN OBJECT MOVES IT IS GOING THROUGH

SOME KIND OF GEOMETRIC TRANSFORMATION.

LET'S LOOK AT THE MOTIONS

OF ROLLER COASTER CARS GEOMETRICALLY.

WHEN THE CARS ARE TRAVELING IN A STRAIGHT LINE,

AS SHOWN HERE,

THEN THE CARS ARE EXPERIENCING A TRANSLATION.

A TRANSLATION INVOLVES MOVING A POINT

OR A COLLECTION OF POINTS TO A DIFFERENT LOCATION.

FOR SIMPLICITY, THINK OF THE ROLLER COASTER CARS

AS A RECTANGLE MOVING IN SPACE.

A RECTANGLE IS MADE UP OF

AN INFINITE NUMBER OF POINTS.

LET'S USE THE POINT WHERE

THE DIAGONALS OF THE RECTANGLE INTERSECT

TO STUDY THE TRANSLATION OF THE WHOLE OBJECT.

THE ROLLER COASTER CAR STARTS AT POSITION A

WITH COORDINATES (x1, y1).

THE CAR IS THEN TRANSLATED TO POSITION B

WITH COORDINATES (X2, Y2).

THE COASTER CARS ARE DISPLACED A CERTAIN

DISTANCE (d) AND AT ANGLE THETA.

A VECTOR IS A GEOMETRIC OBJECT THAT DESCRIBES

THE SIZE AND DIRECTION OF THE TRANSLATION.

THE DISPLACEMENT VECTOR THAT DESCRIBES

THE COASTER CAR'S TRANSLATION

STARTS AT POSITION A AND ENDS AT POSITION B.

THE VECTOR POINTS FROM ONE LOCATION TO ANOTHER.

THE DISTANCE FROM POINT A TO POINT B

IS THE MAGNITUDE OF THE DISPLACEMENT VECTOR.

THE ORIENTATION OF THE VECTOR RELATIVE TO THE

X AXIS IS THE DIRECTION OF THE VECTOR.

WE CAN USE THE TI-NSPIRE TO EXPLORE

THE GEOMETRIC TRANSLATION OF AN OBJECT.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

LET THE GRAPH OF (y=2) SIMULATE

A STRAIGHT PIECE OF TRACK.

AT THE FUNCTION ENTRY LINE INPUT 2 AND PRESS ENTER.

NOW CONSTRUCT A CIRCLE TO SIMULATE A ROLLER COASTER.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO POSITION (-7,3) AND PRESS ENTER.

USE THE UP ARROW TO MOVE THE POINTER.

STOP WHEN THE RADIUS OF THE CIRCLE IS ONE UNIT.

PRESS ENTER AGAIN.

MAKE SURE THE SECOND POINT THAT DEFINES THE CIRCLE

DOESN'T INTERSECT THE HORIZONTAL LINE.

DISPLAY THE COORDINATES OF THE CENTER POINT.

PRESS MENU AND UNDER ACTIONS

SELECT COORDINATES AND EQUATIONS.

MOVE THE POINTER ABOVE THE CENTER OF THE CIRCLE

AND PRESS ENTER ONCE TO DISPLAY THE COORDINATES.

THEN MOVE THE POINTER ABOVE THE CIRCLE AND PRESS ENTER

ONE MORE TIME TO PLACE THE COORDINATES ONSCREEN.

MAKE SURE THE COORDINATES ARE (-7,3).

IF NOT, PRESS ESCAPE.

HOVER OVER THE COORDINATES

AND PRESS ENTER TO EDIT AND CHANGE THE VALUES.

WE WANT TO TRANSLATE THE ROLLER COASTER CAR

ALONG THE SIMULATED TRACK.

LET'S CONSTRUCT THE DISPLACEMENT VECTOR

THAT REPRESENTS THE MOVEMENT OF THIS CAR.

PRESS MENU AND UNDER POINTS AND LINES

SELECT VECTOR.

MOVE THE POINTER ABOVE THE POINT

WITH COORDINATES (-7,3) AND PRESS ENTER.

NEXT, USE THE RIGHT ARROW KEY TO MOVE THE POINTER

TO COORDINATE (2,3) AND PRESS ENTER AGAIN.

YOU SHOULD NOW SEE THE DISPLACEMENT VECTOR.

IT STARTS AT COORDINATES (-7,3)

AND ENDS AT COORDINATES (2,3).

THE ENDPOINT IS INDICATED BY THE ARROWHEAD WHICH ALSO

INDICATES THE DIRECTION OF THE TRANSLATION.

TO DISPLAY THE ENDPOINT COORDINATES,

MOVE THE POINTER ABOVE THE POINT.

PRESS MENU AND UNDER ACTIONS

SELECT COORDINATES AND EQUATIONS.

PRESS ENTER.

MOVE THE POINTER TO THE SIDE OF THE POINT

AND PRESS ENTER AGAIN.

NOW MOVE THE ROLLER COASTER CAR

ALONG THIS VECTOR.

PRESS MENU AND UNDER TRANSFORMATION

SELECT TRANSLATION.

FIRST MOVE THE POINTER ABOVE THE CIRCLE.

WAIT UNTIL YOU SEE THE ONSCREEN LABEL "CIRCLE"

AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE VECTOR

AND PRESS ENTER AGAIN.

YOU'LL SEE THE ROLLER COASTER CAR MOVE TO THE

NEW POSITION INDICATED BY THE DISPLACEMENT VECTOR.

SINCE THE y COORDINATES DIDN'T CHANGE FROM THE

INITIAL POSITION TO THE FINAL POSITION,

THEN THE DISTANCE TRAVELED IS SIMPLY

THE ABSOLUTE VALUE OF THE DIFFERENCE

IN THE x COORDINATES FOR A TOTAL OF NINE UNITS.

BUT AS YOU KNOW, A ROLLER COASTER

DOESN'T ALWAYS MOVE ALONG A FLAT LINE.

IN FACT, WHEN A ROLLER COASTER FIRST STARTS

THERE'S USUALLY A SLOW, SLANTING MOVEMENT UPWARD.

LET'S MODEL THIS ON THE NSPIRE.

PRESS CONTROL AND G TO BRING UP THE FUNCTION ENTRY LINE.

USE THE UP ARROW TO GO TO THE f1 ENTRY LINE.

CHANGE THE CURRENT FUNCTION (y=2) TO (y=x)

AND PRESS ENTER.

NEXT, MOVE THE POINTER ABOVE THE FIRST CIRCLE.

PRESS AND HOLD THE CLICK KEY TO SELECT THE CIRCLE.

MOVE THE CIRCLE SO THAT ITS CENTER IS NOW (0,1).

PRESS ESCAPE.

MOVE THE OTHER CIRCLE TO COORDINATES (4,5).

NOTICE HOW THE DISPLACEMENT IS RESIZED AND REORIENTED TO

REFLECT THE NEW TRANSLATION OF THE ROLLER COASTER CAR.

TO FIND THE DISTANCE FROM ONE POINT TO ANOTHER,

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE VECTOR

AND PRESS ENTER TO RECORD THE MEASUREMENT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN

AND PRESS ENTER AGAIN.

THE DISTANCE, 5.66 UNITS, CAN ALSO BE CALCULATED

USING THE DISTANCE FORMULA

AND COORDINATES OF THE POINTS.

THAT SOLUTION IS SHOWN HERE -

THE SAME RESULT AS THE NSPIRE MEASUREMENT.

BUT REMEMBER, A VECTOR HAS A LENGTH

AND AN ANGLE ASSOCIATED WITH IT.

SO PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

MOVE THE POINTER ABOVE (4,5) AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE (0,1) AND PRESS ENTER AGAIN.

THEN USE THE RIGHT ARROW KEY TO MOVE THE POINTER

HORIZONTALLY AND PRESS ENTER ONE MORE TIME.

THE ANGLE MEASURE DISPLAYS.

IF THE MEASURE IS DISPLAYED IN RADIANS,

THEN UNDER DOCUMENT SETTINGS CHANGE THE

ANGLE DISPLAY FOR THE GRAPH WINDOW TO DEGREES.

THE ANGLE MEASURE IS 45 DEGREES.

BUT THIS STILL DOESN'T CAPTURE ALL THE MOTIONS

OF A ROLLER COASTER.

THE PEAKS AND VALLEYS THAT MAKE UP

THE ACCELERATING EXCITEMENT OF THE RIDE

ARE ALSO EXAMPLES OF GEOMETRIC TRANSFORMATIONS.

LET'S MODEL A SIMPLE HILL AND VALLEY ON THE NSPIRE.

PRESS THE HOME KEY TO CREATE A NEW GRAPH WINDOW.

LET'S GRAPH A POLYNOMIAL FUNCTION, A CUBIC,

THAT INTERSECTS THE X AXIS

AT X=-3, X=0, AND X=4.

SINCE THESE ARE THE ROOTS OF THE CUBIC, WE CAN WRITE

THE CUBIC FUNCTION y=x(x+3)(x-4).

AT THE f2 FUNCTION ENTRY LINE INPUT THE

POLYNOMIAL FUNCTION SHOWN AND PRESS ENTER.

TO SEE THE RELEVANT PARTS OF THE GRAPH,

CHANGE THE WINDOW SETTINGS.

PRESS MENU AND UNDER WINDOW/ZOOM

SELECT WINDOW SETTINGS.

CHANGE THE SETTINGS TO THESE:

XMIN=-5, XMAX=5, YMIN=-30, YMAX=30.

TAB YOUR WAY FROM ONE ENTRY FIELD TO ANOTHER.

WHEN YOU ARE DONE, TAB TO THE "OK" BUTTON

AND PRESS ENTER.

THIS CURVE SIMULATES THE PATH OF A ROLLER COASTER.

LET'S CONSTRUCT A TRIANGLE

TO SIMULATE A ROLLER COASTER CAR.

PRESS MENU AND UNDER SHAPES SELECT TRIANGLE.

MOVE THE POINTER NEAR THE TOP POINT OF THE CURVE,

THE MAXIMUM.

PRESS ENTER TO CREATE ONE VERTEX OF THE TRIANGLE,

THEN PRESS THE RIGHT ARROW

TO CONSTRUCT THE HORIZONTAL BASE.

PRESS ENTER AGAIN.

THEN PRESS THE UP ARROW TO COMPLETE THE TRIANGLE

AND PRESS ENTER AGAIN.

NEXT, CONSTRUCT THE DISPLACEMENT VECTOR.

PRESS MENU AND UNDER POINTS AND LINES

SELECT VECTOR.

MOVE THE POINTER WITHIN THE TRIANGLE

AND PRESS ENTER.

THEN MOVE THE POINTER TOWARD THE BOTTOM

OF THE HILL AND PRESS ENTER AGAIN.

NOW TRANSLATE THE COASTER.

PRESS MENU AND UNDER TRANSFORMATION

SELECT TRANSLATION.

CLICK ON THE TRIANGLE AND THEN CLICK ON THE VECTOR.

YOU SHOULD NOW SEE THE TRIANGLE

AT THE OTHER END OF THE VECTOR.

BUT THE ACTUAL POSITION OF THE CAR

SHOULD NO LONGER BE HORIZONTAL.

WHAT IS NEEDED IS A ROTATION OF THE CAR

TO TAKE THIS INTO ACCOUNT.

PRESS CONTROL AND Z TO UNDO THE TRANSLATION.

LET'S APPLY A ROTATION TO THIS COASTER CAR.

PRESS MENU AND UNDER TRANSFORMATIONS

SELECT ROTATION.

WITH THE ROTATION TOOL

YOU NEED TO SPECIFY THREE THINGS:

1 - THE OBJECT BEING ROTATED.

2 - THE POINT THE OBJECT IS ROTATED AROUND.

3 - THE ANGLE OF ROTATION.

IN FACT, WHENEVER AN OBJECT IS ROTATED

THESE THREE ITEMS NEED TO BE SPECIFIED TOO.

SO MOVE THE POINTER ABOVE THE TRIANGLE.

MAKE SURE YOU SEE THE ONSCREEN LABEL "TRIANGLE"

AND PRESS ENTER.

NEXT, MOVE THE POINTER TO THE INSIDE OF THE TRIANGLE

IN THE MIDDLE AND PRESS ENTER.

FINALLY, TO SPECIFY THE ANGLE OF ROTATION

YOU NEED TO DEFINE THE THREE POINTS

THAT DEFINE THE ANGLE.

SO MOVE THE POINTER SEVERAL UNITS

ABOVE THE VECTOR AND PRESS ENTER.

NEXT, USE THE DOWN ARROW TO MOVE THE POINTER

TO THE VECTOR ITSELF AND PRESS ENTER.

THEN MOVE THE POINTER TO THE ENDPOINT OF THE VECTOR

AND PRESS ENTER.

BASICALLY YOU ARE DEFINING THE ANGLE

THAT THE VECTOR MAKES WITH THE X AXIS.

THE TRIANGLE IS NOW ROTATED.

YOU CAN TRANSLATE THIS TRIANGLE.

PRESS MENU AND UNDER TRANSFORMATION

SELECT TRANSLATION.

MOVE THE POINTER OVER THE TRIANGLE AND PRESS ENTER.

THEN MOVE THE POINTER OVER THE VECTOR

AND PRESS ENTER AGAIN.

AS THE ROLLER COASTER CARS TRAVEL ALONG THE TRACKS,

THERE ARE A NUMBER OF TRANSLATIONS

AND ROTATIONS THAT THEY GO THROUGH.

THE SPEED OF THE CARS ALONG THE TRACK VARIES.

AS THE CARS MOVE FORWARD, THE ANGLE THAT

THE CARS MAKE WITH THE HORIZON CHANGES.

WHICH IS ALSO AN INDICATION OF THE

CHANGING SPEED OF THE CARS.

HERE'S ONE WAY TO SEE THIS

VARYING ANGLE THE CARS MAKE.

RETURN TO THE GRAPH WINDOW.

PRESS ESCAPE.

THEN SELECT THE TWO TRIANGLES, THE VECTOR,

AND THE OTHER MISCELLANEOUS POINTS AND DELETE THEM,

KEEPING ONLY THE POLYNOMIAL GRAPH.

PRESS MENU AND UNDER POINTS AND LINES

SELECT TANGENT.

MOVE THE POINTER OVER THE POLYNOMIAL GRAPH AND PRESS

ENTER TWICE TO CREATE A TANGENT LINE TO THE GRAPH.

PRESS ESCAPE AND MOVE THE POINTER OVER THE POINT OF

INTERSECTION BETWEEN THE TANGENT LINE AND THE GRAPH.

PRESS AND HOLD THE CLICK KEY TO SELECT THE POINT.

THEN USE THE NAVIGATION ARROWS TO MOVE THE POINT.

YOU'LL SEE HOW THE ANGLE THE TANGENT MAKES

WITH THE HORIZONTAL CONSTANTLY CHANGES.

THIS CORRESPONDS TO A CHANGING SLOPE OF THE y.

TO TRACK THE SLOPE, PRESS MENU

AND UNDER MEASUREMENT SELECT SLOPE.

MOVE THE POINTER OVER THE TANGENT LINE

AND PRESS ENTER ONCE TO RECORD THE MEASUREMENT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN

AND PRESS ENTER AGAIN.

AS YOU MOVE THE POINT ALONG THE CURVE YOU'LL SEE

POSITIVE, NEGATIVE, AND ZERO VALUES FOR THE SLOPE.

THE POSITIVE SLOPES ARE FOR UPWARD MOVEMENT

AND THE NEGATIVE SLOPES ARE FOR DOWNWARD MOVEMENT.

SO AS YOU CAN SEE, THERE IS A GREAT DEAL OF GEOMETRY

TO A ROLLER COASTER RIDE.

SOME OF THE GEOMETRY IS SUBTLE, BUT SOME IS NOT.

THE PORT OF LOS ANGELES IS ONE OF THE BUSIEST

IN THE COUNTRY.

EVERY YEAR CARGO SHIPS LOADED WITH PRODUCTS

FROM AROUND THE WORLD ARRIVE AT THIS PORT,

AND JUST AS MANY SHIPS LOADED WITH

AMERICAN PRODUCTS LEAVE FROM THIS PORT.

THE MULTI-COLORED BOXES THAT YOU SEE ON THE SHIP

AND ON THE DOCKS ARE CARGO CONTAINERS.

EACH CONTAINER IS FILLED WITH

THE PRODUCTS BEING TRANSPORTED.

ONCE A CONTAINER HAS ARRIVED AT THE PORT,

THEN IT IS TRANSPORTED TO ITS FINAL DESTINATION

BY MEANS OF A TRUCK OR TRAIN.

MANY OF THE PRODUCTS THAT YOU USE AND EVEN

SOME OF THE FOOD YOU EAT MADE ITS WAY TO YOUR HOME

BY WAY OF A CARGO SHIP.

THIS PRACTICE OF USING CARGO CONTAINERS FOR

DELIVERING PRODUCTS IS CALLED CONTAINERIZATION,

AND CONTAINERIZATION HAS CONTRIBUTED GREATLY

TO THE GROWTH OF THE GLOBAL ECONOMY.

A FULLY LOADED CARGO SHIP CAN HAVE HUNDREDS

OF CONTAINERS STACKED ATOP ONE ANOTHER.

WHAT MAY LOOK LIKE A RANDOM ORGANIZATION OF CONTAINERS

IS ACTUALLY A WELL ORGANIZED ARRANGEMENT OF DATA.

FOR EXAMPLE, SUPPOSE THIS CARGO SHIP IS LOADED WITH

CONTAINERS HEADED FOR HAWAII AND AUSTRALIA.

IT WILL UNLOAD SOME CARGO IN HAWAII

AND THE REST IN AUSTRALIA.

SO IT MAKES SENSE TO HAVE THE CONTAINERS

BOUND FOR AUSTRALIA AT THE BOTTOM OF THE STACK

AND THOSE FOR HAWAII AT THE TOP OF THE STACK.

THIS WAY WHEN THE CARGO SHIP MAKES ITS FIRST STOP

IN HAWAII THE CONTAINERS THAT NEED TO BE UNLOADED

ARE WITHIN EASY REACH AND CAN BE UNLOADED QUICKLY.

NOW AMONG THE CONTAINERS HEADED TO EACH DESTINATION,

SOME CONTAINERS ARE HEAVIER THAN OTHERS.

THE CONTAINERS ARE SUCH THAT YOU CAN STACK THEM

ATOP ONE ANOTHER

REGARDLESS OF THE WEIGHT OF THE CONTAINER.

BUT TO MAKE IT EASIER TO LOAD AND UNLOAD CONTAINERS,

THE HEAVIER CONTAINERS SHOULD BE AT THE BOTTOM

AND THE LIGHTER ONES AT THE TOP.

THERE ARE MANY OTHER CRITERIA THAT GO INTO

DETERMINING THE MOST EFFICIENT WAY

OF LOADING A CARGO SHIP.

THE BASIC NOTION IS THAT THE LAST ITEMS LOADED

ARE THE FIRST TO BE UNLOADED.

THE ACRONYM, LIFO, FOR "LAST IN FIRST OUT"

IS A TYPE OF DATABASE ARRANGEMENT

THAT LEADS TO THE MOST EFFICIENT MEANS

OF LOADING AND UNLOADING A CARGO SHIP.

SUPPOSE YOU WANTED TO CREATE AN AUTOMATED SYSTEM

FOR LOADING CARGO ON A SHIP;

A COMPUTER PROGRAM THAT MANAGES THE CRANES

USED TO LOAD THE CONTAINERS.

YOU HAVE THE DATA ON THE DESTINATION

AND WEIGHT OF EACH CONTAINER.

SO THE PROGRAM WOULD NEED TO SORT THE DATA

INTO A LIFO LIST FOR LOADING THE SHIP.

BUT HOW DOES THE PROGRAM KNOW WHERE TO PLACE

EACH CONTAINER ON THE SHIP?

THIS IS WHERE WE GET TO THE GEOMETRIC CONSIDERATIONS.

AS YOU CAN SEE, EACH CONTAINER IS A

RECTANGULAR PRISM OF A STANDARD SIZE.

THINK OF EACH CONTAINER AS A BUILDING BLOCK

USED TO CREATE A LARGER RECTANGULAR PRISM.

WE CAN USE A THREE-DIMENSIONAL

RECTANGULAR COORDINATE SYSTEM TO MAP THE

COORDINATES OF EACH CONTAINER.

IN A THREE-DIMENSIONAL SYSTEM EACH POSITION

HAS THREE COORDINATES RELATIVE TO X, Y, AND Z.

NOW EACH RECTANGULAR PRISM IS MORE THAN

JUST A SINGLE POINT IN SPACE.

BUT FOR SIMPLICITY WE WILL BE TRACKING

THE COORDINATES THAT CORRESPOND TO

ONE OF THE CORNERS OF THE CONTAINER AS SHOWN HERE.

FURTHERMORE, EVEN THOUGH THE RECTANGULAR PRISM

HAS DIFFERENT LENGTH AND WIDTH DIMENSIONS,

WE WILL ADJUST THE COORDINATE GRID

SO THAT EACH SIDE OF THE CONTAINER

IS ONE UNIT LONG, AS SHOWN HERE.

THIS CONTAINER WITH THIS ORIENTATION

HAS COORDINATES (0,0,0)

WHICH IS THE CORNER POINT HIGHLIGHTED.

SO THE CONTAINERS ARE ARRANGED ON THIS

COORDINATE GRID AS THEY WOULD BE ON A CARGO SHIP.

BY ASSIGNING DISCRETE THREE DIMENSIONAL

COORDINATES TO EACH CONTAINER,

THE AUTOMATED SYSTEM KNOWS WHERE TO PLACE A CONTAINER.

SUPPOSE THAT A SHIP CAN HOLD 27 CONTAINERS

ARRANGED AS A 3X3X3 STACKED SET.

THIS GENERATES 27 xyz COORDINATES.

THIS TABLE SUMMARIZES THE 27 COORDINATES

AND THEY ARE ARRANGED IN THE LIFO FORMAT.

IN OTHER WORDS, THE FIRST COORDINATE

IS THE FIRST CONTAINER TO BE LOADED,

AND THE LAST COORDINATE IS THE

LAST CONTAINER TO BE LOADED,

WHICH IS ALSO THE FIRST CONTAINER TO BE UNLOADED.

AS FOR THE CARGO ITSELF, THIS TABLE SHOWS THE

DESTINATION AND WEIGHT OF EACH CONTAINER.

THERE ARE TWO DESTINATIONS,

HAWAII AND AUSTRALIA, DESIGNATED BY "H" AND "A".

SINCE THE CONTAINERS ARRIVED AT THE DOCK FROM

DIFFERENT LOCATIONS, THEY ARE ARRANGED

IN RANDOM ORDER AND NOT IN THE LIFO FORMAT.

AS A LOGISTICS MANAGER FOR A SHIPPING COMPANY,

YOUR JOB IS TO ARRANGE THE CONTAINERS

IN THE MOST EFFICIENT ORDER POSSIBLE

AND MAP THEM TO THE 27 xyz COORDINATES.

THE AUTOMATED CARGO CRANES WILL THEN ASSEMBLE

THE CONTAINERS USING THE COORDINATES AS THE ENDPOINTS

OF A THREE DIMENSIONAL TRANSLATION.

LET'S USE THE TI-NSPIRE TO SOLVE THIS PROBLEM.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A LIST AND SPREADSHEET WINDOW.

PRESS THE UP ARROW TO MOVE TO THE VERY TOP OF COLUMN A.

INPUT THE HEADING "DESTINATION"

AND PRESS TAB TO GO TO THE NEXT COLUMN.

INPUT THE HEADING "WEIGHT" AND PRESS ENTER.

USE THE NAVIGATION ARROWS TO MOVE TO CELL A1.

INPUT THE FOLLOWING DATA AS SHOWN IN THIS TABLE.

PAUSE THE VIDEO TO INPUT THE DATA.

THE DATA THAT YOU INPUT IS NOT IN THE CORRECT ORDER

FOR LOADING THE CARGO SHIP, BUT SIMPLY REFLECTS

HOW THE CONTAINERS ARE ARRANGED ON THE DOCK.

WE WANT TO ORGANIZE THE DATA BY DESTINATION AND WEIGHT.

DESTINATION IS THE PRIMARY WAY OF SORTING THE DATA,

SO WE WANT TO SORT THE ENTIRE DATA SET.

MOVE THE CURSOR TO THE VERY TOP OF COLUMN A

WHERE THE COLUMN HEADING IS.

THEN PRESS THE UP ARROW ONE MORE TIME TO SELECT

THE ENTIRE COLUMN, WHICH SHOULD BE HIGHLIGHTED.

PRESS THE SHIFT KEY AND THE RIGHT ARROW

TO HIGHLIGHT ALL OF COLUMN B.

WITH ALL OF THE DATA HIGHLIGHTED, SORT IT.

PRESS MENU AND UNDER ACTIONS SELECT SORT.

THE PRIMARY CONSIDERATION IN ARRANGING THE DATA

IS THE DESTINATION.

CONTAINERS GOING TO AUSTRALIA GET LOADED FIRST.

BY DEFAULT, THE SPREADSHEET SORTS THE DATA

IN ASCENDING ORDER WHICH COINCIDES WITH WHAT WE WANT.

THIS IS ONLY BECAUSE THE "A" IN AUSTRALIA

COMES BEFORE THE "H" IN HAWAII.

OTHERWISE YOU WOULD SORT THE DATA IN DESCENDING ORDER

- FOR EXAMPLE, IF THE FINAL DESTINATION WAS CHINA.

TAB YOUR WAY TO "OK" AND PRESS ENTER.

YOUR DATA IS NOW ORGANIZED BY DESTINATION.

SCROLL DOWN THE SPREADSHEET TO VERIFY.

AS YOU DO, YOU'LL NOTICE THAT

WITHIN EACH DESTINATION GROUP THE CONTAINERS

ARE STILL NOT ARRANGED FROM HEAVIEST TO LIGHTEST.

RECALL WE WANT THE HEAVIER CONTAINERS AT THE BOTTOM,

SO THOSE NEED TO BE LOADED FIRST.

AS A RESULT, THE DATA NEEDS WHAT'S CALLED

A SECONDARY SORT.

THE PRIMARY SORT WAS TO ARRANGE THE DATA

BY DESTINATION.

THE SECONDARY SORT IS TO ARRANGE THE DATA

WITHIN THE PRIMARY SORT BY WEIGHT,

BUT THIS TIME IN DESCENDING ORDER.

SO MOVE THE CURSOR TO CELL A1.

HOLD THE SHIFT KEY WHILE PRESSING THE DOWN

AND RIGHT ARROWS TO SELECT ALL THE CELLS

THAT HAVE THE DATA FOR AUSTRALIA.

YOUR SCREEN SHOULD LOOK LIKE THIS.

IF YOU NEED TO START OVER, PRESS ESCAPE,

GO BACK TO CELL A1 AND START HIGHLIGHTING.

WITH THE DATA HIGHLIGHTED

YOU ARE NOW READY FOR THE SECONDARY SORT.

PRESS MENU AND UNDER ACTIONS SELECT SORT.

AS NOTED EARLIER, THE DEFAULT SETTINGS

ARE TO SORT ALONG COLUMN A IN ASCENDING ORDER.

IN THIS CASE YOU WANT TO SORT THE DATA IN COLUMN B

IN DESCENDING ORDER, SO PRESS THE DOWN ARROW

TO SELECT COLUMN B FROM THE DROP DOWN MENU.

PRESS ENTER.

PRESS TAB TO GO TO THE NEXT DROP DOWN MENU.

USE THE DOWN ARROW TO SELECT DESCENDING.

PRESS ENTER.

PRESS TAB TO HIGHLIGHT THE "OK" BUTTON AND PRESS ENTER.

YOUR DATA FOR AUSTRALIA IS NOW ARRANGED

IN DESCENDING ORDER.

REPEAT THIS PROCESS FOR THE HAWAII DATA.

MOVE THE CURSOR TO THE FIRST CELL IN COLUMN A

THAT HAS THE HAWAII DATA.

HOLD THE SHIFT KEY AND PRESS THE DOWN AND RIGHT

ARROW KEYS TO HIGHLIGHT ALL OF THE DATA FOR HAWAII.

WITH THE DATA HIGHLIGHTED, PRESS MENU

AND UNDER ACTIONS SELECT SORT.

PRESS THE DOWN ARROW TO SELECT COLUMN B

FROM THE DROP DOWN MENU.

PRESS ENTER.

PRESS TAB TO GO TO THE NEXT DROP DOWN MENU.

USE THE DOWN ARROW TO SELECT DESCENDING.

PRESS ENTER.

PRESS TAB TO HIGHLIGHT "OK" AND PRESS ENTER.

NOW ALL 27 DATA ITEMS ARE ARRANGED

ACCORDING TO DESTINATION AND WEIGHT.

THIS ARRANGEMENT CAN BE MAPPED TO THE

APPROPRIATE COORDINATES THAT DETERMINE

WHERE ON THE SHIP EACH CONTAINER WILL BE LOCATED.

THESE 27 CONTAINERS ARE STACKED ON THE

LOADING DOCKS AND SO EACH CONTAINER HAS ITS OWN

INITIAL COORDINATES WHERE IT IS LOCATED.

TAKING THESE COORDINATES AND THE COORDINATES FOR

WHERE THE CONTAINER SHOULD BE LOCATED ON THE SHIP,

AN AUTOMATED SYSTEM FOR LOADING THE CONTAINERS

CAN TAKE BOTH SETS OF COORDINATES

AND USE THEM TO LOAD THE SHIP.

OF COURSE, AUTOMATED DOESN'T MEAN

THAT PEOPLE AREN'T INVOLVED.

DOCK WORKERS ARE STILL A KEY PART OF

MAKING SUCH A SYSTEM WORK.

BUT AUTOMATING THE WORK OF LOADING AND UNLOADING

CONTAINERS CREATES AN EFFICIENT SYSTEM

FOR DEALING WITH THE HUGE NUMBER OF CONTAINERS

THAT ARRIVE AT PORTS AROUND THE WORLD.

THE MOVEMENT OF CONTAINERS TO AND FROM A SHIP

IS AN EXAMPLE OF A GEOMETRIC TRANSLATION

IN A THREE DIMENSIONAL COORDINATE SYSTEM.

SUCH AUTOMATED SYSTEMS CAN BE USED

IN MANY DIFFERENT WAYS.

EVEN WITHIN THE CASE OF CONTAINER SHIPS,

IT IS POSSIBLE TO HAVE MULTIPLE DESTINATIONS

FOR CARGO.

FOR EXAMPLE, A CONTAINER SHIP LEAVING THE PORT OF L.A.

MIGHT STOP AT HAWAII, AUSTRALIA, NEW ZEALAND,

JAPAN, AND FINALLY CHINA.

AT EACH PORT CONTAINERS WOULD BE UNLOADED,

BUT ALSO NEW ONES WOULD BE LOADED,

ADDING AN ADDITIONAL COMPLICATION

TO THE SORTING AND ARRANGING OF CONTAINERS.

IN THE REAL WORLD OF IMPORTING AND

EXPORTING CARGO THE DATA MANAGEMENT NEEDS

ARE VERY COMPLEX, SO AUTOMATED SYSTEMS

ARE A NECESSITY FOR TODAY'S GLOBAL ECONOMY.

ON THE SUMMIT OF MAUNA KEA IN HAWAII,

ASTRONOMERS GATHER TO STUDY THE STARS

AND BE DAZZLED BY THE HEAVENS.

A NUMBER OF OBSERVATORIES ARE LOCATED ATOP MAUNA KEA,

AND ONE OF THE MOST RECENT ADDITIONS

IS THE GEMINI OBSERVATORY.

MAUNA KEA PROVIDES SEVERAL ADVANTAGES

OVER OTHER LAND BASED OBSERVATORIES.

THE AIR IS CLEAR AND DRY AND THE ALTITUDE ATOP MAUNA KEA,

OVER 13,000 FEET ABOVE SEA LEVEL,

PROVIDES A CLOUDLESS VIEW OF THE SKY.

AND WHAT A VIEW, AS SEEN IN THIS NIGHT TIME

PHOTOGRAPH OF THE MILKY WAY.

THE GEMINI OBSERVATORY PROVIDES ONE MORE ADVANTAGE.

AS INDICATED BY ITS NAME, GEMINI HAS A TWIN

OBSERVATORY LOCATED IN THE COUNTRY OF CHILE.

ALSO ON A MOUNTAINTOP.

TOGETHER GEMINI NORTH AND SOUTH CAN SEE

ALMOST ALL OF THE STARS IN THE SKY.

AS YOU HAVE SEEN WITH OTHER CIRCULAR STRUCTURES

WE HAVE STUDIED, BUILDINGS IN THESE SHAPES HAVE CERTAIN

ADVANTAGES WHEN IT COMES TO VIEWING THE NIGHT SKY.

THE KIVAS AT CHOCO CANYON PROVIDE AN EXPANSIVE VIEW

OF THE NEW MEXICO SKY.

THE PANTHEON IN ROME PROVIDES A

CELESTIAL MAP OF THE CHANGING SEASONS

THROUGH THE LIGHT SHINING PAST THE OCULUS.

IN BOTH OF THESE CASES

THE ARCHITECTURE OF THE CIRCULAR STRUCTURE

ACCOUNTS FOR THE MOVEMENTS OF THE HEAVENS.

AN OBSERVATORY TAKES THIS IDEA ONE STEP FARTHER.

NOT ONLY DOES THE OBSERVATORY

TAKE THE MOTION OF THE EARTH, THE SOLAR SYSTEM,

AND THE GALAXY INTO ACCOUNT,

THE BUILDING ITSELF MOVES.

BECAUSE OF THE EARTH'S ROTATION, IN ORDER FOR

THE TELESCOPE TO VIEW A STAR, PLANET OR OTHER

CELESTIAL OBJECT FOR AN EXTENDED PERIOD OF TIME,

THE OBSERVATORY NEEDS TO MAINTAIN

A LINE OF SIGHT WITH THE OBJECT.

AN OBSERVATORY DOES THIS BY MEANS OF ROTATION.

AN OBSERVATORY IS BUILT SO THAT ITS BODY,

WHICH IS WHERE THE TELESCOPE IS LOCATED,

CAN ROTATE 360 DEGREES IN A DIRECTION

PARALLEL TO THE EARTH'S SURFACE.

THE TELESCOPE ITSELF CAN ROTATE

IN A VERTICAL DIRECTION.

THE OBSERVATORY'S ABILITY TO ROTATE TAKES ADVANTAGE

OF ANOTHER ASPECT OF A CIRCLE'S GEOMETRY:

ITS SYMMETRY.

THE GEOMETRIC PROPERTY OF SYMMETRY IS ONE THAT

HAS TO DO WITH AN OBJECT'S ABILITY TO MIRROR ITSELF.

FOR EXAMPLE, A SQUARE HAS FOUR LINES OF SYMMETRY.

NOTICE THAT ALONG EACH LINE THE SQUARE

IS SPLIT INTO MIRROR IMAGES OF ITSELF.

A REGULAR HEXAGON HAS SIX LINES OF SYMMETRY.

A REGULAR OCTAGON HAS EIGHT LINES OF SYMMETRY.

AS YOU CAN SEE, THERE IS A PATTERN TO THE NUMBER OF

SIDES AND THE NUMBER OF LINES OF SYMMETRY

FOR AN n SIDED FIGURE.

SO AS n APPROACHES INFINITY,

SO DOES THE NUMBER OF LINES OF SYMMETRY.

AND AS THE NUMBER OF SIDES INCREASES,

THE SHAPE OF A POLYGON

APPROXIMATES THAT OF A CIRCLE.

SO A CIRCLE HAS AN INFINITE NUMBER

OF LINES OF SYMMETRY.

AT ANY ANGLE OF ROTATION THE OBSERVATORY

HAS THE SAME ORIENTATION RELATIVE TO THE SKY.

THE INFINITE LINES OF SYMMETRY ALSO PROVIDE

A PANORAMIC VIEW OF THE SKY.

THE GEMINI OBSERVATORY'S VIEWING TOWER

HAS MOVABLE WALLS THAT EXPAND.

A CIRCULAR BUILDING ALSO HAS ROTATIONAL SYMMETRY.

AS THE NAME SUGGESTS, ROTATIONAL SYMMETRY

HAS TO DO WITH MAINTAINING AN OBJECT'S APPEARANCE

AFTER IT IS ROTATED IN A CERTAIN NUMBER OF DEGREES.

RETURNING TO THE EXAMPLE OF REGULAR POLYGONS,

A SQUARE HAS ROTATIONAL SYMMETRY OF ORDER 4.

IN OTHER WORDS, A SQUARE CAN BE ROTATED

90 DEGREES FOUR TIMES.

EACH TIME THE SQUARE'S APPEARANCE IS INTACT.

A REGULAR HEXAGON HAS ROTATIONAL SYMMETRY

OF ORDER 6.

AS WITH LINE SYMMETRY,

AS THE NUMBER OF SIDES INCREASES,

THE ORDER OF ROTATIONAL SYMMETRY INCREASES.

AS n APPROACHES INFINITY,

SO DOES THE ORDER OF ROTATIONAL SYMMETRY.

SO A CIRCLE HAS INFINITE ROTATIONAL SYMMETRY.

FINALLY, A CIRCULAR BUILDING THAT ROTATES

ABOUT ITS CENTER HAS POINT SYMMETRY.

WITH POINT SYMMETRY A MIRROR IMAGE OF AN OBJECT

IS CREATED ABOUT THE POINT AFTER A 180 DEGREE ROTATION.

SO WITH POINT, LINE, AND ROTATIONAL SYMMETRY,

AN OBSERVATORY OFFERS A GREAT DEAL OF FLEXIBILITY

FOR VIEWING THE NIGHT SKY.

THE NIGHT SKY GOES THROUGH PERIODIC CHANGES.

MOST PEOPLE ARE FAMILIAR WITH THE PHASES OF THE MOON,

BUT THERE ARE CONTINUAL CHANGES

OCCURRING WITH THE STARS.

AS THE EARTH ORBITS AROUND THE SUN,

THE PANORAMA OF STARS

CHANGE THROUGHOUT THE YEAR.

FOR STARS AND CONSTELLATIONS

THAT ARE CLOSER TO THE NORTH STAR,

RATHER THAN SLIDING ACROSS THE SKY

THEY WILL ROTATE AROUND THE NORTH STAR.

THE BIG DIPPER IS AN EXAMPLE OF THIS.

THIS CONSTELLATION IS MADE UP OF SEVEN STARS

THAT ARE CLOSELY ARRAYED NEAR THE NORTH STAR.

THROUGHOUT THE YEAR THE CLUSTER OF STARS

MAINTAINS ITS SHAPE,

BUT IT ALSO ROTATES AROUND THE NORTH STAR.

YOU CAN THINK OF THE NORTH STAR

AS THE AXIS OF ROTATION.

WE CAN USE THE NSPIRE TO CREATE A

MODEL OF ROTATION OF THE BIG DIPPER.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW.

PLACE A POINT IN THE CENTER OF THE SCREEN

TO REPRESENT THE NORTH STAR.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

AND PRESS ENTER.

NOW USE THE POLYGON TOOL

TO CONSTRUCT THE CONSTELLATION.

PRESS MENU AND UNDER SHAPES SELECT POLYGON.

MOVE THE POINTER TO THE LOWER LEFT

QUADRANT OF THE SCREEN AND PRESS ENTER.

THIS DEFINES THE FIRST POINT OF THE POLYGON.

MOVE THE POINTER DOWN TO DEFINE ONE OF THE

VERTICAL SIDES OF THE BIG DIPPER.

PRESS ENTER.

MOVE THE POINTER TO THE LEFT

TO DEFINE THE LOWER PART OF THE DIPPER.

PRESS ENTER.

MOVE THE POINTER UP

TO COMPLETE A QUADRILATERAL SHAPE.

PRESS ENTER.

NOW MOVE THE POINTER DIRECTLY TO THE LEFT

OF THE POINT YOU JUST CREATED

TO SIMULATE THE HANDLE OF THE DIPPER.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU NOW HAVE A SIMULATION OF THE BIG DIPPER

NEAR THE NORTH STAR.

TO ROTATE THIS CONSTELLATION AROUND

THE POINT REPRESENTING THE NORTH STAR,

PRESS MENU AND UNDER TRANSFORMATION

SELECT ROTATION.

MOVE THE POINTER ABOVE THE BIG DIPPER SHAPE

UNTIL YOU SEE THE ONSCREEN LABEL "POLYGON".

PRESS ENTER.

NOTICE THAT THE POINTER CHANGES TO A DIFFERENT ICON,

ONE THAT LOOKS LIKE A POINT

WITH TWO CURVED ARROWS AROUND IT.

THIS ICON IS FOR DEFINING THE CENTER OF ROTATION.

MOVE THE POINTER ABOVE THE SOLITARY POINT YOU CREATED.

PRESS ENTER.

THE POINTER NOW CHANGES TO A PENCIL.

USE THIS PENCIL TO DEFINE A LINE.

PRESS ENTER.

MOVE THE POINTER AND PRESS ENTER AGAIN.

THIS LINE CONTROLS THE ROTATION.

MOVE THE POINTER ACROSS THE SCREEN AND YOU WILL SEE

THE BIG DIPPER ROTATE ABOUT THE NORTH STAR.

NOTICE THAT AS THE DIPPER ROTATES,

THE RELATIVE POSITIONS OF THE STARS REMAIN THE SAME.

IN OTHER WORDS, THE BIG DIPPER RETAINS ITS SHAPE.

SO THE GEMINI TELESCOPE ON MAUNA KEA RELIES ON

THE PROPERTIES OF ROTATION, SYMMETRY AND REFLECTION

TO DO THE IMPORTANT WORK OF EXPLORING THE HEAVENS.