Title: Geometry Applications: Circular Arcs

Title: Geometry Applications: Circular Arcs

[Music]

[Music]

THE YEAR WAS 80 AD, AND THE FLAVIAN AMPHITHEATER

WHICH WE NOW REFER TO AS THE ROMAN COLOSSEUM,

OPENED ITS GATES.

STARTED UNDER THE REIGN OF EMPEROR VESPASIAN,

THE COLOSSEUM WAS FINISHED WHEN HIS SON,

TITUS, BECAME EMPEROR.

THE SEATING CAPACITY OF THE COLOSSEUM WAS 50,000

AN ENORMOUS SIZE FOR THE TIME.

SEEN FROM ABOVE, THE COLOSSEUM IS ELLIPTICAL

IN SHAPE.

IT WAS 188 METERS LONG AND 156 METERS WIDE.

IT WAS ALSO 50 METERS HIGH,

OR ROUGHLY THE HEIGHT OF A 12 STORY BUILDING.

PART OF WHAT HELD THE COLOSSEUM TOGETHER

AND ALLOWED FOR SUCH A GRAND STRUCTURE TO BE BUILT

WAS CONCRETE.

THE ROMANS ORIGINATED THE USE OF CONCRETE

AND THE SUBSTANCE CONTRIBUTED TO THE

LONGEVITY OF MANY ROMAN BUILDINGS.

LONG BEFORE THE COLOSSEUM BECAME THE CENTER OF

BLOODY GLADIATORIAL FIGHTS IT WAS USED TO REENACT

FAMOUS NAVAL BATTLES FROM ANCIENT HISTORY.

THE GROUNDS WERE FLOODED WITH WATER

ALLOWING SHIPS TO FLOAT AND SIMULATE

THE CONDITION OF A BATTLE AT SEA.

YET OF THE MANY BATTLES FOUGHT IN THIS STADIUM

THROUGHOUT THE CENTURIES, THE VERY FIRST BATTLE

FOUGHT ON THESE GROUNDS MAY HAVE BEEN MATHEMATICAL.

HOW WERE THE ROMANS ABLE TO BUILD

SUCH A LARGE ELLIPTICAL STRUCTURE?

WHY NOT A CIRCULAR BUILDING?

AS YOU HAVE SEEN,

CONSTRUCTING A CIRCLE IS EASY.

TAKE A COMPASS, DEFINE A CENTER POINT,

AND SWEEP OUT A CIRCLE.

A ROMAN SURVEYOR USING THE SURVEYING TOOL OF THE TIME,

THE GROMA, COULD ACCURATELY CREATE A STRAIGHT LINE

MARKING THE DISTANCE FROM ONE POINT TO ANOTHER.

THIS WAY A LOCUS OF POINTS DEFINING A CIRCLE

OF WHATEVER SIZE COULD EASILY BE MARKED

AND A CIRCULAR STRUCTURE COULD BE BUILT.

THIS IS WHY CIRCULAR BUILDINGS

ARE RELATIVELY EASY TO CREATE.

IF WE LET THE SURVEYOR REPRESENT THE CENTER

OF THE CIRCLE, THEN THE SURVEYOR CAN DETERMINE

A NUMBER OF RADII WHICH CAN BE USED

AS THE FRAMEWORK FOR THE CIRCULAR STRUCTURE.

IN FACT, MOST CIRCULAR STRUCTURES ARE MADE UP OF

AN EMBEDDED REGULAR POLYGON FRAMEWORK.

THE VERTICES OF THE POLYGON

ARE WHAT THE SURVEYOR MAPPED OUT.

THE EASE OF CREATING A CIRCULAR STRUCTURE

IS IN STARK CONTRAST TO THE DIFFICULTY

OF CREATING AN ELLIPTICAL ONE.

THE GEOMETRY OF THE ELLIPSE

PRESENTS SOME STEEP CHALLENGES.

FIRST, THE ELLIPSE HAS TWO POINTS THAT ARE

THE EQUIVALENT TO THE CIRCLE'S CENTER.

THESE TWO POINTS ARE CALLED THE FOCI.

THE ELLIPSE IS THE LOCUS OF POINTS SUCH THAT

THE SUM OF THE DISTANCES FROM THE FOCI TO THE ELLIPSE

IS A CONSTANT.

HERE IS ONE WAY TO CONSTRUCT AN ELLIPSE.

TAKE A LOOP OF STRING, TWO THUMBTACKS,

A PENCIL AND A SHEET OF PAPER.

TACK THE PAPER ON A BULLETIN BOARD OR SIMILAR SURFACE

WITH THE TWO THUMBTACKS.

THESE TACKS REPRESENT THE FOCI OF THE ELLIPSE

YOU ARE ABOUT TO CONSTRUCT.

PLACE THE LOOP OF STRING AROUND THE TACKS.

TAKE THE PENCIL AND EXTEND THE STRING

SO THAT YOU END UP WITH A TRIANGULAR SHAPE AS SHOWN.

THESE TWO SIDES OF THE TRIANGLE REPRESENT THE

DISTANCES TO THE FOCI FROM THE VERTEX OF THE TRIANGLE.

THIS VERTEX IS A POINT ON THE ELLIPSE.

KEEP THE STRING TAUT AND MOVE THE PENCIL TO DIFFERENT

LOCATIONS, IN THE PROCESS CONSTRUCTING THE ELLIPSE.

AS YOU MOVE THE PENCIL AROUND, THE SIDE LENGTHS

OF THE TRIANGLE CHANGE, BUT THE TOTAL AMOUNT OF STRING

THAT REPRESENTS BOTH SIDES IS CONSTANT.

IN OTHER WORDS, THE SUM OF THE LENGTHS

OF THE TWO SIDES OF THE TRIANGLE IS CONSTANT,

FULFILLING THE REQUIREMENT OF AN ELLIPSE.

CONSTRUCTING AN ELLIPSE THIS WAY IS STRAIGHTFORWARD.

BUT TRANSFERRING THAT TECHNIQUE TO THE WORLD

OF SURVEYING AND THE CONSTRUCTION OF BUILDINGS,

ESPECIALLY DURING THE TIME OF THE ROMANS,

IS A CHALLENGE.

THE SIMPLE SITUATION OF ONE SURVEYOR DEFINING

A NUMBER OF RADII BECOMES TWO SURVEYORS

WHOSE MEASUREMENTS MUST ALIGN AT A THIRD POINT.

AND THE SUM OF THEIR SEPARATE MEASUREMENTS

MUST BE A CONSTANT.

FURTHERMORE, THIS THIRD POINT, THE ONE ON THE

ELLIPSE, IS CONSTANTLY CHANGING ITS LOCATION

AND CHANGING THE TWO SIDE LENGTHS OF THE TRIANGLE.

THIS WOULD HAVE BEEN VERY DIFFICULT TO ACHIEVE

WITHOUT RESULTING IN AN INACCURATE CURVE.

FURTHERMORE, CONSTRUCTING ARCHITECTURAL PLANS

OR BLUEPRINTS FOR AN ELLIPTICAL STRUCTURE

WOULD HAVE BEEN IMPOSSIBLE SINCE THERE IS NO WAY TO

CONSTRUCT AN ELLIPSE USING A COMPASS AND A STRAIGHTEDGE.

SO WHY DID THE ROMANS BUILD AN ELLIPTICAL STADIUM?

AND HOW DID THEY DO IT?

THE FIRST QUESTION IS EASY TO ANSWER.

SINCE AN ELLIPSE IS A WIDER VERSION OF A CIRCLE,

THEN THE LARGER AREA WOULD RESULT IN MORE

AVAILABLE STADIUM SEATS.

FOR EXAMPLE, HERE IS A CIRCLE WITH RADIUS R.

AND HERE IS AN ELLIPSE OF WIDTH 2R AND LENGTH 4R.

THE AREA OF THE CIRCLE IS PI R SQUARED

WHILE THE AREA OF THE ELLIPSE IS 2 PI R SQUARED,

TWICE THE AREA OF THE CIRCLE.

THE PERIMETER OF THE ELLIPSE, ON THE OTHER HAND,

IS ONLY 50% LONGER THAN THAT OF THE CIRCLE.

THIS MEANS THAT THE COST OF MATERIALS TO BUILD AN

ELLIPTICAL STRUCTURE OF TWICE THE AREA OF THE CIRCLE

IS NOT TWICE THE COST.

SO THERE'S AN ECONOMICAL REASON

FOR BUILDING AN ELLIPTICAL STRUCTURE.

THE ANSWER TO THE SECOND QUESTION,

HOW DID THEY DO IT? IS MORE DIFFICULT TO ANSWER.

BECAUSE OF THE CONDITION OF THE COLOSSEUM,

IT MAKES IT DIFFICULT TO MAKE A FINAL DETERMINATION

ABOUT HOW IT WAS BUILT.

THERE IS EVIDENCE TO SUGGEST THAT THE ROMANS USED

CIRCULAR ARCS TO APPROXIMATE THE SHAPE OF AN ELLIPSE.

LET'S ANALYZE HOW THIS WOULD WORK.

HERE IS AN ELLIPSE PROPORTIONAL TO THE

ARENA PORTION OF THE COLOSSEUM.

IT IS POSSIBLE TO OVERLAY A SERIES OF CIRCLES

SO THAT PORTIONS OF EACH CIRCLE

OVERLAP THE CURVE OF THE ELLIPSE.

FOR NOW WE WILL LOOK AT JUST THE PORTION OF THE CURVE

IN THE XY COORDINATE SYSTEM'S QUADRANT 1.

PCING TOGETHER THE THREE ARCS SHOWN REVEALS A CURVE

REMARKABLY CLOSE TO THAT OF THE ELLIPSE.

SINCE THIS ELLIPSE-LIKE SHAPE IS ACTUALLY MADE UP

OF CIRCLES, THEN ONCE AGAIN THE ENGINEERING CHALLENGES

ARE NO DIFFERENT THAN FOR A CIRCULAR STRUCTURE.

THIS WOULD HAVE ELIMINATED THE INSURMOUNTABLE

OBSTACLE TO BUILDING THE COLOSSEUM.

LET'S USE THE TI-NSPIRE TO EXPLORE THIS CONSTRUCTION.

TURN ON THE NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

YOU WILL BE USING THE GEOMETRY TOOLS

WITHIN THE GRAPHING WINDOW.

TO BEGIN WITH, HIDE THE FUNCTION ENTRY LINE.

PRESS CONTROL AND G.

TURN ON THE BACKGROUND GRID BY PRESSING MENU

AND UNDER VIEW SELECTING SHOW GRID.

CREATE A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE ORIGIN OF THE XY COORDINATE GRID.

YOU'LL SEE THE ON SCREEN LABEL "INTERSECTION POINT".

PRESS ENTER.

MOVE THE POINTER TO THE RIGHT

TO CREATE A CIRCLE OF RADIUS 6.

USE THE BACKGROUND GRID AS A GUIDE TO MAKE SURE

THAT THE POINTER IS AT COORDINATE (6,0).

PRESS ENTER.

YOU SHOULD NOW HAVE A CIRCLE OF RADIUS 6 ONSCREEN.

NEXT YOU'LL NEED TO DEFINE THE CENTERS

OF THE THREE CIRCLES YOU'LL BE CONSTRUCTING.

CONSTRUCT A LINE FROM THE LOWEST POINT ON THE CIRCLE

AND CROSS ENDPOINT (2,0).

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MOVE THE POINTER TO (0,-6) AND PRESS ENTER.

THEN MOVE THE POINTER TO (2,0).

PRESS ENTER.

PRESS ESCAPE AND MOVE THE POINTER

TO THE END OF THE LINE IN ORDER TO EXTEND IT.

PRESS AND HOLD THE CLICK KEY SO THAT THE POINTER

CHANGES FROM AN OPEN HAND TO A GRASPING HAND.

MOVE THE POINTER UP AND TO THE RIGHT

SO THAT THE LINE EXTENDS BEYOND THE CIRCLE.

PRESS ENTER.

NEXT, FIND THE MIDPOINT OF THE LINE SEGMENT

BETWEEN THESE TWO POINTS.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

MOVE THE POINTER TO ONE OF THE ENDPOINTS

OF THE SEGMENT AND PRESS ENTER.

THEN MOVE THE POINTER TO THE OTHER ENDPOINT

AND PRESS ENTER AGAIN.

YOU SHOULD NOW SEE THE MIDPOINT OF THE SEGMENT.

MOVE THE POINTER SO THAT IT HOVERS OVER THE MIDPOINT.

YOU WILL NOW CREATE A LINE FROM THIS POINT

TO THE POINT (4,0).

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

PRESS ENTER TO DEFINE THE FIRST ENDPOINT OF THE LINE.

THEN MOVE THE POINTER TO THE OTHER ENDPOINT

AND PRESS ENTER AGAIN.

PRESS ESCAPE AND HIGHLIGHT THE ENDPOINT

OF THE LINE YOU JUST CREATED.

PRESS AND HOLD THE CLICK KEY TO HIGHLIGHT THE POINT.

EXTEND THE LENGTH OF THIS LINE

TO BEYOND THE LARGE CIRCLE.

NEXT, FIND THE INTERSECTION POINT

BETWEEN THIS LINE AND THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINT SO THAT IT HOVERS OVER THE LINE.

PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE CIRCLE

AND PRESS ENTER AGAIN.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU ARE NOW READY TO CONSTRUCT THE

THREE CIRCLES WHOSE ARCS WILL APPROXIMATE

THE SHAPE OF THE ELLIPSE IN THE FIRST QUADRANT.

CREATE THE FIRST CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO COORDINATE (4,0)

WHERE THE SECOND LINE YOU CONSTRUCTED

INTERSECTS THE X AXIS.

PRESS ENTER TO DEFINE THE CENTER OF THE CIRCLE.

NEXT, MOVE THE POINTER TO COORDINATE (6,0).

PRESS ENTER.

BEFORE CONSTRUCTING THE SECOND CIRCLE,

CREATE AN INTERSECTION POINT

WHERE THE CIRCLE YOU JUST CONSTRUCTED

AND THE SECOND LINE YOU CONSTRUCTED MEET.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER OVER THE SMALL CIRCLE

AND PRESS ENTER.

THEN MOVE THE POINTER TO THE LINE

AND PRESS ENTER AGAIN.

YOU'RE NOW READY TO CONSTRUCT THE SECOND CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

FOR THE SECOND CIRCLE, MOVE THE POINTER

TO THE MIDPOINT YOU PREVIOUSLY CREATED.

PRESS ENTER.

MOVE THE POINTER TO THE INTERSECTION POINT

YOU CREATED. PRESS ENTER.

BEFORE CONSTRUCTING THE THIRD CIRCLE,

CREATE AN INTERSECTION POINT

WHERE THE SECOND CIRCLE INTERSECTS

THE FIRST LINE YOU CREATED.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER ABOVE THE SECOND CIRCLE.

PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE LINE

AND PRESS ENTER AGAIN.

NOW YOU ARE READY TO CONSTRUCT THE THIRD CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

FOR THE THIRD CIRCLE,

MOVE THE POINTER TO THE POINT (0,-6).

PRESS ENTER.

THEN MOVE THE POINTER TO THE INTERSECTION POINT

YOU JUST CREATED.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU NOW HAVE THE THREE CIRCLES

WHOSE ARCS APPROXIMATE THE SHAPE OF THE ELLIPSE.

IF WE HIGHLIGHT THOSE ARCS YOU WILL SEE THAT THEY ARE

LIKE PUZZLE PCES THAT SNAP TOGETHER SEAMLESSLY.

HOW GOOD A FIT ARE THESE ARCS?

LOOKING AT THE TANGENTS TO THE CIRCLES

AT THIS POINT WILL TELL US.

IT IS POSSIBLE FOR A LINE TO INTERSECT A CIRCLE

AT TWO POINTS, IN WHICH CASE

SUCH A LINE IS CALLED A SECANT.

A TANGENT, ON THE OTHER HAND,

INTERSECTS THE CIRCLE AT A POINT THAT IS

PERPENDICULAR TO THE RADIUS AT THAT POINT.

THE POINT WE ARE INTERESTED IN

IS WHERE THE SMALL CIRCLE INTERSECTS THE LARGER ONE.

CREATE TANGENT LINES TO THESE CIRCLES AT THIS POINT.

LET'S START BY ZOOMING IN ON THE POINT.

PRESS MENU AND UNDER WINDOW/ZOOM SELECT ZOOM-IN.

MOVE THE POINTER, WHICH SHOULD LOOK LIKE

A MAGNIFYING GLASS, AND PRESS ENTER ONCE OR TWICE

TO ZOOM IN SUFFICIENTLY TO SEE THE POINT CLEARLY

IN THE MIDDLE PART OF THE SCREEN.

NEXT, PRESS MENU AND UNDER "POINTS & LINES"

SELECT TANGENT.

MOVE THE POINTER ABOVE THE SMALLER CIRCLE.

FOR NOW DON'T WORRY ABOUT

CLICKING ON THE INTERSECTION POINT,

BUT JUST ON THE PART OF THE CIRCLE

AWAY FROM THE INTERSECTION.

PRESS ENTER.

YOU'LL SEE THE TANGENT LINE APPEAR.

NEXT, MEASURE THE SLOPE OF EACH TANGENT.

PRESS MENU AND UNDER MEASUREMENT SELECT SLOPE.

MOVE THE POINTER ABOVE THE FIRST TANGENT

AND PRESS ENTER TO MEASURE THE SLOPE.

MOVE THE POINTER TO THE SIDE OF THE TANGENT

AND PRESS ENTER AGAIN

TO RECORD THE SLOPE MEASUREMENT.

REPEAT THIS PROCESS WITH THE OTHER TANGENT LINE.

PRESS ESCAPE AND MOVE THE POINTER

ABOVE THE FIRST TANGENT YOU CREATED.

PRESS AND HOLD THE CLICK KEY

TO HIGHLIGHT THE TANGENT LINE.

MOVE THE TANGENT POINT ALONG THE CIRCLE UNTIL IT

OVERLAPS THE INTERSECTION POINT OF THE TWO CIRCLES.

PRESS ENTER.

REPEAT THIS PROCESS FOR THE OTHER TANGENT.

ZOOM IN IF YOU NEED TO IN ORDER TO MAKE SURE

THAT ALL THREE POINTS OVERLAP.

YOU'LL SEE THAT THE SLOPES ARE NEARLY IDENTICAL

AND MAY ONLY DIFFER BY A SMALL DECIMAL AMOUNT

DUE TO THE FACT THAT THE POINTS

MAY NOT COMPLETELY OVERLAP.

IN FACT, THE TANGENT LINES HAVE THE SAME SLOPE.

WHY IS THIS?

NOTICE THAT ALTHOUGH THE TWO CIRCLES

DO NOT HAVE THE SAME CENTER,

THE TWO CENTER POINTS ARE ON THE SAME LINE.

WHEN RADII ARE COLLINEAR THIS WAY,

THE TANGENT LINES THAT INTERSECT THESE RADII

WILL BE PARALLEL TO EACH OTHER

AND THIS MEANS THAT THE TANGENTS

WILL HAVE THE SAME SLOPE.

HAVING THE SAME SLOPE MEANS THAT

TWO CURVES MEET SMOOTHLY.

THIS HELPS IN PCING TOGETHER AN ELLIPSE

FROM A SET OF CIRCULAR ARCS.

NOW LOOK AT WHERE THE SECOND

AND THIRD CIRCLES INTERSECT.

WITHOUT EVEN MEASURING THE SLOPES

OF THE TANGENTS ALONG THE INTERSECTION POINT,

YOU KNOW THAT THE TANGENTS ARE PARALLEL

SINCE THE RADII ARE COLLINEAR.

PCEWISE APPROXIMATIONS OF CURVES

IS A COMMON TECHNIQUE USED TO BUILD THE

FRAMEWORK OF A BUILDING WITH A MORE COMPLEX SHAPE.

AS YOU CAN SEE, THE RESULTS ARE SURPRISINGLY ACCURATE.

HOW ACCURATE?

NOW THAT YOU KNOW HOW TO CONSTRUCT THE COLOSSEUM

USING CIRCULAR ARCS, YOU CAN CONSTRUCT

YOUR OWN SCALE MODEL OF THE COLOSSEUM

USING POPSICLE STICKS OR LEGO BLOCKS.

FOR THIS ACTIVITY USE A COMPASS, STRAIGHTEDGE,

AND ENOUGH BUILDING MATERIALS TO AT LEAST

CONSTRUCT ONE SECTION OF THE COLOSSEUM.

USE THE COMPASS TO CONSTRUCT A LARGE CIRCLE.

USE THE STRAIGHTEDGE TO CONSTRUCT A DIAMETER.

DIVIDE THIS RADIUS INTO THREE PARTS.

USE A RULER TO MEASURE THE RADIUS.

CONSTRUCT A LINE THAT CROSSES THESE TWO POINTS

AND MAKE SURE THAT THE LINE EXTENDS

BEYOND THE LARGE CIRCLE.

FIND THE MIDPOINT OF THIS SEGMENT AND HAVE THAT

BE THE START OF A LINE THAT CROSSES THIS POINT

AND EXTENDS BEYOND THE CIRCLE.

NOW USE THE COMPASS TO CONSTRUCT THESE CIRCLES.

HIGHLIGHT THE THREE ARCS AS SHOWN

AND BEGIN BUILDING A STADIUM SECTION

AROUND THESE ARCS.

ADD AS MANY LEVELS AS NECESSARY

AND YOU WILL SOON SEE THE OUTLINES

OF THE SCALE MODEL COMING INTO SHAPE.

YOU WILL ALSO BEGIN TO APPRECIATE WHY THE ROMANS

HAVE SUCH A REPUTATION FOR CLEVER ENGINEERING

AND CREATING MONUMENTS THAT STAND THE TEST OF TIME.