TITLE: Lines in Space

TITLE: Lines in Space

TITLE: Lines in Space

SUNRISE OVER THE HOUSTON SKYLINE.

SEEN IN PROFILE, THE SKYLINE IS A JAGGED OUTLINE.

BUT SEEN FROM ABOVE, THE CITY REVEALS AN ORDERLY

ARRANGEMENT OF PARALLEL AND PERPENDICULAR LINES.

MANY CITIES ARE ARRANGED THIS WAY.

MANY CITIES HAVE BEEN ARRANGED THIS WAY

FOR CENTURIES.

WHAT PROPERTIES OF LINES MAKE THIS A PREFERRED WAY

OF ORGANIZING A CITY OR COMMUNITY?

IN THE PREVIOUS SECTION YOU LEARNED ABOUT LINES

IN THE CONTEXT OF POINTS.

RECALL EUCLID'S DEFINITION.

A LINE IS BREADTHLESS LENGTH.

A LINE IS MADE UP OF AN INFINITE NUMBER

OF COLLINEAR POINTS.

IN FACT, THE SHORTEST DISTANCE BETWEEN TWO POINTS

IS A LINE.

HOW DO WE KNOW THIS?

HERE ARE POINTS A AND B.

WE KNOW THAT THERE IS ONLY ONE LINE

THAT CONNECTS THESE POINTS.

BUT HOW DO WE KNOW THAT THE SHORTEST DISTANCE

BETWEEN THE POINTS IS DEFINED BY THIS LINE?

LET'S ASSUME THAT THE LINE ISN'T THE SHORTEST DISTANCE.

THEN IT FOLLOWS THAT THE SHORTEST DISTANCE

WOULD GO THROUGH ANOTHER PATH

AND CROSS ANOTHER POINT: C.

POINT C IS NOT ON THE LINE.

SO THE SHORTEST DISTANCE FROM A TO B

WOULD GO THROUGH C.

BUT THE LINE FROM A TO C CANNOT BE

THE SHORTEST PATH FROM A TO C

USING THE SAME LOGIC AS THE PATH FROM A TO B.

THERE MUST BE ANOTHER PATH

THAT CROSSES ANOTHER POINT, D,

NOT ON THE LINE CONNECTING A AND C.

BUT ONCE AGAIN THE LINE CONNECTING POINTS A AND D

CANNOT POSSIBLY BE THE SHORTEST DISTANCE

BETWEEN THESE TWO POINTS.

AS YOU CAN SEE, WE ARE CONTINUALLY HAVING TO ADD

NEW POINTS AND THIS WOULD CONTINUE AD INFINITUM.

NOT ONLY THAT...

NOTICE THAT AS EACH NEW POINT IS ADDED,

THE ORIGINAL DISTANCE FROM A TO B KEEPS INCREASING.

IN FACT, AS THE NUMBER OF POINTS APPROACHES INFINITY,

SO DOES THE DISTANCE BETWEEN A AND B.

SO USING LOGICAL REASONING WE CONCLUDE THAT

THE SHORTEST DISTANCE BETWEEN TWO POINTS IS A LINE.

ANY OTHER NON-COLLINEAR PATH WOULD BE A LONGER DISTANCE.

THIS HAS IMPORTANT IMPLICATIONS

FOR CITY PLANNING.

WHY? LET'S GO BACK TO HOUSTON.

TO TRAVEL FROM ONE PART OF THE CITY TO ANOTHER

INVOLVES WALKING A SERIES OF STRAIGHT LINES.

BUT SUPPOSE THE CITY WAS LAID OUT

ALONG A CIRCULAR GRID LIKE THIS.

IN SOME CASES GOING FROM POINT A TO B

WOULD BE SHORTER THAN A STRAIGHT PATH.

LET'S EXAMINE THIS ON THE TI-NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

CLICK ON MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

MOVE THE POINTER TO THE MIDDLE PART OF THE SCREEN

AND PRESS ENTER TO CREATE A POINT.

MOVE THE POINTER UP.

PRESS ENTER TO DEFINE THE FIRST LINE SEGMENT.

PRESS ENTER AGAIN ON TOP OF THE POINT YOU JUST CREATED.

MOVE THE POINTER TO THE RIGHT.

PRESS ENTER ONCE MORE TO DEFINE THE SECOND SEGMENT.

NOW LABEL EACH POINT.

PRESS ESCAPE AND MOVE THE CURSOR

ABOVE THE LAST POINT YOU CREATED.

PRESS CONTROL AND MENU AND SELECT THE LABEL OPTION.

PRESS THE CAPS KEY AND THE LETTER C TO LABEL THE POINT.

REPEAT THE LABELING PROCESS WITH THE OTHER POINT.

LABEL IT B.

REPEAT THE LABELING PROCESS WITH THE THIRD POINT.

LABEL IT A.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

THEN PRESS ESCAPE.

WE KNOW THAT THE SHORTEST DISTANCE BETWEEN A AND B

IS THE LINE THAT PASSES THROUGH THEM

AND THAT THE SHORTEST DISTANCE BETWEEN B AND C

IS THE LINE CONNECTING THOSE POINTS.

SO IF THIS WERE A CITY BLOCK,

THIS WOULD BE THE QUICKEST WAY TO GO FROM A TO C.

NOW SUPPOSE THE CITY WAS LAID OUT IN A CIRCULAR GRID.

WE'LL USE THE ARC TOOL TO CONNECT POINTS A AND C.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT THE CIRCLE ARC TOOL.

MOVE THE POINTER ABOVE POINT A.

PRESS ENTER.

NOW MOVE THE POINTER MIDWAY BETWEEN A AND C.

PRESS ENTER AGAIN.

FINALLY MOVE THE POINTER TO POINT C.

PRESS ENTER ONE MORE TIME.

YOU'LL SEE A CURVED ARC FROM POINT A TO C.

DEPENDING ON THE SIZE OF THE ARC,

THE DISTANCE FROM A TO C CAN BE SHORTER OR LONGER

THAN THE STRAIGHT LINE DISTANCE FROM A TO C.

TO MEASURE THE LENGTHS, PRESS MENU

AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE ARC.

PRESS ENTER TO SEE THE MEASUREMENT.

PRESS ENTER AGAIN TO PLACE THE MEASUREMENT

ON THE SCREEN.

REPEAT THIS PROCESS FOR THE LINE SEGMENTS.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

AFTER YOU ARE DONE MEASURING THE LENGTHS,

PRESS ESCAPE.

MOVE THE POINTER ABOVE THE MIDDLE POINT ON THE ARC.

MOVE TO VARIOUS LOCATIONS ON SCREEN.

YOU'LL SEE THAT THE DISTANCE FROM A TO C

WILL VARY AND WILL BE SHORTER OR LONGER

THAN THE STRAIGHT LINE DISTANCES.

YOU CAN ALSO CREATE AN ON-SCREEN FORMULA

FOR CALCULATING THE CHANGING LENGTHS

OF THE STRAIGHT SIDES.

PRESS MENU AND UNDER ACTIONS SELECT TEXT.

MOVE THE POINTER AND PRESS THE CLICK KEY.

YOU SHOULD SEE THE TEXT CURSOR.

INPUT THE FORMULA AB+BC AND PRESS ENTER.

LINK THIS FORMULA TO THE VALUES

OF THE STRAIGHT SIDE LENGTHS.

PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

MOVE THE POINTER ABOVE THE TEXT FORMULA

AND PRESS ENTER.

YOU WILL BE ASKED TO FIRST LINK AB AND THEN BC.

FOR AB MOVE THE POINTER TO THE LENGTH MEASUREMENT

OF SIDE AB.

PRESS ENTER.

REPEAT FOR SIDE BC.

WHEN YOU ARE DONE PRESS ESCAPE.

SO IT IS POSSIBLE TO CREATE A CIRCULAR STREET GRID

THAT HAS SHORTER DISTANCES BETWEEN TWO POINTS.

WHY AREN'T MORE CITIES ARRANGED IN A CIRCULAR GRID?

THE ANSWER HAS MORE TO DO WITH PHYSICS

THAN WITH GEOMETRY.

TRAVELING IN A STRAIGHT LINE TAKES LESS ENERGY

THAN TRAVELING ALONG A CURVE.

IF A CITY WERE LAID OUT IN A CIRCULAR GRID

ALL THE CARS AND TRUCKS TRAVELING ALONG THOSE ROADS

WOULD USE MUCH MORE FUEL THAN IF THOSE SAME VEHICLES

WERE TRAVELING ALONG A STRAIGHT LINE GRID.

A STRAIGHT LINE IS NOT ONLY

THE SHORTEST DISTANCE BETWEEN TWO POINTS,

IT IS ALSO THE MOST FUEL EFFICIENT.

THE SIMPLE ELEGANT LINES OF A STREET GRID

ALSO REVEAL AN UNDERLYING EFFICIENCY.

SINCE THE CENTER OF A CITY LIKE HOUSTON

IS ALSO THE CENTER OF COMMERCE WITH LOTS OF

PEOPLE AND VEHICLES MOVING BACK AND FORTH,

THEN THE MOST EFFECTIVE ARRANGEMENT

INVOLVES AN UNDERLYING STRAIGHT-LINE GRID.

BUT THERE ARE DIFFERENT WAYS OF ARRANGING

A STRAIGHT LINE GRID... LIKE THE ONE SHOWN HERE.

NOTICE THAT WITH THESE GRIDS

THE LINES ARE NOT ALL PARALLEL.

NOT ONLY DOES IT NOT LOOK AS ORDERLY

AS THE PREVIOUS GRIDS BUT IS THERE A DIFFERENT

TYPE OF INEFFICIENCY INTRODUCED

WITH THIS KIND OF ARRANGEMENT?

LET'S EXPLORE THE GEOMETRY OF PARALLEL LINES.

YOU KNOW THAT FOR ANY TWO POINTS A AND B

THERE IS A UNIQUE LINE THAT CROSSES THE TWO POINTS.

YOU ALSO KNOW THAT FOR ANY THREE

NON-COLLINEAR POINTS, A, B AND C,

THERE IS NO LINE THAT INCLUDES ALL THREE POINTS.

THERE IS, HOWEVER, A GEOMETRIC OBJECT

THAT INCLUDES THESE POINTS:

IT IS THE GEOMETRIC STRUCTURE KNOWN AS A PLANE.

A PLANE IS A TWO DIMENSIONAL FLAT SURFACE

MADE UP OF AN INFINITE NUMBER OF POINTS.

IT ALSO CONTAINS AN INFINITE NUMBER OF LINES.

THE PLANE EXTENDS INDEFINITELY

IN BOTH DIRECTIONS.

BECAUSE THE PLANE IS MADE OF POINTS IT HAS NO THICKNESS.

SO LET'S RESTRICT OURSELVES TO THIS PLANE.

SUPPOSE THERE ARE TWO LINES, L1 AND L2.

WE DEFINE PARALLEL LINES TO MEAN THAT

PARALLEL STRAIGHT LINES ARE STRAIGHT LINES WHICH,

BEING IN THE SAME PLANE AND BEING

PRODUCED INDEFINITELY IN BOTH DIRECTIONS,

DO NOT MEET ONE ANOTHER IN EITHER DIRECTION.

NO MATTER HOW FAR LINES L1 AND L2 EXTEND,

THE TWO LINES WILL NEVER INTERSECT.

YOU'LL SEE THAT A CITY GRID IS MADE UP OF

PARALLEL LINES.

WHAT ADVANTAGE DO PARALLEL LINES HAVE

OVER NON PARALLEL LINES?

LET'S EXPLORE ON THE TI-NSPIRE.

CLEAR THE PREVIOUS DOCUMENT YOU WERE USING

OR CREATE A NEW ONE.

TO CLEAR THE SCREEN PRESS MENU

AND UNDER ACTIONS SELECT "DELETE ALL".

PRESS ENTER TO SELECT OKAY.

CREATE A LINE.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT LINE.

MOVE THE POINTER TO THE MIDDLE LEFT

PART OF THE SCREEN AND PRESS ENTER.

MOVE THE POINTER TO THE RIGHT.

YOU'LL SEE THE LINE TAKING SHAPE.

PRESS ENTER WHEN THE POINTER IS ON THE

OTHER SIDE OF THE SCREEN.

NOW CREATE A PARALLEL LINE.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PARALLEL.

MOVE THE POINTER DOWN ABOUT

A QUARTER OF THE LENGTH OF THE SCREEN.

PRESS ENTER.

THE SECOND LINE YOU'VE CREATED

IS PARALLEL TO THE FIRST.

IF YOU MANIPULATE THE FIRST LINE,

INCLUDING ROTATING IT,

THE SECOND LINE WILL REMAIN PARALLEL.

WE WILL MEASURE THE DISTANCE BETWEEN THE TWO LINES.

ADD TWO POINTS, ONE ON EACH LINE.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

MAKE SURE THE POINTER IS ON ONE OF THE LINES.

PRESS ENTER TO ADD ONE OF THE ENDPOINTS

OF THE LINE SEGMENT.

MOVE THE POINTER TO THE OTHER LINES.

PRESS ENTER AGAIN TO ADD THE OTHER ENDPOINT.

YOU WILL NOW CREATE MULTIPLE COPIES OF THIS SEGMENT

USING THE TRANSLATE FEATURE OF THE INSPIRE.

CLICK ON MENU AND UNDER TRANSFORMATIONS

SELECT TRANSLATE.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE SEGMENT YOU'VE CREATED.

PRESS ENTER.

MOVE THE POINTER DOWN TO HOVER OVER THE BOTTOM POINT.

PRESS ENTER AGAIN.

NOW MOVE THE POINTER TO THE LEFT.

WHAT YOU'RE DOING IS TRANSLATING,

OR MOVING A COPY OF THE ORIGINAL SEGMENT

TO A NEW LOCATION.

PRESS ENTER.

REPEAT THIS PROCESS SEVERAL MORE TIMES

TO CREATE SEVERAL SEGMENTS.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW MEASURE THE LENGTH OF THE SEGMENT.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER TO EACH OF THE SEGMENTS.

YOU'LL SEE THE MEASUREMENT OF THE SEGMENTS

APPEAR ON SCREEN.

YOU'LL NOTICE THAT EACH SEGMENT HAS THE SAME LENGTH.

PRESS ENTER TWICE OVER ONE OF THE SEGMENTS

TO RECORD THE LENGTH OF THE SEGMENT.

SO ONE OF THE PROPERTIES OF PARALLEL LINES

IS THAT THEY ARE THE SAME DISTANCE FROM EACH OTHER

THROUGHOUT THE EXPANSE OF THE LINES.

AS YOU CAN SEE FROM THE CITY GRID OF HOUSTON,

MANY OF THE STREETS ARE PARALLEL TO EACH OTHER.

NOW LET'S EXPLORE WHICH LINE SEGMENT IS THE SHORTEST.

FOR THIS WE WILL BE MEASURING THE ANGLE

FORMED BY THE SEGMENT IN ONE OF THE PARALLEL LINES.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

YOU NEED THREE POINTS TO DEFINE A LINE.

USE THE NAV PAD TO MOVE THE POINTER TO THE MIDDLE OF THE

LINE SEGMENT THAT HAS THE LENGTH MEASUREMENT SHOWING.

PRESS ENTER.

MOVE THE POINTER DOWN TO THE POINT

THAT INTERSECTS THE PARALLEL LINE.

PRESS ENTER.

MOVE THE POINTER TO THE RIGHT OF THIS POINT

AND PRESS ENTER AGAIN.

THE ANGLE MEASURE APPEARS AND SHOULD BE 90 DEGREES.

PRESS ESCAPE.

GO TO ONE OF THE ENDPOINTS OF THE LINE SEGMENT.

SELECT THE POINT AND MOVE IT RIGHT AND LEFT.

NOTE HOW THE ANGLE MEASURE

AND THE SEGMENT LENGTH CHANGE.

FOR WHICH VALUES OF THE ANGLE

IS THE SEGMENT LENGTH THE LEAST?

90 DEGREES.

IN OTHER WORDS, A LINE THAT IS PERPENDICULAR

TO THE PARALLEL LINES IS THE SHORTEST DISTANCE.

SO NOW WE GET A BETTER UNDERSTANDING

OF WHY A CITY GRID IS MADE UP OF

PARALLEL AND PERPENDICULAR LINES.

THE USE OF PARALLEL LINES ENSURES THAT CITY BLOCKS

ARE EQUIDISTANT FROM EACH OTHER.

THE USE OF PERPENDICULAR LINES

ENSURES THE SHORTEST PATH FROM ONE BLOCK TO ANOTHER.

AND HERE IS THE RESULT.

LOOK AT THIS DIAGRAM OF A SECTION OF DOWNTOWN.

SUPPOSE YOU WANT TO GO FROM POINT A TO POINT B.

THERE ARE MULTIPLE WAYS OF GETTING FROM A TO B,

SOME OF WHICH ARE SHOWN HERE.

BECAUSE OF THE USE OF PARALLEL

AND PERPENDICULAR LINES

IT DOESN'T MATTER WHICH ROUTE YOU TAKE.

YOU TRAVEL THE SAME DISTANCE.

WHY IS THIS AN ADVANTAGE?

SINCE ANY PATH IS THE SAME DISTANCE,

THIS MEANS THAT TRAFFIC CAN BE EQUALLY DISTRIBUTED.

NO ONE PATH IS SHORTER THAN ANOTHER AND SO

TRAFFIC WON'T CLUSTER IN SOME AREAS OVER OTHERS.

THIS IS AN EFFICIENT WAY FOR PEOPLE IN VEHICLES

TO MOVE THROUGH THE DOWNTOWN AREA.

FINALLY, LET'S LOOK AT ANOTHER PROPERTY

OF PARALLEL LINES.

YOU SAW THAT THE SHORTEST DISTANCE

FROM ONE PARALLEL TO ANOTHER IS A LINE

PERPENDICULAR TO THE TWO PARALLEL LINES.

NOW LET'S INVESTIGATE WHAT HAPPENS WHEN

THE INTERSECTING LINE IS NOT PERPENDICULAR.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT THE LINE TOOL.

MOVE THE POINTER TO THE LOWER PARALLEL LINE

TO A PLACE WHERE THERE ISN'T ALREADY A POINT.

PRESS ENTER.

NOW MOVE THE POINTER TO THE OTHER PARALLEL LINE

AND MAKE SURE THAT THE LINE YOU CREATE

IS AT A SLANT LIKE THE ONE SHOWN.

PRESS ENTER.

WHAT YOU NOW HAVE ARE TWO PARALLEL LINES

CUT BY A TRANSVERSAL.

COMPARE THE ANGLE MEASURES

FOR THE HIGHLIGHTED ANGLE SHOWN.

USE THE ANGLE MEASURE TOOL.

PRESS MENU AND UNDER MEASUREMENTS SELECT ANGLE.

REMEMBER THAT WITH THE ANGLE MEASUREMENT TOOL

YOU NEED TO DEFINE THREE POINTS.

STARTING WITH THE FIRST ANGLE

IN THE UPPER RIGHT-HAND SECTION,

MOVE THE POINTER ABOVE THE TRANSVERSAL

AND PRESS ENTER.

MOVE THE POINTER TO THE INTERSECTION POINT

OF THE TRANSVERSAL AND ONE OF THE PARALLEL LINES

AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE PARALLEL LINE.

REPEAT THIS FOR THE OTHER ANGLES.

PAUSE THE VIDEO TO MEASURE THE ANGLES.

WHEN YOU ARE DONE, PRESS THE ESCAPE KEY.

NOW SELECT ONE OF THE INTERSECTION POINTS

OF THE TRANSVERSAL AND ONE OF THE PARALLEL LINES.

MOVE IT LEFT OR RIGHT TO CHANGE THE ORIENTATION

OF THE LINE.

NOTICE THAT THESE PAIRS OF ANGLES REMAIN EQUAL,

OR CONGRUENT, TO EACH OTHER.

FURTHERMORE, THE OTHER PAIR OF ANGLES

ALSO REMAIN CONGRUENT.

FINALLY, THE SUM OF ADJACENT ANGLES IS 180 DEGREES.

AS WE LOOK ON THE HOUSTON SKYLINE

WE KNOW THAT ITS IRREGULAR CONTOURS

REVEAL AN ORDERED GEOMETRY AT ITS CORE.

THIS ORDER IS BASED ON THE PROPERTIES

OF PARALLEL AND PERPENDICULAR LINES.