Title: Geometry Applications: Translations and Rotations

Title: Geometry Applications: Translations and Rotations

[Music]

[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 PCE 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 FLD 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.