Exploring the Bezier curves.docx

Exploring the Bezier curves.docx

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ExploringtheBeziercurves

ExploringtheBeziercurves

ABeziercurveisaparametriccurvewhichusuallyusedincomputergraphicstomodelsmoothcurvesthatcanbescaledindefinitelyandalsousedinanimationasatooltocontrolmotion.‘Beziercurvesarealsousedinthetimedomain,particularlyinanimationandinterfacedesign,e.g.,BeziercurvescanbeusedtospecifythevelocityovertimeofanobjectsuchasaniconmovingfromAtoB,ratherthansimplymovingatafixednumberofpixelsperstep.Whenanimatorsorinterfacedesignerstalkaboutthe"physics"or"feel"ofanoperation,theymaybereferringtotheparticularBeziercurvesusedtocontrolthevelocityovertimeofthemoveinquestion.’（http:

//en.wikipedia.org/wiki/B%C3%A9zier_curve）Thisinvestigationisaimedtousetheknowledgeofvectorsandparametricequationstoexplorethemathematicsofcomputeraideddesigncurves.

Inthisinvestigation,coordinateaxes,graphs,includingaxesintercepts;vectors,includingvectoraddition,scalarmultiplicationandratiodivision;quadraticandcubicpolynomialswillbeused.First,asimpleexamplewillbeusedtoshowhowtheparametricequationworks.Then,ourknowledgeandcalculatorwillbeusedtoexploremoreaboutBeziercurves.

Tobegintogiveasimpleexplanationabouthowtheparametricequationworks.ThereisalinesegmentstartfrompointA（-3,4）topointB（2,5）.Onthislinesegment,thereisapointPmovingfromAtoB.SoPdividedABintheratiotto1-t,and

.Whent=0,PisattheinitialpointA.Whent=1,itisattheendpointB.

t

A

P

B

O

Accordingtothediagram,andusethedivisionformula,wecanfindthat

.BecauseitisaarbitrarypointonAB,soit’sparametricequationofABis

.Showitingraphiccalculatoris

A

B

Then,usethesamemethodandfollowthesamesteptofindanotherlinesegmentBC,whereCis（4,6）.

Now,QisthemovingpointonBC.Usethesamemethod,wecanfindthat

.SotheparametricequationofBCis

.Showitingraphiccalculatoris

C

B

A

Inthegraph,bothpointsPandQmovefromtheinitialpointtotheendpoint,andthereisaninvisiblelinesegmentbetweenPandQ.ThelinesegmentcreatedbyPQismovingtoo.AssumethatthereisanotherpointSliesonPQ,andSmovesfromPtoQjustlikePmovesfromAtoBandQmovesfromBtoCwithtchangingfrom0to1.ItmeansthatwhenPstartsatA（-3,4）,SalsostartsatA（-3,4）,andwhenQarrivesatC（4,-6）,SalsoarrivesatC（4,-6）.InordertogetthepathofS,previousresultswillbeused.

SinceSalsodividesPQintheratiooftto1-t,

SotheparametricequationforthearcACis

Thenentertheequationonthegraphiccalculation,andgraphthecurve.Wecanget

C

B

A

Fromthegraph,wecanfindthatABandBCarethetangentlinesofarcAC.

Usetheparametricequation,thecoordinatesofthecurvewhereitcutstheaxescanbefound.

Whenx=0:

SoitcutstheY-axesat（0,10/3）

Wheny=0:

SoitcutstheX-axesat（-14/3,0）

NowwearegoingtousetheknowledgewegetpreviouslytodrawapictureofNemo.Thepictureisacombinationofsixarcs.

Thecoordinateofpointsare:

H（-5,1）;I（-1,5）;J（4,2）;K（1,-6）;L（-1,-2）;M（-2,-4）;N（1,-3）;R（2,1.5）;T（1,-2）;V（5,6）;W（7,-1）;X（4,-2）

R

J

W

V

I

T

X

L

H

M

N

K

Takethedorsalfinintoconsiderationfirst.IthasaninitialpointE（-2,3）andendpointG（2,3）.Then,useourruler,wecanfindthesharppointF（3,8）.WecanimagethatPisalwaysthepointmovingfrominitialpointtosharppointandQisalwaysmovesfromsharppointtoendpoint.ThenSisalwaysmovesfromPtoQ.

Usingthesamemethodusedabove,theparametricequationofEFandGFcanbefound.Andtherefore,theparametricequationforarcEGcanbefound.

SoEF=

SoFG=

SotheparametricequationofarcEGis

Entertheequationintothecalculatorwillget

G

F

E

Repeattheprocessfortheremainingandgraphallsixarcswiththecalculator.

SotheparametricequationofarcHJis

SotheparametricequationofarcHJ2is

SotheparametricequationofarcLNis

SotheparametricequationofarcTNis

SotheparametricequationofarcVWis

EnterthesesixarcsintocalculatorwillgetapictureofNemo.

Becauseallcurvesabovearepartsofparabolas,theyarecalledparabolicarcs.

Nextstepisgoingtodomoreabouttheparaboliccurves.

GiventhatinitialpointisL（-2,4）,sharppointM（0,-4）andendpointN（2,4）.

Usethesamemethod:

EnteritintocalculatorandaddthelinesegmentLMandMNcanget:

M

N

L

Fromthegraph,LM，MNbotharethetangentlineofcurveLN.

SoarcLNisapartofparaboliccurve

AnotherparabolicarchasinitialpointU（0,0）,sharppointV（0,1）andendpointW（4,2）.

EnteritintocalculatorandaddthelinesegmentUVandVWcanget:

W

U

V

LinesegmentsUVandVWaretangentlinesofcurveUW.

Fromtheparametricequation,Cartesianequationcanbefound.

SotheCartesianequationis

Becauseparabolicarcsareallhavethreepoints,sotheparametricequationcanbefoundinaregularway.

AssumethatthereisaparabolicarcwithinitialpointA

sharppointB

andendpointC

.Theparametricequationcanbefound.

Sotheparametricequationis

Thisequationcanbeusedtochecktheworkwehavedone.

ArcAC=

ArcEG=

ArcLN=

ArcUW=

Thisequationalsocanbeusedtofindanewarcwithinitialpoint（-1,5）,sharppoint（1,5）andendpoint（3,1）.

Soitisapartoftheparabola:

Allofthearcsfoundabovearecalledparabolicarcs.Theyareallcontrolledbythreepoints.Inordertocreateamoreflexiblecurve,anotherpointcanbeaddedtothegraph.Andtheycancreateanon-parabolicarccalledBeziercurve.

FirstarccanbeusedagaintofindaBeziercurve.ThenanotherpointD（6,0）willbeadded.

UsetheequationtofindarcBD:

ArcACis

Thenenterthemintocalculate.

B

D

C

A

AssumethatthereisapointTmovesfromBtoDonarcBDandSfromAtoConarcAC.Astincreasesfrom0to1,linesegmentSTstartsasABwhent=0andfinishesasCDwhent=1.NowassumeagainthatthereisapointZonlinesegmentSTanddividingSTintheratiotto1-t.Thenusethesamemethod,theparametricequationofarcADcanbefound.

SotheparametricequationforarcADis:

C

D

B

A

ThearcADiscalledtheBeziercurvewithinitialpointA,sharppointBandCandendpointD.

Withtheworkdonebefore,thegeneralequationalsocanbedetermined.

LetDbe

Sis

Tis

Sotheparametricequationis:

Inconclusion,ifanarciscontrolledbyaninitialpoint,asharppointandanendpoint,wecallthisarcparabolicarc.Itsequationis

.

Ifthereisafourthpointalsocontrolonearc,thearccanbemoreflexibleandnon-parabolic.ItiscalledBeziercurve.Anditsequationis

Bibliography:

http:

//en.wikipedia.org/wiki/B%C3%A9zier_curve