Algebraic and geometric characterizations of a class of planar quartic curves with rational offsets

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Computer Aided Geometric Design

www.elsevier.com/locate/cagd

Algebraic and geometric characterizations of a class of planar quartic curves with rational offsets

Kai Hormann

^a

^,

^b

^,∗ , Jianmin Zheng

^b

aUniversitàdellaSvizzeraitaliana,Lugano,Switzerland bNanyangTechnologicalUniversity,Singapore

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Availableonline6May2020

Keywords:

QuarticBéziercurves Offsetcurves Pythagoreanhodograph

Planar polynomial curves have rational offset curves, if they are either Pythagorean- hodograph(PH)orindirectPythagorean-hodograph(iPH)curves.Inthispaper,wederive an algebraic and two geometric characterizations for planar quartic iPH curves. The characterizationsaregivenintermsofquantitiesrelatedtotheBéziercontrolpolygonof thecurve,andnaturallyextendtoquarticandcubicPHandquadraticiPHcurves.

©^{2020ElsevierB.V.Allrightsreserved.}

1. Introduction

Letr

(

t

) = (

x

(

t

),

y

(

t

))

be aparametric curve intheplane.The offsetcurveto ratsome (signed)distanced

∈

R canbe writtenasrd

(

t

) =

^r

(

t

) +

^dn

(

t

)

,wheren

(

t

) = (

y

(

t

), −

^x

(

t

))/ σ (

t

)

istheunitnormal ofthecurve r

(

t

)

and

σ (

t

) =

_r

(

t

) =

x

(

t

)

²

+

^y

(

t

)

² isits speed.Offset curvesariseina varietyofapplications,includingCNC machining,railwaydesign,and shapeblending (TillerandHanson,1984;Pham,1992;Maekawa,1999),butduetothesquarerootinthedefinitionof

σ (

t

)

, theyareusuallynotrationalandcanthusnotberepresentedexactlyincommonCADsystems (FaroukiandSakkalis,1990;

Farouki,1992).Therefore,researchonconditionsforpolynomialcurvestohaverationaloffsetsandmethodsforconstructing suchcurveshasattractedalotofattention (FaroukiandSakkalis,1990;Farouki,1992;Lü,1994,1995),andthetheoretical analysisoftherationalityofgeneralizedoffsetstoirreduciblehypersurfaceswas studied,too (Arrondoetal.,1997; Sendra andSendra,2000).

Especiallyincomputer-aidedgeometricdesignandmanufacturing (Farinetal.,2002),planarpolynomialcurvesareoften expressedintheBézierformasr

: [

⁰

,

1

] →

R²,

r

(

t

) =

n

i=⁰

p_iBⁿ_i

(

t

),

(1)

where Bⁿ_i

(

t

) =

_n

i

(

1

−

^t

)

ⁿ⁻ⁱtⁱ aretheBernsteinbasispolynomialsofdegreen

∈

Nand p₀

, . . . ,

p_n

∈

R² arethecontrolpoints.

Connecting the control points forms the controlpolygon that provides an intuitive approximation anddescription of the curve.Asubsetofthesecurveshasexactlyrepresentablerationaloffsetcurves.

ThefirstclassofpolynomialcurveswithrationaloffsetsarePythagorean-hodograph(PH)curves,whichwereintroduced by Farouki and Sakkalis (1990). For these curves, the speed

σ (

t

)

is a polynomial, so that the two components of the curve’sfirstparametric derivativeorhodographr

(

t

) = (

x

(

t

),

y

(

t

))

and

σ (

t

)

forma polynomialPythagoreantriple,that is,

*

Correspondingauthorat:UniversitàdellaSvizzeraitaliana,Lugano,Switzerland.

E-mailaddresses:kai.hormann@usi.ch(K. Hormann),ASJMZheng@ntu.edu.sg(J. Zheng).

https://doi.org/10.1016/j.cagd.2020.101873 0167-8396/©^{2020ElsevierB.V.Allrightsreserved.}

Fig. 1.ExamplesofC¹Hermiteinterpolants(top),theirspeeds(middle),andcurvatures(bottom).Notethattheprescribedtangentsarescaledbyafactor of1/5 tobetterfittheimages.

x

(

t

)

²

+

^y

(

t

)

²

= σ (

t

)

².PHcurvesandtheirapplicationshavebeenstudiedintensively,andwerefertheinterestedreaderto thebookbyFarouki(2008) andthereferencestherein.

The second classof polynomialcurves thathaverational offsetswithrespectto aproperly chosen reparameterization was discoveredbyLü(1994,1995).Forsucha(non-PH)offset-rationalorindirectPythagorean-hodograph(iPH)curve (Luet al., 2016), thereexistsasuitablerationalquadraticparametertransformationt

: [

⁰

,

1

] → [

⁰

,

1

]

^witht

(

0

) =

^0,t

(

1

) =

^1,and t

(

s

) >

0 fors

∈ [

⁰

,

1

]

^{,suchthatthespeed}

σ ˜ (

s

)

ofthereparameterized curver

˜ (

s

) =

^r

(

t

(

s

))

andthetwocomponentsx

˜

(

s

)

and

˜

y

(

s

)

ofitsfirstderivativeformarationalPythagoreantriple (Lü,1995;Luetal.,2016).

In thispaper,we studya class ofquarticiPHcurves,which hasbeenshowntobe capableof interpolatingfirst-order Hermite datawith up to four differentsolutions (Lü, 1995), akin to quinticPH curves (Farouki,1992; Farouki andNeff, 1995);seeFig.1.Wefocusonproperlyparameterizedcurves,forwhichtheparametervaluetandthecurvepointr

(

t

)

arein one-to-onecorrespondenceforallt

∈

R,exceptforparametervaluescorrespondingtoself-intersectionsofr(Farouki,2008).

Foran improperly parameterizedpolynomialorrationalcurve,it isalwayspossible tomakeitproperlyparameterized by reparameterization (Sederberg,1984,1986). Following Farouki(1994),weusethecomplexrepresentationofplanar Bézier curves toanalyse thestructure ofthesequarticiPH curvesandderivea simplealgebraic characterization, giveninterms of thecomplexformof thecontrol edgevectors(see Section 3),which turnsout tobe usefulfortheconstruction ofC¹ Hermiteinterpolants(seeSection3.4).Wetheninvestigatetwogeometriccharacterizations,wheretheconditionsarestated intermsofquantitiesrelatedtothecontrolpolygon,whichcanbeusedtoparameterizethisclassofquarticiPHcurvesin anintuitiveway(seeSection4).

1.1. Relatedwork

Variousgeometricandalgebraiccharacterizationsoflowerdegreecurveswithrationaloffsetshavebeenderived,usually intermsofquantitiesrelatedtotheirBéziercontrolpolygons,withmostworkfocussingonPHcurves.

We knowthatacubic curveisaPHcurve, ifandonlyifthetwointeriorangles

θ

1 and

θ

2 betweenadjacentedges of theBéziercontrolpolygon(seeFig.2)arethesameandthelengths Ei

=

ei

ofthecontroledgevectorsei

=

p_i+1

−

p_i are in geometricprogression, that is, E₁

= √

E₀E₂ (FaroukiandSakkalis, 1990), whichis equivalentto thecondition that the triangles

[

^p0

,

p₁

,

p₂

]

^and

[

^p1

,

p₂

,

p₃

]

^aresimilar (Pelosietal.,2007).Interpretingtheedgevectorsei ascomplexnumbers,

Fig. 2.Notation for the characterization of cubic (left) and quartic (right) PH curves in Bézier form.

thesegeometric conditionscanbe combinedtothe singlecomplex constrainte²₁

=

e0e2 (Farouki,1994),andwe derivea similaralgebraicconditionforquarticiPHcurvesinSection3.

Accordingto theanalysisin Wang andFang(2009),a quarticcurve isa PHcurve, ifthereexists a line,which passes through p₂ andintersectswiththelines p₀p₁andp₃p₄ ats1ands2,respectively,suchthattheangles

(

p₁

−

^s1

,

p₂

−

^s1

)

and (^p2

−

^s2

,

p₃

−

^s2

)

are equal (see Fig. 2) and the lengths F_i

=

f_i of the vectors f₀

=

^s1

−

^p1, f₁

=

^p2

−

^s1, f₂

=

^s2

−

^p2, f₃

=

^p3

−

^s2,togetherwiththelengthsE₀ andE₃satisfythethreeconstraintsE₀F₂

=

^3F0F₁,E₃F₁

=

^3F2F₃, andF1F2

=

^4F0F3.InSection4,weshowthatageneralizedversionoftheseconditionscharacterizesquarticiPHcurves.

Furtherinvestigationshaverevealedanadditionalcondition foraquarticPHcurvetohavemonotoniccurvature (Zheng andWang,2018), algebraic (Farouki,1994) andgeometric (Fangand Wang,2018) characterizations ofquintic PHcurves, threemethodsforidentifyingsexticPHcurves (Wangetal.,2017),aswellasgeometricpropertiesforPHcurvesofdegree seven (Lietal.,2019;Zhengetal.,2016).

1.2. Contributions

MuchlessisknownforiPHcurves,apartfromthefactthat quadraticcurvesareiPH,butnotPHcurves,aslongasthe controlpoints p₀,p₁, p₂ arenotcollinear (FaroukiandSederberg, 1995; Lü,1995),andanin-depthanalysisofgeometric conditionsforproperlyparameterizedcubiciPHcurves (Luetal.,2016).Tothebestofourknowledge,thispaperisthefirst totake acloserlookatquarticiPHcurvesandtoderivealgebraic aswell asgeometric characterizationsforanimportant subsetofthesecurves.

2. Preliminaries

FortheanalysisofPHandiPHcurves,ithasturnedouttobeusefultoexploitthecomplexrepresentationofR²(Farouki, 1994).Wefollowthisapproachandthroughoutthispaperidentifytheplanarpoint

(

x

,

y

) ∈

R² withthecomplexnumber x

+

^iy

∈

Candlikewiseforvectorsandplanarcurves.Thestarting pointofourinvestigationsisanecessaryandsufficient conditionthatwasdiscoveredbyLü(1995).

Theorem1.Aproperlyparameterizedpolynomialcurver

(

t

)

hasrationaloffsets,ifandonlyifitshodographcanbewritteninthe complexformas

r

(

t

) =

^p

(

t

)(

1

+

^kt

)

w

(

t

)

²

,

(2)

wherep

(

t

)

isarealpolynomial,kisacomplexconstant,andw

(

t

) =

^x

(

t

) +

^iy

(

t

)

isacomplexpolynomialwithx

(

t

)

andy

(

t

)

relatively prime.

Withoutlossofgenerality,wecanassumetheleadingcoefficientofp

(

t

)

in (2) tobeequalto1,sinceallothercasescan bereducedtothissituationbymultiplying w

(

t

)

withthesquarerootofthisleadingcoefficient.Moreover,thecurver

(

t

)

is aPHcurve,ifIm

(

k

) =

^{0,andaniPHcurve,otherwise.}

In thispaper, we focus onquartic iPH curves.For thesecurves, we havedeg

(

1

+

^kt

) =

^{1 and} consequently deg

(

r

) =

deg

(

r

) +

¹

=

^deg

(

p

) +

^2deg

(

w

) +

²

=

^4,andwedistinguishthefollowingcases:

•

^ClassI:deg

(

p

) =

^{2 anddeg}

(

w

) =

^0.

Inthiscase,wecanrewritethehodographin (2) as r

(

t

) =

(

t

−

^a

)

²

+

^b

(

1

+

^kt

)

w (3)

for some a

,

b

∈

R andk

,

w

∈

C with Im

(

k

) =

^{0 and w}

=

^{0. Thesecurves havea linearly varying} normal (Ahn and Hoffmann,2011),andwefurtherrecognizethreesub-cases:

– ClassI.0:b

>

0.Thesecurvesareregular.

– ClassI.1:b

=

^{0.Thesecurvesaresingular,buttangentcontinuousatt}

=

^a.

– ClassI.2:b

<

0.Thesecurveshavetwocuspsatt

=

^a

± √

−

^b.

•

^ClassII:deg

(

p

) =

^{0 anddeg}

(

w

) =

^1.

Inthiscase,wecanrewritethehodographin (2) as

r

(

t

) = (

1

+

^kt

)(

w₀

(

1

−

^t

) +

^w1t

)

² (4)

forsomek

,

w0

,

w1

∈

CwithIm

(

k

) =

^0,w0

=

^0,w1

=

^0,andIm

(

w1

/

w0

) =

^{0.Thesecurvesareregular.}

Inparticular,weareinterestedinclassIIquarticiPHcurves,sincetheycanbeusedforsolvingtheC¹ Hermiteinterpo- lationproblem (Lü,1995),butwewillseethatclassI.1quarticiPHcurves,quarticandcubicPHcurves,andquadraticiPH curvesplayaroleinthiscontext,too(seeSection3.4).

3. Algebraicconsiderations 3.1. ClassIIquarticiPHcurves

Ourfirst observation, whichwas alsomentionedin Lü(1995), butwithoutfurtherexplanation,is thatthere existsan alternativerepresentationforthehodographofclassIIquarticiPHcurves.

Corollary1.Aproperlyparameterizedquarticcurver

(

t

)

isaniPHcurveofclassII,ifandonlyifitshodographcanbewritteninthe complexformas

r

(

t

) =

u₀

(

1

−

^t

) +

^u1 a²t

((

1

−

^t

) +

^at

)

²

,

(5)

whereu0

,

u1

,

a

∈

Cwithu0

=

^0,u1

=

^0,Im

(

a

) =

^0,andIm

(

a²u0

/

u1

) =

^0.

Proof. Ifr

(

t

)

isgivenasin (4),thenwecanrewriteitasin (5) bylettingu0

=

^w20,u1

= (

1

+

^k

)

w²₁,a

=

^w1

/

w0.Viceversa, wecanconvertahodographfromtheformin (5) totheformin (4) bylettingk

=

^u1

/(

u₀a²

) −

^{1, w}0

= ± √

u₀,w₁

=

^w0a.

Notethattheadditionalconditionsonk,w0,w1 in (4) andonu0,u1,ain (5) implyeachother.

InordertoderiveanalgebraiccharacterizationofclassIIquarticiPHcurvesintermsofthecontroledges e_i

=

^pi+1

−

^pi fromtheirBézierrepresentation,werecallthatthehodographofaquarticcurver

(

t

)

canbewritten,bydifferentiatingits Bézierrepresentationin (1),as

r

(

t

) =

⁴

e0B³₀

(

t

) +

^e1B³₁

(

t

) +

^e2B³₂

(

t

) +

^e3B³₃

(

t

)

.

(6)

Comparingthetwoexpressionsofr

(

t

)

in (5) and (6) fort

=

^{0 andt}

=

^{1,weobservethat} e₀

=

¹

4u₀

,

e₃

=

¹

4u₁

,

(7)

and,bysimilarlycomparingtheirderivatives,thatis,thetwoformsofr

(

t

)

thattheyinduce, e₁

=

²

3ae₀

+

¹

3a⁻²e₃

,

e₂

=

¹

3a²e₀

+

²

3a⁻¹e₃

.

(8)

Moreover,weneedtointroducetheconceptofnon-degenerateBéziercontrolpolygons.

Definition1.We saythat theBézier control polygonofa quarticcurve r

(

t

)

isnon-degenerate,ifandonly ifthefirstand last controledgedonotvanishandthecontrolpointsarenotcollinear,thatis,e0

=

^0,e3

=

^0,andIm

(

ei

/

e0

) =

^{0 forsome} i

∈ {

¹

,

2

,

3

}

^.

Thiskindofnon-degeneracyrulesoutcurvesthataresingularateithert

=

^{0 ort}

=

^{1 andthosethatdescribea(possibly} improperlyparameterized)straightline.Wearenowreadytopresentourmainresult.

Theorem2.Aproperlyparameterizedquarticcurver

(

t

)

isaniPHcurveofclassII,ifandonlyifitsBéziercontrolpolygonisnon- degenerateanditscontroledgessatisfy

(

e₀e₃

−

^e1e₂

)

²

=

⁴

(

e₀e₂

−

^e21

)(

e₁e₃

−

^e22

)

(9) andeither

e₀e₃

=

^e1e₂ (10)

or Im

2 e₁e₃

−

^e2₂ e0e3

−

^e1e2

=

^{0 and Im}

4

_e

1e₃

−

^e2₂ e0e3

−

^e1e2

2

e0

e3

=

⁰

.

(11)

Proof. Letusstartby assuming that r

(

t

)

isa properlyparameterized class II quarticiPH curve. Toshow that theBézier controlpolygonofr

(

t

)

isnon-degenerate,wefirstrecallfromCorollary1thatu0

=

^0,u1

=

^0,andIm

(

a

) =

^0,hencea

=

^0.It thenfollowsfrom (7) thate₀

=

^{0 ande}3

=

^{0.Now assumethatall e}i areparallel,thatis,e_i

= λ

ie₀fori

=

¹

,

2

,

3 andsome

λ

1

, λ

2

, λ

3

∈

R

\ {

⁰

}

^{.By (8),wethenhave}

λ

2

=

^a

(

2

λ

1

−

^a

)

and

λ

3

=

^a2

(

3

λ

1

−

^2a

)

.SinceIm

(

a

(

2

λ

1

−

^a

)) =

^2Im

(

a

)(λ

1

−

^Re

(

a

))

andIm

(

a

) =

^{0,we concludethat}

λ

1

=

^Re

(

a

)

,because

λ

2 wouldotherwise notbe a realnumber. Withthis, however,we findthat Im

(

a²

(

3

λ

1

−

^2a

)) =

^2Im

(

a

)

³

=

^{0, which}contradicts the assumptionthat

λ

3

∈

R.We now proceedto prove the algebraicconditions.Itfollowsfrom (8) that

e₀e₃

−

^e1e2

= −

²

9a

(

ae0

−

^a−2e3

)

²

,

(12a)

e₀e₂

−

^e21

= −

¹

9

(

ae₀

−

^a−2e3

)

²

,

(12b)

e₁e₃

−

^e2₂

= −

¹

9

(

a²e₀

−

^a−1e3

)

²

,

(12c)

whichimmediatelyimplies (9).By (12a),condition (10) isequivalenttoa³e0

=

^e3,becausea

=

^{0,andusing (8),weseethat} moregenerallye_i

=

^aei−1

=

^aie0 fori

=

¹

,

2

,

3 in thisspecialcase, whichmeans thatthe controledges are ingeometric progression,justlikethecontroledgesofacubicPHcurve.Otherwise,e0e3

=

^e1e2,anditfollowsfrom (12a) and (12c) that

2 e₁e₃

−

^e2₂ e0e3

−

^e1e2

=

^a

.

By (7),wefurtherfindthate0

/

e3

=

^u0

/

u1,andtheconditionsin (11) thenfollowfromCorollary1,whichguaranteesthat Im

(

a

) =

^{0 andIm}

(

a²u₀

/

u₁

) =

^0.

Letusnow assume that r

(

t

)

isa properly parameterizedquartic curve witha non-degenerate Béziercontrol polygon andcontroledgesei thatsatisfyconditions (9) and (10).Then,eithere0e2

=

^e21 ore1e3

=

^e22,butasbothidentitiesactually implyeach other,by (9) and underthenon-degeneracyassumption ofthecontrol polygon,whichguarantees e₀

=

^{0 and} e3

=

^{0,we can safely assume both ofthem to be true.}Consequently, e2

=

^e21

/

e0 ande3

=

^e31

/

e²₀, whichmeans that the control edges are in geometric progression, and it is not hard to verify that both conditions in (8) hold fora

=

^e1

/

e₀. Consequently,andinview of (7), wecan writethehodographofr

(

t

)

asin (5) with u0

=

^4e0

=

^{0 andu}1

=

^4e3

=

^0,and itactually simplifiesto r

(

t

) =

^u0

((

1

−

^t

) +

^at

)

³ inthiscase.Toshow that Im

(

a

) =

^{0,let usassumethe opposite,that is,} Im

(

e₁

/

e₀

) =

^{0.Wethen haveIm}

(

e₂

/

e₀

) =

^Im

(

a²

) =

^{0 andIm}

(

e₃

/

e₀

) =

^Im

(

a³

) =

^0,thus contradictingtheassumptionthat theei arenotallparalleltoeachother.Therefore,Im

(

a

) =

^{0,whichalsoimpliesIm}

(

a²u0

/

u1

) =

^Im

(

a⁻¹

) =

^0.

Finally,letusassumethat (11) holdsinsteadof (10),andnotethate₀e₃

=

^e1e₂,by (9),impliese₀e₂

=

^e2₁^.Letting a₁

=

^e0e3

−

^e1e₂

2

(

e₀e₂

−

^e2₁

) ,

a₂

=

²

(

e1e3

−

^e2₂

)

e₀e₃

−

^e1e₂

,

itthenfollowsfrom (9) thata1

=

^a2,anditcanfurtherbeverifiedthat 4a²₁e1

= (

e0e3

−

^e1e2

)

²

(

e₀e₂

−

^e2₁

)

²

e1

=

^e20e1e23

−

^2e0e²₁e₂e₃

+

^e3₁^e2₂

(

e₀e₂

−

^e2₁

)

²

,

2a²₁a₂e₀

= (

e₀e₃

−

^e1e₂

)(

e₁e₃

−

^e2₂

) (

e0e2

−

^e2₁

)

²

e₀

=

^e20e1e23

−

^e2₀^e2₂^e3

−

^e0e²₁e2e3

+

^e0e1e³₂

(

e0e2

−

^e2₁

)

²

,

a₁a₂e₁

= (

e0e2

−

e²₁

)(

e1e3

−

e²₂

) (

e₀e₂

−

^e2₁

)

²

e₁

=

^e0e21e2e3

−

e0e1e³₂

−

e⁴₁e3

+

e³₁e²₂

) (

e₀e₂

−

^e2₁

)

²

.

Therefore, 4a²₁e1

=

^2a21a2e0

+

^a1a2e1

+

^e3,and we concludethat the condition for e1 in (8) holds fora

=

^a1

=

^a2.The conditionfore₂ canbecheckedsimilarly.Asinthepreviouscase,we canthenwritethehodographofr

(

t

)

asin (5) with u0

=

^4e0

=

^{0 andu}1

=

^4e3

=

^{0,andtheconditionsIm}

(

a

) =

^{0 andIm}

(

a²u0

/

u1

) =

^{0 followdirectlyfrom (11).}

Thekey ingredienttothecharacterizationofclassIIquarticiPHcurvesinTheorem2certainly iscondition (9),andwe will nowshow that thiscondition isalso validforotherquarticBézier curveswithrationaloffsets,whichare notiPH of classII.

3.2. CubicPHandquadraticiPHcurves

Ifr

(

t

)

isacubicBéziercurvewithcontroledgesd0,d1,d2,thenthecontroledgesafterdegreeelevationare e₀

=

³

4d₀

,

e₁

=

¹ 4d₀

+

¹

2d₁

,

e₂

=

¹ 2d₁

+

¹

4d₂

,

e₃

=

³

4d₂

,

(13)

andastraightforwardcalculationrevealsthat

(

e0e3

−

e1e2

)

²

−

4

(

e0e2

−

e²₁

)(

e1e3

−

e²₂

) =

³

64

(

d0d2

−

d²₁

)(

d0

−

2d1

+

d2

)

²

.

Therefore,condition (9) is true,ifd²₁

=

^d0d2 ord1

=

¹₂

(

d0

+

^d2

)

,thatis,d1 mustbethegeometricorthearithmeticmean ofd₀ andd₂.Notethatthefirstcaseholds,ifr

(

t

)

isacubicPHcurve, whilethesecond caseoccurs,ifandonlyifr

(

t

)

is a degree-raisedquadratic curve.Thissuggeststhattheremightbe characterizationsofcubic PHandquadraticiPH curves that aresimilartotheoneinTheorem 2,butbeforewe gettotheadditionalalgebraicconditionsthatareneededforthis purpose,letusstatetheequivalentsofCorollary1.

Corollary2.Aproperlyparameterizedcubiccurver

(

t

)

isaPHcurve,ifandonlyifitshodographcanbewritteninthecomplexform asin(5)withu₀

=

^0,u1

=

^0,Im

(

a

) =

^0,anda2u0

=

^u1.

Proof. Wefirstrecallfrom FaroukiandSakkalis(1990) thatthehodographofaproperlyparameterizedcubicPHcurvecan writtenasin (2) withk

=

^0,deg

(

p

) =

^0,anddeg

(

w

) =

^1,thatis,as

r

(

t

) = (

w0

(

1

−

t

) +

w1t

)

² (14)

forsome w0

,

w1

∈

C with w0

=

0, w1

=

0,andIm

(

w1

/

w0

) =

0.Itfollowsdirectly,thatwecanthenrewrite r

(

t

)

inthe form (5) bylettingu₀

=

^w2₀^,u1

=

^w2₁^,anda

=

^w1

/

w₀.Viceversa,ifr

(

t

)

isgivenasin (5),thenwegetbacktotheform in (14) byletting w0

= ± √

u0 and w1

=

^w0a.Notethat theadditionalconditionson w0,w1 in (14) andon u0,u1,ain thestatementimplyeachother.

Corollary3.Aproperlyparameterizedquadraticcurver

(

t

)

isaniPHcurve,ifandonlyifitshodographcanbewritteninthecomplex formasin(5)withu0

=

^0,u1

=

^0,a

=

^1,andIm

(

a²u0

/

u1

) =

^Im

(

u0

/

u1

) =

^0.

Proof. ProperlyparameterizedquadraticiPHcurveshaveahodographasin (2) withIm

(

k

) =

^0,deg

(

p

) =

^0,anddeg

(

w

) =

^0, thatis, w

(

t

)

²

≡

^u0forsome u0

∈

C withu0

=

^{0,whichcanberewrittenasin (5) bylettingu}1

= (

1

+

^k

)

u0

=

^{0 anda}

=

^1, andviceversa.

In view ofCorollaries 2and3,our previous discovery isactually not surprising, becausethe identitiesin (12), which imply (9),followonlyfromtherepresentationofthehodographin (5),butdonotdependonthespecificconditionsonu₀, u1,anda.LetusnowpresenttheanaloguesofTheorem2.

Theorem3.Aproperlyparameterizedcubiccurver

(

t

)

isaPHcurve,ifandonlyifitsdegree-raised,quarticBéziercontrolpolygonis non-degenerateanditscontroledgessatisfy(9),e0e3

=

^e1e2,and

Im

2 e₁e₃

−

^e2₂ e₀e₃

−

^e1e₂

=

0 and 4

_e

1e₃

−

^e2₂ e₀e₃

−

^e1e₂

2

e0

e₃

=

1

.

(15)

Proof. Toprove the necessityofthe conditions,let usassume that r

(

t

)

isa cubic PHcurve witha hodographasstated in Corollary 2. The non-degeneracy of the degree-raised, quarticBézier control polygon then follows asin the proof of Theorem 2,becausewe still haveu₀

=

^{0, u}1

=

^0,andIm

(

a

) =

^{0,andthevalidity of (9) wasderived above. Now,ife}0e₃ were equalto e1e2,then a³e0

/

e3

=

^{1,by (12a), butit alsofollows}from (7) and Corollary2 thata²e0

/

e3

=

^a2u0

/

u1

=

^1, whichimpliesa

=

^1,thuscontradictingthefactthatIm

(

a

) =

^{0.Expressingthee}iintermsofthecubicBéziercontroledges d_iasin (13) andsubstitutingd₂

=

^d21

/

d₀,astraightforwardcalculationshowsthat

2 e1e3

−

^e2₂ e₀e₃

−

^e1e₂

=

^d1

d₀

=

^d2

d₁

,

(16)

whichhasanon-vanishingimaginarypart,ifandonlyifalldi andthusalsoalleiareparallel,butwejustshowedthatthe lattercannot happen.Theremaining identity in (15) follows directlyfrom (16),because e₀

/

e₃

=

^d0

/

d₂.The sufficiencyof theconditionscanbeshownwiththesameargumentsasintheproofofTheorem2.

Theorem4.Aproperlyparameterizedquadraticcurver

(

t

)

isaniPHcurve,ifandonlyifitstwicedegree-raised,quarticBéziercontrol polygonisnon-degenerateanditscontroledgessatisfy(9),e₀e₃

=

^e1e₂,and

2 e₁e₃

−

^e2₂

e0e3

−

e1e2

=

1 and Im

4

_e

1e₃

−

^e2₂ e0e3

−

e1e2

2

e0

e3

=

Im

e0

e3

=

0

.

(17)

Proof. Letusfirstassumethatr

(

t

)

isaquadraticiPHcurvewithahodographasstatedinCorollary3andquadraticBézier controledgesc0,c1.Afterdegreeelevation,thecontroledgesofthequarticBéziercontrolpolygonare

e₀

=

¹

2c₀

,

e₁

=

¹ 3c₀

+

¹

6c₁

,

e₂

=

¹ 6c₀

+

¹

3c₁

,

e₃

=

¹

2c₁

.

(18)

Asbefore,itisclearthate₀

=

^{0 ande}3

=

^{0.Moreover,sinceIm}

(

e₀

/

e₃

) =

^Im

(

u₀

/

u₁

) =

0 by (7) andCorollary3,weconclude that atleast e0 ande3 are notparallel,hencethequarticcontrol polygonisnon-degenerate.We alreadydiscussedabove that (9) holds;toshowtheremainingconditions,weuse (18) toget

e₀e₃

−

^e1e₂

=

²

(

e₀e₂

−

^e21

) =

²

(

e₁e₃

−

^e2₂

) = −

¹

18

(

c₁

−

^c0

)

²

andrecallIm

(

e0

/

e3

) =

^Im

(

c0

/

c1

) =

^{0,whichimpliesc}0

=

^c1andfurthere0e3

=

^e1e2,aswellas (17).Thesufficiencyofthe conditionscanagainbeshownwiththesameargumentsasintheproofofTheorem2.

3.3. QuarticPHandclassI.1quarticiPHcurves

LetusnowextendtheresultsabovetoquarticPHandclassI.1quarticiPHcurves.WefirstobservethatCorollary1can beextendedtobothkindsofcurves,ifweexcludetheoccurrenceofsingularitiesatt

=

^{0 andt}

=

^1.

Corollary4.Aproperlyparameterizedquarticcurver

(

t

)

isaPHcurvewithr

(

0

) =

^0andr

(

1

) =

^{0,ifandonlyifitshodographcan} bewritteninthecomplexformasin(5)withu0

=

^0,u1

=

^0,Im

(

a

) =

^0,Im

(

a²u0

/

u1

) =

^0,anda2u0

=

^u1.

Proof. Wefirstrecall from FaroukiandSakkalis (1990) that thehodographofaproperly parameterizedquarticPHcurve canbewrittenasin (2) witheitherk

=

^0,deg

(

p

) =

^{1 ork}

=

^0,Im

(

k

) =

^0,deg

(

p

) =

^0,anddeg

(

w

) =

^1,thatis,as

r

(

t

) = (

t

−

^a

)(

w₀

(

1

−

^t

) +

^w1t

)

² (19)

forsome a

∈

R and w₀

,

w₁

∈

C with w₀

=

^{0, w}1

=

^{0, andIm}

(

w₁

/

w₀

) =

^0.This representationshowsthat quartic PH curveshaveacusp att

=

^a,hencea

∈ { /

⁰

,

1

}

^{,becauseofthe}restrictions statedabove.Itthen followsthatwe canrewrite r

(

t

)

inthe form (5) by letting u0

= −

^aw2₀^{, u}1

= (

1

−

^a

)

w²₁,anda

=

^w1

/

w0. Viceversa,ifr

(

t

)

is givenasin (5), then we get back to the formin (19) byletting a

=

^a2u0

/(

a²u₀

−

^u1

)

, w₀

= ±

u₁

−

^a2u0

/

a, and w₁

=

^aw0. Notethat the additionalconditionsona,w0,w1 aboveandonu0,u1,ainthestatementimplyeachother.

Corollary5.Aproperlyparameterizedquarticcurver

(

t

)

isaniPHcurveofclassI.1withr

(

0

) =

^0andr

(

1

) =

^{0,ifandonlyifits} hodographcanbewritteninthecomplexformasin(5)withu0

=

^0,u1

=

^0,Im

(

a

) =

^0,a

∈ { /

⁰

,

1

}

^,andIm

(

a²u0

/

u1

) =

^0.

Proof. Ifr

(

t

)

is given asin (3) with a

∈ { /

⁰

,

1

}

^{and b}

=

^{0, then we can rewrite it as in (5) by letting u}0

=

^{a2w, u}1

= (

1

−

^a

)

²

(

1

+

^k

)

w, anda

= (

a

−

¹

)/

a.Viceversa,wecan convertahodographfromthe formin (5) tothe formin (3) by lettinga

=

¹

/(

1

−

^a

)

,b

=

^0,k

= (

u1

−

^a2u0

)/(

a²u0

)

,andw

= (

1

−

^a

)

²u0.Asinthepreviousproofs,theadditionalconditions implyeachother.

Asbefore,thesecorollariespavethewayforanalgebraiccharacterizationofthesekindofquarticcurvesintermsofthe Béziercontroledges.

Theorem5.Aproperlyparameterizedquarticcurver

(

t

)

isaPHcurvewithr

(

0

) =

^0andr

(

1

) =

^{0,ifandonlyifitsBéziercontrol} polygonisnon-degenerateanditscontroledgessatisfy(9),e0e3

=

^e1e2,and

Im

2 e1e3

−

e²₂ e0e3

−

^e1e2

=

^{0 and 4}

e1e3

−

e²₂ e0e3

−

^e1e2

2

e₀

e3

∈ R \ {

¹

} .

(20)