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PreliminariesTheoretical Results

Constructions and Examples

Rational Curves with Rational Rotation Minimizing

Frames from Pythagorean-Hodograph Curves

G. R. Quintana2,3

Joint work with the Prof. Dr. B. Juettler1, Prof. Dr. F. Etayo2

and Prof. Dr. L. Gonzalez-Vega2

1Institut f ur Angewandte GeometrieJohannes Kepler University, Linz, Austria

2Departamento de MATematicas, EStadıstica y COmputacionUniversity of Cantabria, Santander, Spain

3This work has been partially supported by the spanish MICINN grant

MTM2008-04699-C03-03 and the project

CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones

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Contents

1 PreliminariesInvolutes and evolutes of space curves

2 Theoretical ResultsRelationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship between R3MF curves and RDPH curves

3 Constructions and Examples


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Constructions and ExamplesInvolutes and evolutes of space curves

Definition (PH curves)

Polynomial Pythagorean-Hodograph ( PH) space curves are polynomial parametric curves with the property that their hodographs p(u) = ( p1(u), p2(u), p3(u)) satisfy the Pythagoreancondition

( p1(u))2 + ( p2(u))2 + ( p3(u))2 = (σ(u))2

for some polynomial σ(u).

Spatial PH curves satisfy p(u) × p(u)2 = σ2(u)ρ(u) whereρ(u) = p(u)2 − σ2(u)**.

**From Farouki, Rida T., Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable Springer, Berlin,

2008.


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Definition (RPH curves)

Rational Pythagorean-Hodograph ( RPH) space curves are rational parametric curves with the property that their hodographs p(u) = ( p1(u), p2(u), p3(u)) satisfy the Pythagorean condition

( p1(u))2 + ( p2(u))2 + ( p3(u))2 = (σ(u))2

for some piecewise rational function σ(u).

Definition (RM vector field)

A unit vector field v over a curve q is said to be Rotation

Minimizing ( RM) if it is contained in the normal plane of q and v(u) = α(u)q(u), where α is a scalar-valued function.

**(from Corollary 3.2 in Wang, Wenpin; Juttler, Bert; Zheng, Dayue; Liu, Yang, Computation of Rotation

Minimizing Frames , ACM Trans. Graph. 27,1, Article 2, 2008).


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Definition (RM vector field)

A unit vector field v over a curve q is said to be RotationMinimizing ( RM) if it is contained in the normal plane of q and v(u) = α(u)q(u), where α is a scalar-valued function.

Consequences:

Given v RM vector field over q, any unitary vector wperpendicular to q and v is a RM vector field over q**.

The ruled surface D(u, λ) = q(u) + λv(u) is developable.

**(from Corollary 3.2 in Wang, Wenpin; Juttler, Bert; Zheng, Dayue; Liu, Yang, Computation of Rotation

Minimizing Frames , ACM Trans. Graph. 27,1, Article 2, 2008).


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Definition (RMF curve)

A Rotation Minimizing Frame RMF in a curve is defined by a unit tangent vector tangent and two mutually orthogonal RM vectors.

Definition (R2MF, resp. R3MF, curve)

A polynomial (resp. rational) space curve is said to be a curve witha Rational Rotation Minimizing Frame (an R2MF curve; resp. an

R3MF curve) if there exists a rational RMF over the curve.


Preliminaries

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Definition (DPH, resp. RDPH, curve)

A polynomial (resp. rational) space curve p is said to be apolynomial (resp. rational) Double Pythagorean-Hodograph( DPH, resp. RDPH) curve if p and p × p are both

piecewise polynomial (resp. rational) functions of t, i.e., if the conditions

1 p(u)2 = σ2(u)

2 p(u) × p(u)2 = (σ(u)ω(u))2

are simultaneously satisfied for some piecewise polynomials (resp.rational functions) σ(u), ω(u).


Preliminaries

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Definition (SPH curve)

A rational curve is said to be a Spherical Pythagorean Hodograph( SPH) curve if it is RPH and it is contained in the unit sphere.

Definition (Parallel curves)

Two rational curves p, ˆ p : I → Rn are said to be parallel curves if

there exists a rational function λ = 0 such that

p

(u) = λ(u) ˆ p

(u), , ∀u ∈ I

Equivalence relation → [ p] the equivalence class generated by p.


Preliminaries

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Theorem

Let p and ˆ p be rational parallel curves

1 If p is RPH then ˆ p is also RPH.

2 If p is RDPH then ˆ p is also RDPH.3 If p is R3MF then ˆ p is also R3MF.

Consequently If a curve p is RPH (resp. RDPH, R3MF) then the curves in [ p] are RPH (resp. RDPH, R3MF).


Preliminaries

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Relationships illustrated

Theorem

Let p and p be rational parallel curves

1 If p is RPH then p is also RPH.

2 If p is RDPH then p is also RDPH.

3 If p is R3MF then p is also R3MF.

Consequently If a curve p is RPH (resp. RDPH, R3MF) then the curves in [p] are RPH (resp. RDPH, R3MF).


Preliminaries

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Theoretical ResultsConstructions and Examples

Involutes and evolutes of space curves

Given p and q curves in R3,

p is an evolute of q and q is an involute of p if the tangent linesto p are normal to q.

Let p : I = [a, b] → R

3 be a PH space curve;

s(u) = u

0 p(t)dt, the arc-length function;

q, an involute of p:

q(u) = p(u) − s(u)p(u)

p(u)


Preliminaries

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Lemma

The vector field v(u) = p(u)p(u) is a RM vector field over the

involute q(u).

Geometric proof: since q · v=0,

1 v is RM vector field over qiff the ruled surface q + λvdevelopable; and

2 q + λv is the tangentsurface of p.


Preliminaries

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Lemma

**Given a PH space curve p, we consider q an involute of p. The

frame defined by q(u)

q(u) ,v(u),w(u)

is an ( RMF) over q, where v(u) = p(u)

p(u) and

w(u) = q(u)q(u) × v(u).

Proposition

If p is a spatial PH curve then v(u) = p(u)

p

(u)

, the involute

q(u) = p(u) − s(u)v(u) and w(u) = q(u)

q(u) × v(u) are piecewise rational.

Generality: ”any curve is the involute of another curve” from Eisenhart, Luther Pfahler, A Treatise on Differential

Geometry of Curves and Surfaces , Constable and Company Limited, London, 1909.


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Lemma

Every curve p satisfies

p

p

= p×p

p .

Note:

For a PH curve it is reduced to p

p = ρ

p

p

is piecewise rational but

p

p

is not.

PropositionGiven a curve p, the vector field b(u) = p

(u)×p(u)p(u)×p(u) is a RM vector

field with respect to the involute q(u) = p(u) − s(u) p(u)

p(u) .


PreliminariesTh i l R l I l d l f

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Lemma

Consider a curve p and its involute q = p− s p

p . Then

p(u) × p(u)

p

(u) × p

(u)=

q(u) × p(u)

q

(u) × p

(u)

RMF over the involute q:

q

q ,p

p , b =p

× p

p × p


PreliminariesTh ti l R s lts

Relationship between planar RPH curves and SPH curvesR l ti shi b t DPH s d SPH s

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Relationship between DPH curves and SPH curvesRelationship between R3MF curves and RDPH curves

Theorem

The image s of a rational planar PH curve r = (r1, r2, 0) by the M obius transformation

Σ : x → 2

x + z

x + z2 − z

where z = (0, 0, 1), is a SPH curve and vice versa.

Note that Σ ◦ Σ =Id. Then by direct computations the necessaryand the sufficient conditions hold.



Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curves

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Proof .Necessary condition. r is PH,

r

2 = σ2, σ rational. Then,

s = Σ(r) =1

r21 + r22 + 1(2r1, 2r2, 1 − r21 − r22)

By direct computation s = 1. Since

s

=(−2(r2

1r1

+ 2r1r2r2− r

1− r

1r22

), 2(−2r2r1r1− r

22r2

+ r2

+ r2r21

),−4(r1r1

+ r2r2

))

(r21

+ r2

2+ 1)2

it holds s = 2r/r + z2 = 2σ(r + z)−2.Sufficient condition. let s = (s1, s2, s3) such that s21 + s22 + s23 = 1 and

s21 + s

22 + s

23 = σ

2

for σ rational. Then r = Σ(s) = s1

s3+1 ,s2

s3+1 , 0 ⇒

r contained in z = 0.Differentiating r = − s

3

(s3+1)3(s1, s2, 0) + 1

s3+1(s1, s2, 0).

Substituting s1s1 + s2s2 = −s3s3 ⇒ r = σ

s3+1 .




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Theorem

1 Given a SPH curve r(u)/w(u) : I → R3 where v and w are

polynomial functions of the parameter then the

integrated-numerator curve p(u) = r(u)du is DPH.

2 If a space curve p(u) is RDPH then the unit-hodograph curve p(u)

p(u) is SPH.




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Proof .(1) r/w is spherical, r/w = 1 so r21 + r22 + r23 = w2. Derivating2(r1r1 + r2r2 + r3r3) = 2ww. From PH curve def.

(r/w) = σ, σ

rational. This gives rw − rw = w2σ. Direct comput.r2 = w2σ2 + w2. p is DPH because p = r = w and

p

× p

2 =

r

×r

2 = (σw2)2.

(2) By hypothesis p2 = σ2 and p × p2 = σ2 p2 − σ2

= δ2,

σ and δ rational. Since Lemma** holds for rational space curves we havethat

p

p

2

= p2 − σ2 =

δ

σ

2

Lemma**: Every curve p satisfies

p

p

=p×p

p.




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Theorem

1 Given a SPH curve r(u)/w(u) : I → R

3

where v and w are polynomial functions of the parameter then the integrated-numerator curve p(u) =

r(u)du is DPH.

2 If a space curve p(u) is RDPH then the unit-hodograph curve p(u)

p(u) is SPH.

Corollary

1 If p is a DPH curve then the unit-hodograph p

p is an SPH curve

and additionally ( p/ p

) =

p

× p

/ p

, polynomial.

2 If p is an RPH curve then p × p2 = σ2ρ, where p = σ and ρ = p2 − σ2.

3 If p is RDPH then p × p2 = σ2ω2, where ω2 = ρ.




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Constructions and Examplesp

Relationship between R3MF curves and RDPH curves

Theorem

1 Let p be a DPH curve and consider an involute q. The vectors

q(u)

q(u) and b(u) =p(u) × p(u)

p(u) × p(u)

are piecewise rational, where q is an involute of p. Thus q is R3

MF.2 If a rational space curve q is R 3MF then we can find a space curve p(u) such that p(u) is RDPH and q(u) is an involute of p(u).

Proof .

(1)Initial lemmas.(2)Basically construction of the involute in

Do Carmo, Manfredo P, Geometrıa Diferencial de Curvas y Superficies ,

Alianza Editorial, S. A., Madrid,1990.




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Constructions and Examples Relationship between R3MF curves and RDPH curves

The R3MF curve from pevious Theorem (1) q has piecewise polynomial

arc-length function: q

= |s| p

p and then

q = |s| p × p p = |s|σω

σ= |s|ω

Note that the previous property does not hold in general for R

3

MFcurves.

Lemma

If two curves p and ˆ p are parallel, then the corresponding involutes q, qare also parallel.

Theorem

Every R3MF curve is parallel to the involute of a DPH curve.



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Construction of an R2MF curve of degree 9

Degree 9 R2MF curve from a polynomial planar PH curve. Thederivative of the PH curve is defined from two linear univariatepolynomials a(t) = a1t + a0 and b(t) = b1t + b0:

r(t) = (a2(t)

−b2(t), 2a(t)b(t), 0)

The SPH curve s is image of r by the transformation described in theprevious Theorem getting s(t) = (s1, s2, s3), where

s1 = (6(a21t3 + 3a1t2a0 + 3a20t − b21t3 − 3b1t2b0 − 3b20t + 3c1))/(9 +24a1t4a0b1b0 + 6a21t5b1b0 + 6a1t5a0b21 + 18a1t3a0b20 + 18a1t2a0c1 +

18a2

0t3

b1b0 − 18b1t2

b0c1 + 12a1b1t3

c2 + 18t2

a0b1c2 + 18t2

a1b0c2 +36a0b0tc2 + a41t6 + 9a40t2 + b41t6 + 9b40t2 + 9c21 + 9c22 + 3a21t4b20 +2a21t6b21 + 6a21t3c1 + 6a31t5a0 + 15a21t4a20 + 18a1t3a30 + 3a20t4b21 +18a20t2b20 + 18a20tc1 + 6b31t5b0 + 15b21t4b20 − 6b21t3c1 + 18b1t3b30 − 18b20tc1)



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s2 = (6(2a1b1t

3

+ 3t

2

a0b1 + 3t

2

a1b0 + 6a0b0t + 3c2))/(9 +24a1t4a0b1b0 + 6a21t5b1b0 + 6a1t5a0b21 + 18a1t3a0b20 + 18a1t2a0c1 +18a20t3b1b0 − 18b1t2b0c1 + 12a1b1t3c2 + 18t2a0b1c2 + 18t2a1b0c2 +36a0b0tc2 + a41t6 + 9a40t2 + b41t6 + 9b40t2 + 9c21 + 9c22 + 3a21t4b20 +2a21t6b21 + 6a21t3c1 + 6a31t5a0 + 15a21t4a20 + 18a1t3a30 + 3a20t4b21 +18a20t2b20 + 18a20tc1 + 6b31t5b0 + 15b21t4b20

−6b21t3c1 + 18b1t3b30

−18b20tc1)

s3 = −(−9 + 24a1t4a0b1b0 + 6a21t5b1b0 + 6a1t5a0b21 + 18a1t3a0b20 +18a1t2a0c1 + 18a20t3b1b0 − 18b1t2b0c1 + 12a1b1t3c2 + 18t2a0b1c2 +18t2a1b0c2 + 36a0b0tc2 + a41t6 + 9a40t2 + b41t6 + 9b40t2 + 9c21 + 9c22 +3a21t4b20 + 2a21t6b21 + 6a21t3c1 + 6a31t5a0 + 15a21t4a20 + 18a1t3a30 +3a20t4b21 + 18a20t2b20 + 18a20tc1 + 6b31t5b0 + 15b21t4b20

−6b21t3c1 + 18b1t3b30

−18b20tc1)/(9 + 24a1t4a0b1b0 + 6a21t5b1b0 + 6a1t5a0b21 + 18a1t3a0b20 +18a1t2a0c1 + 18a20t3b1b0 − 18b1t2b0c1 + 12a1b1t3c2 + 18t2a0b1c2 +18t2a1b0c2 + 36a0b0tc2 + a41t6 + 9a40t2 + b41t6 + 9b40t2 + 9c21 + 9c22 +3a21t4b20 + 2a21t6b21 + 6a21t3c1 + 6a31t5a0 + 15a21t4a20 + 18a1t3a30 + 3a20t4b21 +18a20t2b20 + 18a20tc1 + 6b31t5b0 + 15b21t4b20 − 6b21t3c1 + 18b1t3b30 − 18b20tc1)



C t ti d E l

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s depends on the parameters that define the initial polynomials a and band on the integration constants obtained when we integrate r: c1, c2,c3. Integrating the numerator of the spherical curve we obtain a DPH

curve ˆ p. Its involute q is R3MF. We take now the minimum degreepolynomial curve q such that [q] = [q]. Once done we find an R2MF

curve of degree 9 q = (q1, q2, q3) whereq1 = (a61 + a41b21 − a12b41 − b61)t9 + (9a51a2 + 9a31a2b21 − 9a21b31b2 −9b51b2)t8 + (36a41a22 − (36/7)a41b22 + (72/7)a31a2b1b2 + (216/7)a21a22b21 −(216/7)a21b21b22 − (72/7)a1a2b31b2 + (36/7)a22b41 − 36b41b22)t7 + (81a31a32 −27a31a2b22 + 63a21a22b1b2−45a21b1b32 + 45a1a32b21−63a1a2b21b22 + 27a22b31b2−81b31b32)t6+(108a21a42−(216/5)a21a22b22−(108/5)a21b42+(648/5)a1a32b1b2−(648/5)a1a2b1b32 + (108/5)a42b21 + (216/5)a22b21b22 − 108b21b42)t5 +(81a1a52 − 81a1a2b42 + 81a42b1b2 − 81b1b52)t4 +(27a62 + 27a42b22 − 27a22b42 −27b62 − 27a21 + 27b21)t3 + (−81a1a2 + 81b1b2)t2 + (−81a22 + 81b22)t




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q2 = (2a51b1 + 4a31b31 + 2a1b51)t9 + ((9/2)a51b2 + (27/2)a41a2b1 +18a31b21b2 + 18a21a2b31 + (27/2)a1b41b2 + (9/2)a2b51)t8 + (36a41a2b2 +36a31a22b1 + (180/7)a31b1b22 + (648/7)a21a2b21b2 + (180/7)a1a22b31 +36a1b31b22 + 36a2b41b2)t7 + (117a31a22b2 + 9a31b32 + 45a21a32b1 +

153a21a2b1b

22 + 153a1a

22b

21b2 + 45a1b

21b

32 + 9a

32b

31 + 117a2b

31b

22)t

6

+((972/5)a21a32b2 + (324/5)a21a2b32 + (108/5)a1a42b1 + (1512/5)a1a22b1b22 +(108/5)a1b1b42 + (324/5)a32b21b2 + (972/5)a2b21b32)t5 + (162a1a42b2 +162a1a22b32 + 162a32b1b22 + 162a2b1b42)t4 − 162a2b2t + (54a52b2 +108a32b32 + 54a2b52 − 54a1b1)t3 + (−81a1b2 − 81a2b1)t2

q3 = (9a4

1+18a2

1b2

1+9b4

1)t6

+(54a3

1a0+54a2

1b1b0+54a1a0b2

1+54b3

1b0)t5

+(135a21a20 + 27a21b20 + 216a1a0b1b0 + 27a20b21 + 135b21b20)t4 + (162a1a30 +162a1a0b20 + 162a20b1b0 + 162b1b30)t3 + (81a40 + 162a20b20 + 81b40)t2




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Construction of a R2MF quintic

We consider the R2MF quintic (introduced in Farouki, Rida T.; Gianelli,Carlotta; Manni, Carla; Sestini, Alessandra, 2009. Quintic Space Curves

with Rational Rotation-Minimizing Frames. Computer Aided GeometricDesign 26, 580–592) q =−8 t3 − 245 t5 + 12 t4 − 4 t2

√2 + 8 t3

√2 − 8 t4

√2 + 16

5 t5√

2, −2 t2√

2 − 4 t3 +

6 t4 − 6 t4√

2 − 4 t5 + 165 t5

√2, −10 t + 20 t2 − 10 t2

√2 − 28 t3 +

20 t3√

2 + 22 t4 − 16 t4√

2 −8 t5 + 245 t5

√2




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Construction of a R2MF quintic

Using our method we can obtain the previous curve from the planar PH

curve




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Work still in process.......... Any suggestions???




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p

Thank you!


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