CVC Seminar

PreliminariesTheoretical Results

Constructions and Examples

Rational Curves with Rational Rotation MinimizingFrames 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 fur 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



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



Constructions and ExamplesInvolutes and evolutes of space curves

Definition (PH curves)

Polynomial Pythagorean-Hodograph ( PH) space curves arepolynomial parametric curves with the property that theirhodographs p′(u) = (p′1(u), p′2(u), p′3(u)) satisfy the Pythagoreancondition

(p′1(u))2 + (p′2(u))2 + (p′3(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 − σ′2(u)**.

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

2008.




Definition (RPH curves)

Rational Pythagorean-Hodograph ( RPH) space curves are rationalparametric curves with the property that their hodographsp′(u) = (p′1(u), p′2(u), p′3(u)) satisfy the Pythagorean condition

(p′1(u))2 + (p′2(u))2 + (p′3(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 RotationMinimizing ( RM) if it is contained in the normal plane of q andv′(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).




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 andv′(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).




Definition (RMF curve)

A Rotation Minimizing Frame RMF in a curve is defined by a unittangent 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. anR3MF curve) if there exists a rational RMF over the curve.




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 bothpiecewise polynomial (resp. rational) functions of t, i.e., if theconditions

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).




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, p : I → Rn are said to be parallel curves ifthere exists a rational function λ 6= 0 such that

p′(u) = λ(u)p′(u), , ∀u ∈ I

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




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 thecurves in [p] are RPH (resp. RDPH, R3MF).




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).




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]→ R3 be a PH space curve;

s(u) =∫ u0 ‖p

′(t)‖dt, the arc-length function;

q, an involute of p:

q(u) = p(u)− s(u)p′(u)

‖p′(u)‖




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.




Lemma

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




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.

Proposition

Given 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)‖ .




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′′‖

}




Relationship between planar RPH curves and SPH curvesRelationship 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 theMobius transformation

Σ : x→ 2x + z

‖x + z‖2− 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.





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(r21r′1 + 2r1r2r

′2 − r′1 − r′1r

22), 2(−2r2r1r

′1 − r22r

′2 + r′2 + r′2r

21),−4(r1r

′1 + r2r

′2))

(r21 + r22 + 1)2

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

s′21 + s′22 + s′23 = σ2 for σ rational. Then r = Σ(s) =(

s1s3+1 ,

s2s3+1 , 0

)⇒

r contained in z = 0.Differentiating r′ = − s′3

(s3+1)3 (s1, s2, 0) + 1s3+1 (s′1, s

′2, 0).

Substituting s1s′1 + s2s

′2 = −s3s′3 ⇒ ‖r′‖ = σ

s3+1 .





Theorem

1 Given a SPH curve r(u)/w(u) : I → R3 where v and w arepolynomial functions of the parameter then theintegrated-numerator curve p(u) =

∫r(u)du is DPH.

2 If a space curve p(u) is RDPH then the unit-hodograph curvep′(u)‖p′(u)‖ is SPH.





Proof.(1) r/w is spherical, ‖r/w‖ = 1 so r21 + r22 + r23 = w2. Derivating2(r1r

′1 + r2r

′2 + r3r

′3) = 2ww′. From PH curve def.

∥∥(r/w)′∥∥ = σ, σ

rational. This gives ‖r′w − rw′‖ = w2σ. Direct comput.‖r′‖2 = w2σ2 + w′2. p is DPH because ‖p′‖ = ‖r‖ = w and‖p′ × p′′‖2 = ‖r × r′‖2 = (σw2)2.(2) By hypothesis ‖p′‖2 = σ2 and ‖p′ × p′′‖2 = σ2

(‖p′′‖2 − σ′2

)= δ2,

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

(p′

‖p′‖

)′∥∥∥∥∥2

= ‖p′′‖2 − σ′2 =

(δ

σ

)2

Lemma**: Every curve p satisfies

∥∥∥∥( p′‖p′‖

)′∥∥∥∥ =‖p′×p′′‖‖p′‖ .





Theorem

1 Given a SPH curve r(u)/w(u) : I → R3 where v and w arepolynomial functions of the parameter then theintegrated-numerator curve p(u) =

∫r(u)du is DPH.

2 If a space curve p(u) is RDPH then the unit-hodograph curvep′(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′ × p′′‖2 = σ2ρ, where ‖p′‖ = σ andρ = ‖p′′‖2 − σ′2.

3 If p is RDPH then ‖p′ × p′′‖2 = σ2ω2, where ω2 = ρ.





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 R3MF.

2 If a rational space curve q is R3MF then we can find a space curvep(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.





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|σωσ

= |s|ω

Note that the previous property does not hold in general for R3MFcurves.

Lemma

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

Theorem

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




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 + 3a1t

2a0 + 3a20t− b21t3 − 3b1t2b0 − 3b20t+ 3c1))/(9 +

24a1t4a0b1b0 + 6a21t

5b1b0 + 6a1t5a0b

21 + 18a1t

3a0b20 + 18a1t

2a0c1 +18a20t

3b1b0 − 18b1t2b0c1 + 12a1b1t

3c2 + 18t2a0b1c2 + 18t2a1b0c2 +36a0b0tc2 + a41t

6 + 9a40t2 + b41t

6 + 9b40t2 + 9c21 + 9c22 + 3a21t

4b20 +2a21t

6b21 + 6a21t3c1 + 6a31t

5a0 + 15a21t4a20 + 18a1t

3a30 + 3a20t4b21 +

18a20t2b20 + 18a20tc1 + 6b31t

5b0 + 15b21t4b20 − 6b21t

3c1 + 18b1t3b30 − 18b20tc1)





s2 = (6(2a1b1t3 + 3t2a0b1 + 3t2a1b0 + 6a0b0t+ 3c2))/(9 +

24a1t4a0b1b0 + 6a21t

5b1b0 + 6a1t5a0b

21 + 18a1t

3a0b20 + 18a1t

2a0c1 +18a20t

3b1b0 − 18b1t2b0c1 + 12a1b1t

3c2 + 18t2a0b1c2 + 18t2a1b0c2 +36a0b0tc2 + a41t

6 + 9a40t2 + b41t

6 + 9b40t2 + 9c21 + 9c22 + 3a21t

4b20 +2a21t

6b21 + 6a21t3c1 + 6a31t

5a0 + 15a21t4a20 + 18a1t

3a30 + 3a20t4b21 +

18a20t2b20 + 18a20tc1 + 6b31t

5b0 + 15b21t4b20 − 6b21t

3c1 + 18b1t3b30 − 18b20tc1)

s3 = −(−9 + 24a1t4a0b1b0 + 6a21t

5b1b0 + 6a1t5a0b

21 + 18a1t

3a0b20 +

18a1t2a0c1 + 18a20t

3b1b0 − 18b1t2b0c1 + 12a1b1t

3c2 + 18t2a0b1c2 +18t2a1b0c2 + 36a0b0tc2 + a41t

6 + 9a40t2 + b41t

6 + 9b40t2 + 9c21 + 9c22 +

3a21t4b20 + 2a21t

6b21 + 6a21t3c1 + 6a31t

5a0 + 15a21t4a20 + 18a1t

3a30 +3a20t

4b21 + 18a20t2b20 + 18a20tc1 + 6b31t

5b0 + 15b21t4b20− 6b21t

3c1 + 18b1t3b30−

18b20tc1)/(9 + 24a1t4a0b1b0 + 6a21t

5b1b0 + 6a1t5a0b

21 + 18a1t

3a0b20 +

18a1t2a0c1 + 18a20t

3b1b0 − 18b1t2b0c1 + 12a1b1t

3c2 + 18t2a0b1c2 +18t2a1b0c2 + 36a0b0tc2 + a41t

6 + 9a40t2 + b41t

6 + 9b40t2 + 9c21 + 9c22 +

3a21t4b20 + 2a21t

6b21 + 6a21t3c1 + 6a31t

5a0 + 15a21t4a20 + 18a1t

3a30 + 3a20t4b21 +

18a20t2b20 + 18a20tc1 + 6b31t

5b0 + 15b21t4b20 − 6b21t

3c1 + 18b1t3b30 − 18b20tc1)





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 DPHcurve p. Its involute q is R3MF. We take now the minimum degreepolynomial curve q such that [q] = [q]. Once done we find an R2MFcurve of degree 9 q = (q1, q2, q3) where

q1 = (a61 + a41b21 − a12b41 − b61)t9 + (9a51a2 + 9a31a2b

21 − 9a21b

31b2 −

9b51b2)t8 + (36a41a22 − (36/7)a41b

22 + (72/7)a31a2b1b2 + (216/7)a21a

22b

21 −

(216/7)a21b21b

22 − (72/7)a1a2b

31b2 + (36/7)a22b

41 − 36b41b

22)t7 + (81a31a

32 −

27a31a2b22 +63a21a

22b1b2−45a21b1b

32 +45a1a

32b

21−63a1a2b

21b

22 +27a22b

31b2−

81b31b32)t6+(108a21a

42−(216/5)a21a

22b

22−(108/5)a21b

42+(648/5)a1a

32b1b2−

(648/5)a1a2b1b32 + (108/5)a42b

21 + (216/5)a22b

21b

22 − 108b21b

42)t5 +

(81a1a52− 81a1a2b

42 + 81a42b1b2− 81b1b

52)t4 + (27a62 + 27a42b

22− 27a22b

42−

27b62 − 27a21 + 27b21)t3 + (−81a1a2 + 81b1b2)t2 + (−81a22 + 81b22)t





q2 = (2a51b1 + 4a31b31 + 2a1b

51)t9 + ((9/2)a51b2 + (27/2)a41a2b1 +

18a31b21b2 + 18a21a2b

31 + (27/2)a1b

41b2 + (9/2)a2b

51)t8 + (36a41a2b2 +

36a31a22b1 + (180/7)a31b1b

22 + (648/7)a21a2b

21b2 + (180/7)a1a

22b

31 +

36a1b31b

22 + 36a2b

41b2)t7 + (117a31a

22b2 + 9a31b

32 + 45a21a

32b1 +

153a21a2b1b22 + 153a1a

22b

21b2 + 45a1b

21b

32 + 9a32b

31 + 117a2b

31b

22)t6 +

((972/5)a21a32b2 + (324/5)a21a2b

32 + (108/5)a1a

42b1 + (1512/5)a1a

22b1b

22 +

(108/5)a1b1b42 + (324/5)a32b

21b2 + (972/5)a2b

21b

32)t5 + (162a1a

42b2 +

162a1a22b

32 + 162a32b1b

22 + 162a2b1b

42)t4 − 162a2b2t+ (54a52b2 +

108a32b32 + 54a2b

52 − 54a1b1)t3 + (−81a1b2 − 81a2b1)t2

q3 = (9a41+18a21b21+9b41)t6+(54a31a0+54a21b1b0+54a1a0b

21+54b31b0)t5+

(135a21a20 + 27a21b

20 + 216a1a0b1b0 + 27a20b

21 + 135b21b

20)t4 + (162a1a

30 +

162a1a0b20 + 162a20b1b0 + 162b1b

30)t3 + (81a40 + 162a20b

20 + 81b40)t2




Construction of a R2MF quintic

We consider the R2MF quintic (introduced in Farouki, Rida T.; Gianelli,Carlotta; Manni, Carla; Sestini, Alessandra, 2009. Quintic Space Curveswith Rational Rotation-Minimizing Frames. Computer Aided GeometricDesign 26, 580–592) q =(−8 t3 − 24

5 t5 + 12 t4 − 4 t2

√2 + 8 t3

√2− 8 t4

√2 + 16

5 t5√

2,−2 t2√

2− 4 t3 + 4 t3√

2+

6 t4 − 6 t4√

2− 4 t5 + 165 t

5√

2,−10 t+ 20 t2 − 10 t2√

2− 28 t3 +

20 t3√

2 + 22 t4 − 16 t4√

2 −8 t5 + 245 t

5√

2)




Construction of a R2MF quintic

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

curve




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




Thank you!


Education

CVC Seminar