The presentation talk I gave to the people at CVC group at UT, last year. http://cvcweb.ices.utexas.edu/cvcwp/

1. Preliminaries
Theoretical 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. Ju¨ttler1 , Prof. Dr. F. Etayo2
e
and Prof. Dr. L. Gonz´lez-Vega2
a
1 Institut f¨r Angewandte Geometrie
u
Johannes Kepler University, Linz, Austria
2 Departamento de MATem´ticas, EStad´
a ıstica y COmputaci´n
o
University of Cantabria, Santander, Spain
3 This work has been partially supported by the spanish MICINN grant
MTM2008-04699-C03-03 and the project
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

2. Preliminaries
Theoretical Results
Constructions and Examples
Contents
1 Preliminaries
Involutes and evolutes of space curves
2 Theoretical Results
Relationship between planar RPH curves and SPH curves
Relationship between DPH curves and SPH curves
Relationship between R3 MF curves and RDPH curves
3 Constructions and Examples
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

3. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
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 Pythagorean
condition
(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 = σ 2 (u)ρ(u) where
ρ(u) = p (u) 2 − σ 2 (u)**.
**From Farouki, Rida T., Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable Springer, Berlin,
2008.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

4. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
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; J¨ttler, Bert; Zheng, Dayue; Liu, Yang, Computation of Rotation
u
Minimizing Frames, ACM Trans. Graph. 27,1, Article 2, 2008).
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

5. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
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.
Consequences:
Given v RM vector field over q, any unitary vector w
perpendicular 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; J¨ttler, Bert; Zheng, Dayue; Liu, Yang, Computation of Rotation
u
Minimizing Frames, ACM Trans. Graph. 27,1, Article 2, 2008).
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

6. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
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 (R2 MF, resp. R3 MF, curve)
A polynomial (resp. rational) space curve is said to be a curve with
a Rational Rotation Minimizing Frame (an R2 MF curve; resp. an
R3 MF curve) if there exists a rational RMF over the curve.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

7. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Definition (DPH, resp. RDPH, curve)
A polynomial (resp. rational) space curve p is said to be a
polynomial (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 = σ 2 (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).
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

8. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
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 if
ˆ
there exists a rational function λ = 0 such that
p (u) = λ(u)p (u), , ∀u ∈ I
ˆ
Equivalence relation → [p] the equivalence class generated by p.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

9. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Theorem
ˆ
Let p and p be rational parallel curves
1 ˆ
If p is RPH then p is also RPH.
2 ˆ
If p is RDPH then p is also RDPH.
3 If p is R3 MF then p is also R3 MF.
ˆ
Consequently If a curve p is RPH (resp. RDPH, R3 MF) then the
curves in [p] are RPH (resp. RDPH, R3 MF).
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

10. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Relationships illustrated
Theorem
ˆ
Let p and p be rational parallel curves
1 ˆ
If p is RPH then p is also RPH.
2 ˆ
If p is RDPH then p is also RDPH.
3 If p is R3 MF then p is also R3 MF.
ˆ
Consequently If a curve p is RPH (resp. RDPH, R3 MF) then the curves in [p] are RPH (resp. RDPH, R3 MF).
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

11. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Given p and q curves in R3 ,
p is an evolute of q and q is an involute of p if the tangent lines
to p are normal to q.
Let
p : I = [a, b] → R3 be a PH space curve;
u
s(u) = 0 p (t) dt, the arc-length function;
q, an involute of p:
p (u)
q(u) = p(u) − s(u)
p (u)
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

12. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Lemma
p (u)
The vector field v(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 q
iff the ruled surface q + λv
developable; and
2 q + λv is the tangent
surface of p.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

13. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Lemma
**Given a PH space curve p, we consider q an involute of p. The
frame defined by
q (u)
, v(u), w(u)
q (u)
p (u)
is an ( RMF) over q, where v(u) = p (u) and
q (u)
w(u) = q (u) × v(u).
Proposition
p (u)
If p is a spatial PH curve then v(u) = p (u) , the involute
q (u)
q(u) = p(u) − s(u)v(u) and w(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.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

14. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Lemma
p p ×p
Every curve p satisfies p = p .
Note:
p
For a PH curve it is reduced to p =ρ
p p
p is piecewise rational but p is not.
Proposition
p (u)×p (u)
Given a curve p, the vector field b(u) = p (u)×p (u) is a RM vector
p (u)
field with respect to the involute q(u) = p(u) − s(u) p (u) .
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

15. Preliminaries
Theoretical Results Involutes and evolutes of space curves
Constructions and Examples
Lemma
p
Consider a curve p and its involute q = p − s p . Then
p (u) × p (u) q (u) × p (u)
=
p (u) × p (u) q (u) × p (u)
RMF over the involute q:
q p p ×p
, ,b =
q p p ×p
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

16. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
Theorem
The image s of a rational planar PH curve r = (r1 , r2 , 0) by the
M¨bius transformation
o
x+z
Σ:x→2 2
−z
x+z
where z = (0, 0, 1), is a SPH curve and vice versa.
Note that Σ ◦ Σ =Id. Then by direct computations the necessary
and the sufficient conditions hold.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

17. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
Proof.
2
Necessary condition. r is PH, r = σ 2 , σ rational. Then,
1 2 2
s = Σ(r) = 2 2 (2r1 , 2r2 , 1 − r1 − r2 )
r1 + r2 + 1
By direct computation s = 1. Since
2 2 2 2
(−2(r1 r1 + 2r1 r2 r2 − r1 − r1 r2 ), 2(−2r2 r1 r1 − r2 r2 + r2 + r2 r1 ), −4(r1 r1 + r2 r2 ))
s =
2 2
(r1 + r2 + 1)2
it holds s = 2 r / r + z 2 = 2σ( r + z )−2 .
Sufficient condition. let s = (s1 , s2 , s3 ) such that s2 + s2 + s2 = 1 and
1 2 3
s1 s2
s12 + s22 + s32 = σ 2 for σ rational. Then r = Σ(s) = s3 +1 , s3 +1 , 0 ⇒
r contained in z = 0.
s3
Differentiating r = − (s3 +1)3 (s1 , s2 , 0) + s31 (s1 , s2 , 0).
+1
Substituting s1 s1 + s2 s2 = −s3 s3 ⇒ r = s3σ . +1
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

18. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
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.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

19. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
Proof.
(1) r/w is spherical, r/w = 1 so r1 + r2 + r3 = w2 . Derivating
2 2 2
2(r1 r1 + r2 r2 + r3 r3 ) = 2ww . From PH curve def. (r/w) = σ, σ
rational. This gives r w − rw = w2 σ. Direct comput.
r 2 = w2 σ 2 + w 2 . p is DPH because p = r = w and
p × p 2 = r × r 2 = (σw2 )2 .
(2) By hypothesis p 2 = σ 2 and p × p 2 = σ 2 p 2 − σ 2 = δ 2 ,
σ and δ rational. Since Lemma** holds for rational space curves we have
that
2 2
p δ
= p 2−σ2 =
p σ
p p ×p
Lemma**: Every curve p satisfies = .
p p
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

20. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
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.
Corollary
p
1 If p is a DPH curve then the unit-hodograph p is an SPH curve
and additionally (p / p ) = p ×p / p , polynomial.
2
2 If p is an RPH curve then p × p = σ 2 ρ, where p = σ and
ρ = p 2 − σ 2.
2
3 If p is RDPH then p × p = σ 2 ω 2 , where ω 2 = ρ.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

21. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
Theorem
1 Let p be a DPH curve and consider an involute q. The vectors
q (u) p (u) × p (u)
and b(u) =
q (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 R3 MF 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, Geometr´ Diferencial de Curvas y Superficies,
ıa
Alianza Editorial, S. A., Madrid,1990.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

22. Preliminaries Relationship between planar RPH curves and SPH curves
Theoretical Results Relationship between DPH curves and SPH curves
Constructions and Examples Relationship between R3 MF curves and RDPH curves
The R3 MF curve from pevious Theorem (1) q has piecewise polynomial
p
arc-length function: q = |s| p and then
p ×p σω
q = |s| = |s| = |s|ω
p σ
Note that the previous property does not hold in general for R3 MF
curves.
Lemma
ˆ ˆ
If two curves p and p are parallel, then the corresponding involutes q, q
are also parallel.
Theorem
Every R3 MF curve is parallel to the involute of a DPH curve.
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

23. Preliminaries
Theoretical Results
Constructions and Examples
Construction of an R2 MF curve of degree 9
Degree 9 R2 MF curve from a polynomial planar PH curve. The
derivative of the PH curve is defined from two linear univariate
polynomials a(t) = a1 t + a0 and b(t) = b1 t + 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 the
previous Theorem getting s(t) = (s1 , s2 , s3 ), where
s1 = (6(a2 t3 + 3a1 t2 a0 + 3a2 t − b2 t3 − 3b1 t2 b0 − 3b2 t + 3c1 ))/(9 +
1 0 1 0
24a1 t4 a0 b1 b0 + 6a2 t5 b1 b0 + 6a1 t5 a0 b2 + 18a1 t3 a0 b2 + 18a1 t2 a0 c1 +
1 1 0
18a2 t3 b1 b0 − 18b1 t2 b0 c1 + 12a1 b1 t3 c2 + 18t2 a0 b1 c2 + 18t2 a1 b0 c2 +
0
36a0 b0 tc2 + a4 t6 + 9a4 t2 + b4 t6 + 9b4 t2 + 9c2 + 9c2 + 3a2 t4 b2 +
1 0 1 0 1 2 1 0
2a2 t6 b2 + 6a2 t3 c1 + 6a3 t5 a0 + 15a2 t4 a2 + 18a1 t3 a3 + 3a2 t4 b2 +
1 1 1 1 1 0 0 0 1
18a2 t2 b2 + 18a2 tc1 + 6b3 t5 b0 + 15b2 t4 b2 − 6b2 t3 c1 + 18b1 t3 b3 − 18b2 tc1 )
0 0 0 1 1 0 1 0 0
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

24. Preliminaries
Theoretical Results
Constructions and Examples
Construction of an R2 MF curve of degree 9
s2 = (6(2a1 b1 t3 + 3t2 a0 b1 + 3t2 a1 b0 + 6a0 b0 t + 3c2 ))/(9 +
24a1 t4 a0 b1 b0 + 6a2 t5 b1 b0 + 6a1 t5 a0 b2 + 18a1 t3 a0 b2 + 18a1 t2 a0 c1 +
1 1 0
18a2 t3 b1 b0 − 18b1 t2 b0 c1 + 12a1 b1 t3 c2 + 18t2 a0 b1 c2 + 18t2 a1 b0 c2 +
0
36a0 b0 tc2 + a4 t6 + 9a4 t2 + b4 t6 + 9b4 t2 + 9c2 + 9c2 + 3a2 t4 b2 +
1 0 1 0 1 2 1 0
2a2 t6 b2 + 6a2 t3 c1 + 6a3 t5 a0 + 15a2 t4 a2 + 18a1 t3 a3 + 3a2 t4 b2 +
1 1 1 1 1 0 0 0 1
18a2 t2 b2 + 18a2 tc1 + 6b3 t5 b0 + 15b2 t4 b2 − 6b2 t3 c1 + 18b1 t3 b3 − 18b2 tc1 )
0 0 0 1 1 0 1 0 0
s3 = −(−9 + 24a1 t4 a0 b1 b0 + 6a2 t5 b1 b0 + 6a1 t5 a0 b2 + 18a1 t3 a0 b2 +
1 1 0
18a1 t2 a0 c1 + 18a2 t3 b1 b0 − 18b1 t2 b0 c1 + 12a1 b1 t3 c2 + 18t2 a0 b1 c2 +
0
18t2 a1 b0 c2 + 36a0 b0 tc2 + a4 t6 + 9a4 t2 + b4 t6 + 9b4 t2 + 9c2 + 9c2 +
1 0 1 0 1 2
3a2 t4 b2 + 2a2 t6 b2 + 6a2 t3 c1 + 6a3 t5 a0 + 15a2 t4 a2 + 18a1 t3 a3 +
1 0 1 1 1 1 1 0 0
3a2 t4 b2 + 18a2 t2 b2 + 18a2 tc1 + 6b3 t5 b0 + 15b2 t4 b2 − 6b2 t3 c1 + 18b1 t3 b3 −
0 1 0 0 0 1 1 0 1 0
18b2 tc1 )/(9 + 24a1 t4 a0 b1 b0 + 6a2 t5 b1 b0 + 6a1 t5 a0 b2 + 18a1 t3 a0 b2 +
0 1 1 0
18a1 t2 a0 c1 + 18a2 t3 b1 b0 − 18b1 t2 b0 c1 + 12a1 b1 t3 c2 + 18t2 a0 b1 c2 +
0
18t2 a1 b0 c2 + 36a0 b0 tc2 + a4 t6 + 9a4 t2 + b4 t6 + 9b4 t2 + 9c2 + 9c2 +
1 0 1 0 1 2
3a2 t4 b2 + 2a2 t6 b2 + 6a2 t3 c1 + 6a3 t5 a0 + 15a2 t4 a2 + 18a1 t3 a3 + 3a2 t4 b2 +
1 0 1 1 1 1 1 0 0 0 1
18a2 t2 b2 + 18a2 tc1 + 6b3 t5 b0 + 15b2 t4 b2 − 6b2 t3 c1 + 18b1 t3 b3 − 18b2 tc1 )
0 0 0 1 1 0 1 0 0
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

25. Preliminaries
Theoretical Results
Constructions and Examples
Construction of an R2 MF curve of degree 9
s depends on the parameters that define the initial polynomials a and b
and on the integration constants obtained when we integrate r : c1 , c2 ,
c3 . Integrating the numerator of the spherical curve we obtain a DPH
curve p. Its involute q is R3 MF. We take now the minimum degree
ˆ ˆ
polynomial curve q such that [q] = [q]. Once done we find an R2 MF
ˆ
curve of degree 9 q = (q1 , q2 , q3 ) where
q1 = (a6 + a4 b2 − a12 b4 − b6 )t9 + (9a5 a2 + 9a3 a2 b2 − 9a2 b3 b2 −
1 1 1 1 1 1 1 1 1 1
9b5 b2 )t8 + (36a4 a2 − (36/7)a4 b2 + (72/7)a3 a2 b1 b2 + (216/7)a2 a2 b2 −
1 1 2 1 2 1 1 2 1
(216/7)a2 b2 b2 − (72/7)a1 a2 b3 b2 + (36/7)a2 b4 − 36b4 b2 )t7 + (81a3 a3 −
1 1 2 1 2 1 1 2 1 2
27a3 a2 b2 + 63a2 a2 b1 b2 − 45a2 b1 b3 + 45a1 a3 b2 − 63a1 a2 b2 b2 + 27a2 b3 b2 −
1 2 1 2 1 2 2 1 1 2 2 1
81b3 b3 )t6 +(108a2 a4 −(216/5)a2 a2 b2 −(108/5)a2 b4 +(648/5)a1 a3 b1 b2 −
1 2 1 2 1 2 2 1 2 2
(648/5)a1 a2 b1 b3 + (108/5)a4 b2 + (216/5)a2 b2 b2 − 108b2 b4 )t5 +
2 2 1 2 1 2 1 2
(81a1 a5 − 81a1 a2 b4 + 81a4 b1 b2 − 81b1 b5 )t4 + (27a6 + 27a4 b2 − 27a2 b4 −
2 2 2 2 2 2 2 2 2
27b6 − 27a2 + 27b2 )t3 + (−81a1 a2 + 81b1 b2 )t2 + (−81a2 + 81b2 )t
2 1 1 2 2
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

26. Preliminaries
Theoretical Results
Constructions and Examples
Construction of an R2 MF curve of degree 9
q2 = (2a5 b1 + 4a3 b3 + 2a1 b5 )t9 + ((9/2)a5 b2 + (27/2)a4 a2 b1 +
1 1 1 1 1 1
18a3 b2 b2 + 18a2 a2 b3 + (27/2)a1 b4 b2 + (9/2)a2 b5 )t8 + (36a4 a2 b2 +
1 1 1 1 1 1 1
36a3 a2 b1 + (180/7)a3 b1 b2 + (648/7)a2 a2 b2 b2 + (180/7)a1 a2 b3 +
1 2 1 2 1 1 2 1
36a1 b3 b2 + 36a2 b4 b2 )t7 + (117a3 a2 b2 + 9a3 b3 + 45a2 a3 b1 +
1 2 1 1 2 1 2 1 2
153a2 a2 b1 b2 + 153a1 a2 b2 b2 + 45a1 b2 b3 + 9a3 b3 + 117a2 b3 b2 )t6 +
1 2 2 1 1 2 2 1 1 2
((972/5)a2 a3 b2 + (324/5)a2 a2 b3 + (108/5)a1 a4 b1 + (1512/5)a1 a2 b1 b2 +
1 2 1 2 2 2 2
(108/5)a1 b1 b4 + (324/5)a3 b2 b2 + (972/5)a2 b2 b3 )t5 + (162a1 a4 b2 +
2 2 1 1 2 2
162a1 a2 b3 + 162a3 b1 b2 + 162a2 b1 b4 )t4 − 162a2 b2 t + (54a5 b2 +
2 2 2 2 2 2
108a3 b3 + 54a2 b5 − 54a1 b1 )t3 + (−81a1 b2 − 81a2 b1 )t2
2 2 2
q3 = (9a4 +18a2 b2 +9b4 )t6 +(54a3 a0 +54a2 b1 b0 +54a1 a0 b2 +54b3 b0 )t5 +
1 1 1 1 1 1 1 1
(135a2 a2 + 27a2 b2 + 216a1 a0 b1 b0 + 27a2 b2 + 135b2 b2 )t4 + (162a1 a3 +
1 0 1 0 0 1 1 0 0
162a1 a0 b2 + 162a2 b1 b0 + 162b1 b3 )t3 + (81a4 + 162a2 b2 + 81b4 )t2
0 0 0 0 0 0 0
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

27. Preliminaries
Theoretical Results
Constructions and Examples
Construction of a R2 MF quintic
We consider the R2 MF quintic (introduced in Farouki, Rida T.; Gianelli,
Carlotta; Manni, Carla; Sestini, Alessandra, 2009. Quintic Space Curves
with Rational Rotation-Minimizing Frames. Computer Aided Geometric
Design 26, 580–592) q = √ √ √ √ √
−8 t3 − 24 t5 + 12 t4 − 4 t2 2 + 8 t3 2 − 8 t4 2 + 16 t5 2, −2 t2 2 − 4 t3 +
√
5 √ √ 5
6 t4 − 6 t4 2 − 4 t5 + 16 t5 2, −10 t + 20 t2 − 10 t2 2 − 28 t3 +
√ √5 √
20 t3 2 + 22 t4 − 16 t4 2 −8 t5 + 24 t5 2
5
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

28. Preliminaries
Theoretical Results
Constructions and Examples
Construction of a R2 MF quintic
Using our method we can obtain the previous curve from the planar PH
curve
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

29. Preliminaries
Theoretical Results
Constructions and Examples
Work still in process.......... Any suggestions???
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones

30. Preliminaries
Theoretical Results
Constructions and Examples
Thank you!
CVC seminar, Wed 17 nov 2010 R3 MF curves from PH ones