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The presentation talk I gave to the people at CVC group at UT, last year. http://cvcweb.ices.utexas.edu/cvcwp/
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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
PreliminariesTheoretical Results
Constructions and Examples
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
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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).
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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).
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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).
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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).
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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).
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes 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]→ 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)‖
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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)‖ .
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and ExamplesInvolutes and evolutes of space curves
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′′‖
}
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship between R3MF curves and RDPH curves
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 .
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship between R3MF curves and RDPH curves
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship between R3MF curves and RDPH curves
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′‖ .
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship between R3MF curves and RDPH curves
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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship 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 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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Relationship between planar RPH curves and SPH curvesRelationship between DPH curves and SPH curvesRelationship 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|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.
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
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)
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Construction of an R2MF curve of degree 9
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)
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Construction of an R2MF curve of degree 9
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
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Construction of an R2MF curve of degree 9
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
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
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)
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Construction of a R2MF quintic
Using our method we can obtain the previous curve from the planar PH
curve
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Work still in process.......... Any suggestions???
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
PreliminariesTheoretical Results
Constructions and Examples
Thank you!
CVC seminar, Wed 17 nov 2010 R3MF curves from PH ones
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