Ultraquadrics and its application to the reparametrization of - PowerPoint PPT Presentation

RICAM Computer algebra and polynomials. November 25-29, 2013. Ultraquadrics and its application to the reparametrization of rational complex surfaces C ARLOS V ILLARINO A CKNOWLEDGMENT : T HIS WORK HAS BEEN DONE UNDER THE RESEARCH PROJECT MTM2011-25816-C02-01 (S PANISH M INISTERIO DE E CONOM ´ IA Y C OMPETITIVIDAD ).

RICAM Computer algebra and polynomials. November 25-29, 2013. 1. Introduction – K -algebraic optimality problem (parametric version). – Goals. 2. Preliminaires – i -hypercircles and Weil parametric variety of a curve. – Real reparametrization of space curves. – i -ultraquadrics and Weil parametric variety of a surface. 3. Proper Real Reparametrization of Rational Ruled Surfaces. – Standard form and Theorem of Reparametrization. – Algorithm of reparametrization and examples. 4. Reparametrizing Swung Surfaces over the Reals. – Theorem of Reparametrization. – Algorithm of reparametrization and examples.

1. Introduction: K -Algebraic Optimality Problem (Parametric Version) • Rational parametric representations of algebraic varieties (in particular, of curves and surfaces) are a useful tool in many applied fields, such as CAGD. – J. R. Sendra, F. Winkler, and S. P´ erez-D´ ıaz. Rational algebraic curves: A com- puter algebra approach volume 22 of Algorithms and Computation in Mathematics Springer, Berlin, 2008. – J. Schicho. Rational parametrization of surfaces J. Symbolic Comput. , 26(1):1–29, 1998. • But a rational variety might be described through many different (although related) parametrizations: – P 1 ( s, t ) = ( t, t 2 , s ) – P 2 ( s, t ) = ( t 2 , t 4 , s ) i t, − t 2 , 1 � � – P 3 ( s, t ) = s � � 100 i t t − s , − 10000 t 2 – P 4 ( s, t ) = ( t − s ) 2 , s are parametrizations of the same cylindric surface F ( x, y, z ) = y − x 2 = 0

1. Introduction: K -Algebraic Optimality Problem (Parametric Version) • Given: – K ⊆ L ⊆ F where K is a computable field of characteristic zero (ground field), L = K ( α ) an algebraic extension of K , and F is the algebraic closure of K , – a unirational map ϕ : F m → F n where ϕ = ( ϕ 1 , . . . , ϕ n ) and ϕ i ( T 1 , . . . , T m ) = h i ( T 1 , . . . , T m ) g i ( T 1 , . . . , T m ) ∈ L ( T 1 , . . . , T m ) – V the Zariski closure of ϕ ( F m ) . • Decide: whether V can be parametrized over K . • Find: (in the affirmative case) a K -rational parametrization of V .

1. Introduction: K -Algebraic Optimality Problem (Parametric Version) • For curves the reparametrization problem can be approached by means of hypercircles – C. Andradas, T. Recio, and J. R. Sendra. Base field restriction techniques for para- metric curves. In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation , pages 17–22, 1999. – T. Recio and J. R. Sendra. Real reparametrizations of real curves. J. Symbolic Comput. , 23(2-3):241–254, 1997. – T. Recio, J. R. Sendra, L. F. Tabera, and C. Villarino. Generalizing circles over algebraic extensions. Mathematics of Computation , 79(270):1067–1089, 2010. – L. F. Tabera. Two Tools in Algebraic Geometry: Construction of Configurations in Tropical Geometry and Hypercircles for the Simplification of Parametric Curves. PhD thesis, Universidad de Cantabria, Universit´ e de Rennes I, 2007. – C. Villarino. Algoritmos de optimalidad algebraica y de cuasi-polinomialidad para curvas racionales. PhD thesis, Universidad de Alcal´ a, 2007. • For higher dimensional algebraic varieties, generalizing the ideas for curves, a theoretical frame is established (by means of ultraquadrics): – C. Andradas, T. Recio, J. R. Sendra, and L. F. Tabera. On the simplification of the coefficients of a parametrization. J. Symbolic Comput. , 44(2):192–210, 2009.

1. Introduction: K -Algebraic Optimality Problem (Parametric Version) √ • Let K ⊆ R be a computable field; (for instance, K may by Q , Q ( 2) , etc). • Given a rational parametrization over K ( i ) of a ruled surface S , say P ( T 1 , T 2 ) = ( ϕ 1 ( T 1 ) + T 2 ψ 1 ( T 1 ) , ϕ 2 ( T 1 ) + T 2 ψ 2 ( T 1 ) , ϕ 3 ( T 1 ) + T 2 ψ 3 ( T 1 )) where ϕ i ( T 1 ) , ψ i ( T 1 ) ∈ K ( i )( T 1 ) • Given a rational swung surface parametrized as P ( T 1 , T 2 ) = ( ϕ 1 ( T 1 ) ψ 1 ( T 2 ) , ϕ 1 ( T 1 ) ψ 2 ( T 2 ) , ϕ 2 ( T 1 )) where ϕ i ( T 1 ) , ψ i ( T 2 ) ∈ K ( i )( T i ) • Determine whether S can be reparametrized over K , and in the affirmative case, • Compute a parametrization of S with coefficients in K .

1. Introduction: K -Algebraic Optimality Problem (Parametric Version) • For rational ruled surfaces the reparametrization problem has been solved in – C. Andradas, T. Recio, J. R. Sendra, L. F. Tabera, and C. Villarino. Proper real reparametrization of rational ruled surfaces. In Computer Aided Geometric Design , 28(2), pages 102–113 , 2011. • For rational swung surfaces the reparametrization problem has been solved in – C. Andradas, T. Recio, J. R. Sendra, L. F. Tabera, and C. Villarino. Reparametrizing Swung Surfaces over the reals. Submitted, 2013.

2. Preliminaires: i -Hypercircles • Let u ( t ) be a unit of K ( i )( t ) ; i.e. u ( t ) = ( at + b ) / ( ct + d ) ∈ K ( i )( t ) such that ad − bc � = 0 . • We expand u ( t ) as u ( t ) = φ 0 ( t ) + i φ 1 ( t ) , where φ i ( t ) ∈ K ( t ) , for i = 0 , 1 . The i -hypercircle U generated by u ( t ) is the rational curve in F 2 parametrized by φ ( t ) = ( φ 0 ( t ) , φ 1 ( t )) . t + i = t − i 1 • Let u : Q ( i )( t ) → Q ( i )( t ) be the authomorphism u ( t ) = t 2 + 1 . • Then the i -hypercircle U defined by u is the rational curve parametrized by: t 1 � � φ ( t ) = t 2 + 1 , − . t 2 + 1 • The i -hypercircle is the real circle: x 2 + y 2 + y = 0 .

2. Preliminaires: i -Weil(descent) Parametric Variety of a Curve • Let Φ( t ) = ( ξ 1 ( t ) , . . . , ξ n ( t )) ∈ K ( i )( t ) n be a proper parametrization of a curve C . • We consider, then, the formal substitution Φ( t 0 + i t 1 ) , and we express ξ i ( t 0 + i t 1 ) as ξ i ( t 0 + i t 1 ) = ψ i 0 ( t 0 , t 1 ) + i ψ i 1 ( t 0 , t 1 ) , i = 1 , . . . , n, δ ( t 0 , t 1 ) where ψ ij , δ ∈ K [ t 0 , t 1 ] and gcd( ψ 10 , ψ 11 , . . . , ψ n 1 , δ ) = 1 . • Let Y be the algebraic variety in F 2 defined by the polynomials { ψ i 1 ( t 0 , t 1 ) } i =1 ,...,n (i.e. the imaginary parts of the numerators of ξ i ( t 0 + i t 1 ) ). • Let ∆ be the algebraic variety in F 2 defined by δ ( t 0 , t 1 ) . Then, the i -Weil (descent) parametric variety of C via Φ( t ) is defined as the Zariski closure of Y \ ∆ . We denote it by Weil 1D (Φ) .

2. Preliminaires: i -Weil(descent) Parametric Variety of a Curve The following theorems shows the role that Weil 1D (Φ) plays in the reparametrization prob- lem. In this context we need to recall that the K -definability of a curve (or surface) means that the ideal of the curve (or surface) can be generated by polynomials over K . Theorem 1 It holds that 1. The curve C is K -definable iff Weil 1D (Φ) contains a 1-dimensional component. 2. The curve C can be parametrized over K iff the 1-dimensional component of Weil 1D (Φ) is an i -hypercircle. In this case, if ( χ 1 ( t ) , χ 2 ( t )) is a proper K -parametrization of the i -hypercircle, then Φ( χ 1 ( t ) + i χ 2 ( t )) is a K -parametrization of C Theorem 2 Weil 1D (Φ) contains a component of dimension 1 if and only if gcd( ψ 11 , . . . , ψ n 1 ) � = 1 . If so, Weil 1D (Φ) is defined by the gcd( ψ 11 , . . . , ψ n 1 ) .

2. Preliminaires: Real Reparametrization of Space Curves Algorithm 1 of Real Reparametrization of Space Curves Input: A parametrization ξ = ( ξ 1 , . . . , ξ n ) of a spatial curve with coefficients in K ( i ) . Output: “ C is not a real curve”, else a linear fraction u ( t ) such that ξ ◦ u is real. 1. Find ξ ∗ ( t ) = ( ξ ∗ 1 , . . . , ξ ∗ n ) , a proper parametrization of C . i ( t 0 + i t 1 ) = ψ i 0 ( t 0 ,t 1 )+ i ψ i 1 ( t 0 ,t 1 ) 2. Write ξ ∗ , i = 1 , . . . , n, with ψ i 1 , δ ∈ K [ t 0 , t 1 ] . δ ( t 0 ,t 1 ) 3. Compute ψ ( t 0 , t 1 ) = gcd( ψ 11 , . . . , ψ n 1 ) . 4. If ψ = 1 , return: “ C is not a real curve” 5. If ψ is a linear polynomial (a) Compute a linear real (over K ) parametrization χ = ( χ 1 ( t ) , χ 2 ( t )) of the line defined by ψ . (b) Return t �→ χ 1 ( t ) + i χ 2 ( t )

2. Preliminaires: Real Reparametrization of Space Curves Algorithm 1 of Real Reparametrization of Space Curves 6. Check whether ψ is a real circle. 7. If ψ is not a real circle, return: “ C is not a real curve” 8. Compute a real (over a real field extension of K of degree at most 2) parametrization χ = ( χ 1 , χ 2 ) of the real circle ψ . 9. Return t �→ χ 1 ( t ) + i χ 2 ( t )

2. Preliminaires: Example of Real Reparametrization of Space Curves We consider the space curve given by � 3 t 2 + 1 + 2 i t 3 , − t 2 ( − 1 + 2 i t ) , − 3 t 2 + 1 − 2 t 4 + 5 i t 3 + i t � Φ( t ) = . 1 + 4 t 2 1 + 4 t 2 t (1 + 4 t 2 ) Weil 1D (Φ) contains the curve defined by t 2 0 + t 2 1 − t 1 , that is the circle centered at (0 , 1 / 2) and radius 1 / 2 . Moreover, this circle is parametrized as t 2 − 1 t 2 + 1 , 1 t t 2 + 1 + 1 � � . 2 2 Therefore, we get the real parametrization t 2 − 1 t 2 � t � 1 t 2 + 1 + 1 �� � 1 1 � Φ t 2 + 1 + i = t 2 + 1 , t 2 + 1 , − t ( t 2 + 1) 2 2