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RICAM Computer algebra and polynomials. November 25-29, 2013.

Ultraquadrics and its application to the reparametrization of rational complex surfaces

CARLOS VILLARINO ACKNOWLEDGMENT: THIS WORK HAS BEEN DONE UNDER THE RESEARCH PROJECT MTM2011-25816-C02-01 (SPANISH MINISTERIO DE ECONOM´

IA Y COMPETITIVIDAD).

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: – P1(s, t) = (t, t2, s) – P2(s, t) = (t2, t4, s) – P3(s, t) =

• it, −t2, 1

s

• – P4(s, t) =
• 100it

t−s , −10000t2 (t−s)2 , s

• are parametrizations of the same cylindric surface F (x, y, z) = y − x2 = 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 : Fm → Fn where ϕ = (ϕ1, . . . , ϕn) and ϕi(T1, . . . , Tm) = hi(T1, . . . , Tm) gi(T1, . . . , Tm) ∈ L(T1, . . . , Tm) – V the Zariski closure of ϕ(Fm).

• 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(T1, T2) = (ϕ1(T1) + T2ψ1(T1), ϕ2(T1) + T2ψ2(T1), ϕ3(T1) + T2ψ3(T1)) where ϕi(T1), ψi(T1) ∈ K(i)(T1)

• Given a rational swung surface parametrized as

P(T1, T2) = (ϕ1(T1)ψ1(T2), ϕ1(T1)ψ2(T2), ϕ2(T1)) where ϕi(T1), ψi(T2) ∈ K(i)(Ti)

• 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) = φ0(t) + iφ1(t), where φi(t) ∈ K(t), for i = 0, 1. The i-hypercircle U generated by u(t) is the rational curve in F2 parametrized by φ(t) = (φ0(t), φ1(t)).

• Let u : Q(i)(t) → Q(i)(t) be the authomorphism u(t) =

1 t + i = t − i t2 + 1.

• Then the i-hypercircle U defined by u is the rational curve parametrized by:

φ(t) =

• t

t2 + 1, − 1 t2 + 1

• .
• The i-hypercircle is the real circle: x2 + y2 + y = 0.

• 2. Preliminaires: i-Weil(descent) Parametric Variety of a Curve
• Let Φ(t) = (ξ1(t), . . . , ξn(t)) ∈ K(i)(t)n be a proper parametrization of a curve C.
• We consider, then, the formal substitution Φ(t0 + it1), and we express ξi(t0 + it1) as

ξi(t0 + it1) = ψi0(t0, t1) + iψi1(t0, t1) δ(t0, t1) , i = 1, . . . , n, where ψij, δ ∈ K[t0, t1] and gcd(ψ10, ψ11, . . . , ψn1, δ) = 1.

• Let Y be the algebraic variety in F2 defined by the polynomials {ψi1(t0, t1)}i=1,...,n (i.e.

the imaginary parts of the numerators of ξi(t0 + it1)).

• Let ∆ be the algebraic variety in F2 defined by δ(t0, t1).

Then, the i-Weil (descent) parametric variety of C via Φ(t) is defined as the Zariski closure

• f Y \ ∆. We denote it by Weil1D(Φ).

• 2. Preliminaires: i-Weil(descent) Parametric Variety of a Curve

The following theorems shows the role that Weil1D(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 Weil1D(Φ) contains a 1-dimensional component.
• 2. The curve C can be parametrized over K iff the 1-dimensional component of Weil1D(Φ)

is an i-hypercircle. In this case, if (χ1(t), χ2(t)) is a proper K-parametrization of the i-hypercircle, then Φ(χ1(t) + iχ2(t)) is a K-parametrization of C Theorem 2 Weil1D(Φ) contains a component of dimension 1 if and only if gcd(ψ11, . . . , ψn1) = 1. If so, Weil1D(Φ) is defined by the gcd(ψ11, . . . , ψn1).

• 2. Preliminaires: Real Reparametrization of Space Curves

Algorithm 1 of Real Reparametrization of Space Curves Input: A parametrization ξ = (ξ1, . . . , ξn) 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.

• 2. Write ξ∗

i (t0 + it1) = ψi0(t0,t1)+iψi1(t0,t1) δ(t0,t1)

, i = 1, . . . , n, with ψi1, δ ∈ K[t0, t1].

• 3. Compute ψ(t0, t1) = gcd(ψ11, . . . , ψn1).
• 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 = (χ1(t), χ2(t)) of the line defined by ψ. (b) Return t → χ1(t) + iχ2(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 = (χ1, χ2) of the real circle ψ.

• 9. Return t → χ1(t) + iχ2(t)

• 2. Preliminaires: Example of Real Reparametrization of Space Curves

We consider the space curve given by Φ(t) = 3 t2 + 1 + 2 it3 1 + 4 t2 , −t2 (−1 + 2 it) 1 + 4 t2 , −3 t2 + 1 − 2 t4 + 5 it3 + it t (1 + 4 t2)

• .

Weil1D(Φ) contains the curve defined by t2

0 + t2 1 − t1, that is the circle centered at (0, 1/2)

and radius 1/2. Moreover, this circle is parametrized as

• t

t2 + 1, 1 2 t2 − 1 t2 + 1 + 1 2

• .

Therefore, we get the real parametrization Φ

• t

t2 + 1 + i 1 2 t2 − 1 t2 + 1 + 1 2

• =
• 1

t2 + 1, t2 t2 + 1, − 1 t (t2 + 1)

• 2. Introduction: i-Ultracuadrics

Let θ be an automorphism defined over K(i), θ : F2 → F2: (t, s) → θ(t, s) = (θ1(t, s), θ2(t, s)). Each θi ∈ K(i)(t, s) and can be written uniquely as θi(t, s) = θi0(t, s) + iθi1(t, s) where θi0, θi1 ∈ K(t, s). Then, the i-ultraquadric generated by θ is the rational variety in F4 parametrized by (θ10(t, s), θ11(t, s), θ20(t, s), θ21(t, s)).

• Let θ : C(t, s) → C(t, s) be the authomorphism θ(t, s) =
• s

s + it, t − i s + it

• .
• θ(t, s) =

s2 − ist s2 + t2 , st − t − i(s + t2) s2 + t2

• .
• Then the i-ultraquadric U defined by θ is the rational surface parametrized by:

¯ θ(t, s) =

• s2

s2 + t2, − st s2 + t2, st − t s2 + t2, − s + t2 s2 + t2

• .

• 2. Preliminaires: i-Weil(descent) Parametric Variety of a Surface
• Let Φ(T1, T2) = (ξ1(T1, T2), ξ2(T1, T2), ξ3(T1, T2)) ∈ K(i)(T1, T2)3 be a proper pa-

rametrization of a surface S in F3.

• We introduce new variables Tij, 1 ≤ i ≤ 2, 0 ≤ j ≤ 1, and we consider the formal

substitution Φ(T10 + iT11, T20 + iT21). We express ξi(T10 + iT11, T20 + iT21) as ξi(T10 + iT11, T20 + iT21) = ψi0(T10, T11, T20, T21) + iψi1(T10, T11, T20, T21) δ(T10, T11, T20, T21) for i = 1, 2, 3, where ψij, δ ∈ K[T10, T11, T20, T21] and gcd(ψ10, . . . , ψ31, δ) = 1.

• Let Y be the algebraic variety in F4 defined by the polynomials {ψi1(T10, . . . , T21)}i=1,2,3

(i.e. the imaginary parts of the numerators of ξi(T10 + iT11, T20 + iT21)).

• Let ∆ be the algebraic variety in F4 defined by δ(T10, . . . , T21).

Then, the i-Weil (descent) parametric variety of S via Φ(T1, T2) is defined as the Zariski closure of Y \ ∆. We denote it by Weil2D(Φ).

• 2. Preliminaires: i-Weil(descent) Parametric Variety of a Surface

The following theorem establishes the main properties of Weil2D(Φ). Theorem 3 It holds that

• 1. S is defined over K if and only if Weil2D(Φ) contains a component defined over K that is

K-birational to S.

• 2. S is parametrizable over K if and only if Weil2D(Φ) contains an i-ultraquadric that is

K-birational to S. If this ultraquadric is properly parametrized by (θ10(t, s), θ11(t, s), θ20(t, s), θ21(t, s)) then the authomorphism θ(t, s) = (θ10(t, s) + iθ11(t, s), θ20(t, s) + iθ21(t, s)) verifies Φ(θ(t, s)) ∈ K(t, s)3.

• 3. Reparametrization of ruled surfaces: Standard Form
• Let S be a rational ruled surface in C3, and let

P(T1, T2) = (ϕ1(T1) + T2ψ1(T1), ϕ2(T1) + T2ψ2(T1)ϕ3(T1) + T2ψ3(T1)) be a proper parametrization of S where P(T1, T2) ∈ K(i)(T1, T2)3.

• Assuming that ψ3 = 0, S always admits a proper parametrization of the form

(φ1(T1) + T2χ1(T1), φ2(T1) + T2χ2(T1), T2) ∈ K(i)(T1, T2)3.

• Such a parametrization can be achieved by performing the K(i)-birational transforma-

tion (T1, T2) →

• T1, T2 − ϕ3(T1)

ψ3(T1)

• to P(T1, T2).

• 3. Reparametrization of ruled surfaces: Standard Form
• The parametrization

Q(T1, T2) = P

• T1, T2 − ϕ3(T1)

ψ3(T1)

• is called a standard parametrization of S.
• Moreover, if

Q(T1, T2) = (φ1(T1) + T2χ1(T1), φ2(T1) + T2χ2(T1), T2), the curve in C4 given by the parametrization (φ1(T1), χ1(T1), φ2(T1), χ2(T1)) is the reparametrizing curve associated to Q(T1, T2), denoted by CQ.

• The parametrization (φ1(T1), χ1(T1), φ2(T1), χ2(T1)) of CQ is proper.

• 3. Reparametrization of ruled surfaces: Theorem of Reparametrization

Theorem of Reparametrization of Ruled Surfaces

• Let Q(T1, T2) = (φ1(T1) + T2χ1(T1), φ2(T1) + T2χ2(T1), T2) ∈ K(i)(T1, T2)3 be a

proper parametrization in standard form of S

• and let CQ = (φ1(T1), χ1(T1), φ2(T1), χ2(T1)) be the parametrization of the associ-

ated curve. S can be properly parametrized over a finite real algebraic extension L of K if and only if one

• f the following conditions hold:

(1) CQ is L-parametrizable. (2) The rational functions Im(φi(T1+iT2))

Im(χi(T1+iT2)), i = 1, 2, are well-defined, equal, non-constant, and

the transformation of C2 (T1, T2) →

• T1 + iT2, −Im(φi(T1 + iT2))

Im(χi(T1 + iT2))

• ,

is birational.

• 3. Reparametrization of ruled surfaces: Theorem of Reparametrization

Moreover, in the affirmative case, (i) If (1) holds and T1 → f(T1) reparametrizes CQ over L, then Q(f(T1), T2) is a real proper parametrization of S. (ii) If (2) holds, Q

• T1 + iT2, −Im(φ1(T1 + iT2))

Im(χ1(T1 + iT2))

• is a proper reparametrization of S over K.

• 3. Reparametrization of ruled surfaces: Algorithm of Reparametrization

Algorithm of Reparametrization of Ruled Surfaces Input: A proper parametrization P(T1, T2) of a ruled surface S. Output: “S is not properly parametrizable over the reals” or a change of variables g such that P ◦ g is a real parametrization with coefficients over a extension of degree at most two

• ver K.
• 1. Let i ∈ {1, 2, 3} such that ψi = 0. Permute i and 3 so that ψ3 = 0.
• 2. Q(T1, T2) := P(T1, T2−ϕ3(T1)

ψ3(T1) )

• 3. Write Q(T1, T2) = (φ1(T1) + T2χ1(T1), φ2(T1) + T2χ2(T1), T2)
• 4. Apply Algorithm 1 to Φ(T1) := (φ1(T1), χ1(T1), φ2(T1), χ2(T1))
• 5. If the output in Step 4 is f(T1), then return g : (T1, T2) → (f(T1), T2)

• 3. Reparametrization of ruled surfaces: Algorithm of Reparametrization

Algorithm of Reparametrization of Ruled Surfaces

• 6. h1 := Im(χ1(T1 + iT2)), h2 := Im(χ2(T1 + iT2))
• 7. If h1 = 0 or h2 = 0, then return “S is not properly parametrizable over the reals”.
• 8. A := −Im(φ1(T1+iT2))

Im(χ1(T1+iT2)), B := −Im(φ2(T1+iT2)) Im(χ2(T1+iT2))

• 9. If A = B or A is constant then return “S is not properly parametrizable over the reals”
• 10. g := (T1, T2) → (T1 + iT2, A)
• 11. If g is birational then return g. Else return “S is not properly parametrizable over the

reals”.

• 3. Reparametrization of ruled surfaces: Example 1

We consider the ruled surface S given by the proper parametrization over Q(i) P(T1, T2) =

• T1 + i

iT1 + 1 +

• T 2

1 + 1

• T2

T1 , iT1 + 1 T1 + i + T1T2 T 2

1 + 1, (T1 + i)2

(iT1 + 1)2 +

• T 2

1 + 1

2 T2 T 2

1

• .

The associated standard parametrization is Q(T1, T2) = iT 4

1 + T1 − T 3 1 + i − T2T 3 1 + 2iT2T 2 1 + T1T2

(T 2

1 + 1) (iT1 + 1)2

, iT 9

1 + 3T 8 1 + (T2 + 9)T 6 1 − i(T2 + 3)T 5 1 + (T2 + 3)T 4 1 − i(T2 + 9)T 3 1 − 3iT1 − 1

− (T1 + i) (T 2

1 + 1)3 (iT1 + 1)2

, T2

• .

• 3. Reparametrization of ruled surfaces: Example 1 (cont.)

The curve CQ is parametrized as Φ(T1) = iT 4

1 + T1 − T 3 1 + i

(T 2

1 + 1) (iT1 + 1)2, −T 3 1 + 2 iT 2 1 + T1

(T 2

1 + 1) (iT1 + 1)2,

−3T 8

1 + 9T 6 1 + 3T 4 1 − 1 + i(T 9 1 − 3T 5 1 − 9T 3 1 − 3T1)

(T1 + i) (T 2

1 + 1)3 (iT1 + 1)2

, − −iT 5

1 − iT 3 1 + T 4 1 + T 6 1

(T1 + i) (T 2

1 + 1)3 (iT1 + 1)2

• .

In addition, Weil1D(Φ) contains the real circle T 2

10 + T 2 11 = 1. Therefore,

Q

• 2T1

T 2

1 + 1 + iT 2 1 − 1

T 2

1 + 1, T2

• is a proper parametrization of S. Indeed, it is

T 4

1 − 3 T 2 1 − T2T 2 1 − T2

−4T2 , T 8

1 − (T2 − 3)T 6 1 − 3(T2 − 1)T 4 1 − 3(T2 + 21)T 2 1 − T2

−64T 3

1

, T2

• .

• 3. Reparametrization of ruled surfaces: Example 2

We consider the ruled surface (the plane) S given by the standard proper parametrization Q(T1, T2) = (1 + iT1 + 2iT2, iT1 + (1 + 2i)T2, T2). Observe that CQ = (φ1(T1), χ1(T1), φ2(T1), χ2(T1)) = (1 + iT1, 2i, iT1, 1 + 2i) is not real. However the rational functions Im(φi(T1+iT2))

Im(χi(T1+iT2)), i = 1, 2, in Theorem of reparametriza-

tion (2) are both equal to T1 2 . Moreover, the map (T1, T2) →

• T1 + iT2, −T1

2

• is birational. Therefore,

Q

• T1 + iT2, −T1

2

• =
• 1 − T2, T2 − T1

2 , −T1 2

• parametrizes S.

• 3. Reparametrization of ruled surfaces: Example 3

We consider the ruled surface S given by the standard proper parametrization Q(T1, T2) = (T 2

1 + iT1T2, iT1 + iT 2 1 T2, T2).

The reparametrizing curve C is given by CQ = (φ1(T1), χ1(T1), φ2(T1), χ2(T1)) = (T 2

1 , iT1, iT1, iT 2 1 )

and Weil1D(Φ) does not contain 1-dimensional components. Therefore, CQ is not real. On the other hand, the rational functions Im(φi(T1+iT2))

Im(χi(T1+iT2)), i = 1, 2, in Theorem of reparametriza-

tion (2), although well-defined and non-constant, they are different. Therefore, S cannot be parametrized properly over a real finite field extension of Q.

• 4. Reparametrization of Swung surfaces: Swung Surfaces
• A rational Swung surface is a surface S parametrized as

P(t, s) = (φ1(t)ψ1(s), φ1(t)ψ2(s), φ2(t))

• A rational Revolution surface is a surface S parametrized as

P(t, s) = (φ1(t)s2 − 1 s2 + 1, φ1(t) 2s s2 + 1, φ2(t))

• 4. Reparametrization of Swung surfaces: Theorem of Reparametrization
• Let S be a rational complex surface, other than a plane, parametrized by

P(t, s) = (φ1(t)ψ1(s), φ1(t)ψ2(s), φ2(t)) ∈ K(i)(t, s)3

• where (φ1(t), φ2(t)) ∈ K(i)(t)2 and (ψ1(s), ψ2(s)) ∈ K(i)(s)2 are curves properly

parametrized Then, the following statements are equivalent:

• S is K-parametrizable.
• There exists λ ∈ K(i) \ {0} such that the curves defined by the parametrizations

φλ = (λφ1(t), φ2(t)) and ψλ = ( 1

λψ1(s), 1 λψ2(s)) are K-parameterizable.

• There exists a change of variables:

ξ : K(i)2 → K(i)2 (t, s) →

• a1t+b1

c1t+d1, a2s+b2 c2s+d2

• where aibi − cidi = 0, i = 1, 2, such that P(ξ(t, s)) ∈ K(t, s)3.

• 4. Reparametrization of Swung surfaces: Theorem of Reparametrization
• λ = 1 if (φ1(t), φ2(t)) and (ψ1(s), ψ2(s)) are parametrizables over K.
• λ = i if (iφ1(t), φ2(t)) and (iψ1(s), iψ2(s)) are parametrizables over K.
• λ = −r + i ∈ K(i) where

r = Re(φ1(t0 + it1) Im(φ1(t0 + it1) = − Re(ψ1(s0 + is1) Im(ψ1(s0 + is1) = − Re(ψ2(s0 + is1) Im(ψ2(s0 + is1) ∈ K

• 4. Reparametrization of Revolution surfaces: Theorem of Reparametrization
• Let S be a rational revolution surface, parametrized by

P(t, s) = (φ1(t)s2 − 1 s2 + 1, φ1(t) 2s s2 + 1, φ2(t)) ∈ K(i)(t, s)3, where (φ1(t), φ2(t)) is a proper parametrization of a curve. The following statements are equivalent:

• S is K-parametrizable (but, perhaps, not necessarily with a proper parametrization)
• The curve defined by φ(t) = (φ1(t), φ2(t)) is K-parametrizable.
• There exists a change of parameters with complex coefficients. ξ : K(i) −

→ K(i), where ξ(t) = at + b ct + d and ad − bc = 0, such that P(ξ(t), s) ∈ K(t, s)3.

• 4. Reparametrization of Swung surfaces: Algorithm of Reparametrization

Algorithm of Reparametrization of Swung Surfaces

• Input: A complex parametrization P of a swung surface S, other than a plane,

P(t, s) = (φ1(t)ψ1(s), φ1(t)ψ2(s), φ2(t)) ∈ K(i)(t, s)3 where (φ1(t), φ2(t)) ∈ K(i)(t)2 and (ψ1(s), ψ2(s)) ∈ K(i)(s)2 are curves properly parametrized

• Output: A real parametrization, P′(t, s) ∈ K(t, s)3 of S or “The surface is not K-

parametrizable”

• 4. Reparametrization of Swung surfaces: Algorithm of Reparametrization
• 1. Write φ2(t0 + it1) = B0(t0,t1)+iB1(t0,t1)

B(t0,t1)

• 2. Compute the factors of degree 1 and/or of degree 2 (that correspond to circles) of B1(t0, t1)

in K[t0, t1].

• 3. For each factor f from step 4. do

(a) Compute a real parametrization (v0(t), v1(t)) of the line or circle defined by f. (b) Let v(t) = v0(t) + iv1(t) (c) If there exists a λf ∈ C∗ such that (λf · φ1(v(t)), φ2(v(t))) is real then:

• i. Apply the real reparametrization algorithm for curves to ψλf = (1/λfψ1, 1/λfψ2).
• ii. If ψλf is real and u(s) is an invertible linear fraction such that ψλf(u(s)) is real

then return (v(t), u(s)).

• 4. If no factor f works then return “The surface is not real”.

• 4. Reparametrization of Swung surfaces: Example

Example 1 Let SC be the classical revolution surface given by the parametrization 3 − t2 4 − 2t s2 − 1 s2 + 1, 3 − t2 4 − 2t 2s s2 + 1, −it2 + 4it − 3i 2t − 4

• If we take the φ-curve parametrized by

3 − t2 4 − 2t, −it2 + 4it − 3i 2t − 4

• we obtain that we have to parametrize the circle t2

0 + t2 1 − 4t0 + 3 = 0 (and, thus, the given

curve is real), yielding the associated unit ξ(t) = (t + 3i)/(t + i) If we apply this unit to the original parametrization we get the following real parametrization of SC: t2 + 3 t2 + 1 s2 − 1 s2 + 1, t2 + 3 t2 + 1 2s s2 + 1, 2t t2 + 1