Thierry Bay
email: thierry.bay@uphf.fr

Isabelle Cattiaux-Huillard

Laura Saini
email: laura.saini@junia.com

Hermite interpolation by planar cubic-like ATPH

Keywords: Pythagorean hodograph, Trigonometric functions, Algebraic trigonometric, Trigonometric polynomials, Circle arc, Hermite interpolation Mathematics Subject Classifications (2020): 41A05, 42A15, 51N05, 65D05, 65D07, 65D10, 65D17, 68U07

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed.

Introduction

The most commonly used schemes in CAD and CAGD for the representation of curves and surfaces are specified by polynomial and rational parameterizations, such as Bézier curves, B-splines or NURBS. Even though these models offer many advantages, they have important drawbacks detailed in [START_REF] Mainar | Shape preserving alternatives to the rational Bézier model[END_REF]. For example, on the one hand, polynomial Bézier curves are quite simple to define, compute and manipulate, but exclude important curve families as conics, cycloidal curves, etc. On the other hand, rational parameterization allows to obtain conics and trigonometric curves, but comes at the expense of additional weights for each control point and a heavy process for differentiation. Therefore, looking for an alternative representation is necessary. Moreover, obtaining both straight line segments and circle arcs in the same model is an undeniable advantage in environments associated with Computer Numerical Control Machines and for the conversion of trajectories into G-code. For this purpose, it could be interesting to investigate other curve spaces.

Responding to this problem, Carnicer and Peña [START_REF] Carnicer | Totally positive bases for shape preserving curve design and optimality of B-splines[END_REF] introduced the concept of B-basis to generalize the Bernstein bases for other function spaces, while Mainar et al. [START_REF] Mainar | Shape preserving alternatives to the rational Bézier model[END_REF] provided criteria to find one in a general case. The underlying curves display all the positive properties of the Bézier scheme from variation diminishing property to tangency to the control polygon at the endpoints. These authors gave a particular solution for the mixed Algebraic Trigonometric (AT) curve space P m = span{1, t, {cos(kt), sin(kt)} m k=1 } for m = 1 and m = 2. In these bases, the curves are described on [0, α] unlike the Bézier curves which are drawn on [0, 1]. The real α ∈ ]0, 2π[ can then be used as a shape parameter. An interactive illustration of the B-basis, the curves of P 1 and the influence of α on the shape of the curve is available in [START_REF] Cattiaux | Normalized C-Basis and C-Curves[END_REF].

The Pythagorean Hodograph (PH) polynomial curves, characterized by a polynomial parametric speed, was introduced in [START_REF] Farouki | Pythagorean Hodographs[END_REF]. Combining this feature with the mixed AT space P m , a partial investigation of quintic-like Algebraic Trigonometric Pythagorean Hodograph curves (ATPH) is put forward for m = 2 in [START_REF] Romani | Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation[END_REF]. Generalizing from this concept, we gave in [START_REF] Cattiaux-Huillard | Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves[END_REF] a straightforward and exhaustive description of these curves through their complex expression and proposed an in-depth study of spaces P 1 and P 2 . In our paper, we focus on these P 1 ATPH curves, so-called cubic-like ATPH, to solve the Hermite interpolation problem while taking into account the shape parameter of the B-basis.

Indeed, the existence of polynomial PH solutions of different forms of the Hermite problem was deeply investigated. A cubic solution for the G 1 planar case was proposed with one arc in [START_REF] Meek | Geometric Hermite interpolation with Tschirnhausen cubics[END_REF][START_REF] Byrtus | G 1 Hermite interpolation by PH cubics revisited[END_REF], bi-arc in [START_REF] Bastl | Planar C 1 hermite interpolation with uniform and non-uniform TC-biarcs[END_REF] and as spline in [START_REF] Jüttler | Cubic pythagorean hodograph spline curves and applications to sweep surface modeling[END_REF][START_REF] Šír | C 2 hermite interpolation by pythagorean hodograph space curves[END_REF]. Spatial cubic interpolation was studied in [START_REF] Bastl | C 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs[END_REF]. Quintic PH interpolation was also the subject of many works concerning the planar (see [START_REF] Farouki | Hermite interpolation by pythagorean hodograph quintics[END_REF][START_REF] Moon | Construction and shape analysis of PH quintic Hermite interpolants[END_REF]) and the spatial case (see [START_REF] Zbynek | C 2 Hermite interpolation by Pythagorean Hodograph space curves[END_REF][START_REF] Farouki | Hermite Interpolation by Rotation-Invariant Spatial Pythagorean-Hodograph Curves[END_REF][START_REF] Farouki | Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures[END_REF]). About the ATPH feature, only quintic solutions were considered for planar curves in [START_REF] Romani | Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation[END_REF] and for spatial curves in [START_REF] Romani | Algebraic-Trigonometric Pythagorean-Hodograph Space Curves[END_REF][START_REF] Wu | Geometric Hermite interpolation by a family of spatial algebraic-trigonometric PH curves[END_REF]. But, according to our knowledge, cubic-like ATPH have never been applied to address the first-order Hermite interpolation problem. This article is then devoted to the study of such planar ATPH cubic-like curves, solutions of the G 1 Hermite interpolation problem. A focus is done on detailing the conditions of their existence and some of their properties.

The remainder of this paper is organized as follows. Section 2 recalls some basic facts concerning the mixed AT space and properties of ATPH curves. In Section 3, we start with the study of the construction of cubic-like ATPH Hermite interpolants. Then, we analyze in Section 4 the conditions required for the existence of such solutions. Consequently, in Section 5, we look at the influence of the free parameter α on the curve by working on the control polygon. In Section 6, we focus on a use of the shape parameter α to find the ATPH cubic-like Hermite interpolant that passes through one given additional point, which are illustrated in Section 7 by several examples.

Preliminaries

In [START_REF] Mainar | Shape preserving alternatives to the rational Bézier model[END_REF], the authors proposed B-bases for several AT curve spaces. In particular, they described the normalized P 1 B-basis (Z 0 , Z 1 , Z 2 , Z 3 ), depending on α belonging to the interval ]0, 2π[ and for t ∈ [0, α] as:

Z 3 (t) = t -sin t S α , Z 0 (t) = Z 3 (α -t), Z 2 (t) = M α 1 -cos t C α -Z 3 (t) , Z 1 (t) = Z 2 (α -t), (1)
where

S α = α -sin α, C α = 1 -cos α, R α = 2 sin α -α -α cos α and M α = C α sin α R α if α ̸ = π, M π = 1
otherwise. This formulation is short and elegant but not suitable to perform a change of basis between the canonical basis and the B-basis. So, in [START_REF] Cattiaux-Huillard | Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves[END_REF], we proposed a detailed expression of the matricial form.

P 0 P 1 P 2 P 3 l 0 l 1 l 2 θ 1 θ 2 Figure 1: Cubic-like ATPH polygon data.
In this basis, a curve parameterization R α (t) of P 1 is described by four control points P 0 , P 1 , P 2 and P 3 . This representation admits a complex interpretation, which is especially efficient to characterize PH curves, as proved by Farouki in [START_REF] Farouki | The conformal map z → z 2 of the hodograph plane[END_REF] for the polynomial case. We extended this result to the ATPH curves in [START_REF] Cattiaux-Huillard | Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves[END_REF]. Denoting by r α (t), p 0 , p 1 , p 2 and p 3 the complex forms of R α (t) and its control points1 , its complex expression is

r α (t) = 3 k=0 p k Z k (t). (2)
A cubic-like ATPH curve is characterized by the complex relation:

(∆p 1 ) 2 = K α ∆p 0 ∆p 2 with K α = 4 (α -sin α) 2 2 sin α 2 -α cos α 2 2 , (3)
where ∆p k = p k+1 -p k . This condition can be geometrically rewritten by:

l 2 1 = K α l 0 l 2 and θ 1 = θ 2 , (4)
where θ 1 = ∠(P 0 -P 1 , P 2 -P 1 ), θ 2 = ∠(P 1 -P 2 , P 3 -P 2 ) and, in polar form, ∆p k = l k e iβ k for k = 0, 1, 2 (cf. Figure 1). Afterwards, the K α value appears many times. It is therefore necessary to detail its elementary properties that can be established by some straightforward calculations.

Proposition 1. The term K α is an increasing and continuous function of α. It belongs to the range ]1, 4[ when α is in the interval ]0, 2π[ and K α tends to 1 as α approaches 0. Moreover

K α = 2 α -sin α 2 sin α 2 -α cos α 2 .
The values α = 0 and α = 2π are excluded from our study. On one hand, when α approaches 0, the cubic-like AT curve becomes the cubic polynomial Bézier curve of the same control polygon as shown by Zhang in [START_REF] Zhang | C-curves: An extension of cubic curves[END_REF]. This is consistent with the formula (3) and (4), since when K α = 1, these expressions are identical to the homologous relations given by Farouki in [START_REF] Farouki | The conformal map z → z 2 of the hodograph plane[END_REF].

On the other hand, the second and third functions Z 1 (t) and Z 2 (t) of the B-basis vanish as α = 2π. Therefore the curve parameterized by r α (t), with t ∈ [0, α], tends to the segment [P 0 P 3 ], which obviously constitutes an improper solution.

Cubic-like ATPH solutions for the Hermite problem

The Hermite problem we are going to deal with consists of interpolating two points and two tangent directions by a cubic-like ATPH curve. Therefore, we consider two distinct points P 0 and P 3 and two oriented angles, θ 0 ∈ [-π, 0] and θ 3 ∈ [0, 2π[. The choice of these intervals allows us to study all possible cases without any loss of generality. In the coordinate system (P 0 , i, j) where i = P 3 -P 0 ∥P 3 -P 0 ∥ and j = rot π 2 (i), we define two unit tangent vectors t 0 = (cos θ 0 , sin θ 0 ) and t 3 = (cos θ 3 , sin θ 3 ) (cf. Figure 2). We need to verify:

R α (0) = P 0 , R α (α) = P 3 , dR α dt (0) = λ 0 t 0 , dR α dt (α) = λ 3 t 3 ,
where λ 0 and λ 3 are strictly positive real numbers. Note that if t 0 and t 3 are both colinear with P 3 -P 0 , the solution of Hermite's problem is trivially a line segment and is of little interest. Therefore, we will exclude this case in the rest of the article, and we will denote by H the domain [-π, 0] × [0, 2π[ for (θ 0 , θ 3 ) without couples (-π, 0), (-π, π), (0, 0) and (0, π).

From the data described above, we have θ 1 = π -β 0 + β 1 and θ 2 = π -β 1 + β 2 . Besides, the G 1 Hermite conditions imply β 0 = θ 0 and β 2 = θ 3 . From (4), we therefore deduce

-β 0 + β 1 = -β 1 + β 2 , hence β 1 = 1 2 (β 0 + β 2 ) to within π. So β 1 = 1 2 (θ 0 + θ 3 ) or β 1 = 1 2 (θ 0 + θ 3 ) + π. (5)
To begin with, let us consider the first possibility. Afterwards, we will see how to take into account the second one. It is clear that:

∆p 0 + ∆p 1 + ∆p 2 = p 3 -p 0 , l 0 e iβ 0 + l 1 e iβ 1 + l 2 e iβ 2 = p 3 -p 0 , l 0 e iθ 0 + K α l 0 l 2 e i θ 0 +θ 3 2 + l 2 e iθ 3 = p 3 -p 0 , (6) so
, if we set L = l 2 l 0 , ( 3
) ⇔ e iθ 0 + L K α e i θ 0 +θ 3 2 + L 2 e iθ 3 = p 3 -p 0 l 0 , (7)
ℑm(7) ⇔ sin θ 0 + L K α sin θ 0 + θ 3 2 + L 2 sin θ 3 = 0, (8)
ℜe(7) ⇔ cos θ 0 + L K α cos θ 0 + θ 3 2 + L 2 cos θ 3 = ∥P 3 -P 0 ∥ l 0 . ( 9
) α = 0 P 1 P 2 α = 3 π 4 P 1 P 2 α = 3 π 2 P 1 P 2 P 0 P 3 Figure 3: Solutions for α = 0 (polynomial case), α = 3π 4 and α = 3π 2 .
When the degree of equation ( 8) is 2 with respect to L, i.e. for θ 3 ̸ = 0 and

θ 3 ̸ = π, the discriminant is ∆ = K α sin 2 θ 0 + θ 3 2 -4 sin θ 0 sin θ 3 , (10)
and the solutions, when they exist, are

L ε = 1 2 sin θ 3 -K α sin θ 0 + θ 3 2 + ε √ ∆ , (11)
where ε ∈ {-1, +1}. For the sake of clarity, we will denote them respectively L -and L + .

When the real number

G ε = cos θ 0 + L ε K α cos θ 0 + θ 3 2 + L 2 ε cos θ 3 (12)
is positive, we deduce from ( 9) the value of l 0 and then l 2 . With the previous expressions of β 0 , β 1 and β 2 , we can determine ∆p i , for i = 0, 1, 2 and therefore the control polygon of the solution r α . When θ 3 = 0 or θ 3 = π, equation ( 8) is of degree 1 with respect to L and requires a specific study which is done in Section 4.4.

From these results, we can easily notice that the ATPH provides an infinite number of solutions -depending on α -for a given initial data set. In Part 2, we put forward that the polynomial cubic PH curve can be seen as a particular case of cubic-like ATPH, obtained for α = 0 and K α = 1. This is consistent with our results, since when K α = 1, the expressions ( 10), [START_REF] Farouki | Pythagorean Hodographs[END_REF] and ( 12) are identical to the formulas given by Walton and Meek in [START_REF] Meek | Geometric Hermite interpolation with Tschirnhausen cubics[END_REF] and by Byrtus and Bastl in [START_REF] Byrtus | G 1 Hermite interpolation by PH cubics revisited[END_REF] for the polynomial case. The Figure 3 shows two ATPH solutions and their polynomial equivalent.

Existence of solutions

The solutions described in the previous part do not always exist. We now have to clarify their conditions of existence, i.e. the domains on which the values of ∆, G ε , L ε , defined by the expressions [START_REF] Farouki | Hermite interpolation by pythagorean hodograph quintics[END_REF], [START_REF] Farouki | Pythagorean Hodographs[END_REF], [START_REF] Galetzka | A Simple and Correct Even-Odd Algorithm for the Point-in-Polygon Problem for Complex Polygons[END_REF] are positive. As said previously, the cases θ 3 = 0 and θ 3 = π are different from the general one and will be treated separately.

Sign of the discriminant

Let us denote by D + the domain of H where ∆ ≥ 0, whose boundary is defined by the functions:

f 1 (θ 0 ) = 2π -arccos K 2 α cos θ 0 -4(8 -K α ) √ 4 -K α sin 2 θ 0 K 2 α + 16(4 -K α ) sin 2 θ 0 , f 2 (θ 0 ) = 2π -arccos K 2 α cos θ 0 + 4(8 -K α ) √ 4 -K α sin 2 θ 0 K 2 α + 16(4 -K α ) sin 2 θ 0 . Since K α ∈ ]1, 4
[, these two functions are well defined for any value of α ∈ ]0, 2π[. Of course, the shape of this domain depends on this last parameter (see Figure 4). The function f 1 describes the lower border of the domain, and f 2 the upper border (see Figure 5). So the domain D + is described by

D + = {(θ 0 , θ 3 ) ∈ H | θ 3 ≤ f 1 (θ 0 ) or f 2 (θ 0 ) ≤ θ 3 }.
We can remark that for θ 3 < π, we have ∆ > 0, hence the condition is always verified.

Sign of l 0

Once L ε is determined using [START_REF] Farouki | Pythagorean Hodographs[END_REF], l 0 is obtained from the relation [START_REF] Farouki | Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures[END_REF]. Therefore, l 0 and the real number G ε have the same sign which has to be positive. Moreover, the case G ε = 0 should be excluded, otherwise the length l 0 would be infinite. Here we will study the sign of G ε , as defined in [START_REF] Galetzka | A Simple and Correct Even-Odd Algorithm for the Point-in-Polygon Problem for Complex Polygons[END_REF], using the expression [START_REF] Farouki | Pythagorean Hodographs[END_REF] for L ε .

Denoting Γ + and Γ -the subdomains of H where G + > 0 and G -> 0 respectively, a solution exists if and only if (θ 0 , θ 3 ) belongs to Γ = Γ + ∪ Γ -. The relations [START_REF] Farouki | Pythagorean Hodographs[END_REF] and ( 12) allow us to establish that

4G ε sin 2 θ 3 = A + B, (13)
where:

A = 2 sin θ 3 -θ 0 2 4 sin θ 3 cos θ 3 -θ 0 2 -K α sin θ 0 + θ 3 2 , B = 2ε K α sin θ 3 -θ 0 2 √ ∆.
Then, we have G ε = 0 if and only if sin θ 3 -θ 0 2 = 0 or A ′ -B ′ = 0, with:

A ′ = 4 sin θ 3 cos θ 3 -θ 0 2 -K α sin θ 0 + θ 3 2 , B ′ = ε K α √ ∆.
Evidently, the former equation has for solution θ 3 = θ 0 + 2π in H, which leads to ∆ < 0. From the latter one, we obtain:

A ′2 -B ′2 = 4(2 + 2 cos(θ 3 -θ 0 ) -K α ) sin 2 θ 3 . ( 14
)
The sign of this expression changes for θ 3 = ψ k (θ 0 ) or θ 3 = ψ k (θ 0 ) where:

ψ k (θ 0 ) = θ 0 + arccos K α 2 -1 + 2kπ, (15)
ψ k (θ 0 ) = θ 0 -arccos K α 2 -1 + 2kπ. ( 16
)
Since K α ∈ ]1, 4[ according to Proposition 1, these functions are well defined. Only the lines corresponding to ψ 0 (θ 0 ), ψ 1 (θ 0 ) and ψ 1 (θ 0 ) are in the domain H (cf. Figure 5) and are involved in the description of Γ + and Γ -(cf. Figures 6 and7). These domains are formally described by:

Γ + = {(θ 0 , θ 3 ) ∈ H | θ 3 ≥ π -θ 0 and (f 2 (θ 0 ) < θ 3 < ψ 1 (θ 0 ) or θ 3 < f 1 (θ 0 ))} ∪ {(θ 0 , θ 3 ) ∈ H | θ 3 < π -θ 0 and (θ 3 < ψ 1 (θ 0 )}, Γ -= {(θ 0 , θ 3 ) ∈ H | θ 3 ≥ π -θ 0 and (f 2 (θ 0 ) < θ 3 or ψ 1 (θ 0 ) < θ 3 < f 1 (θ 0 ))} ∪ {(θ 0 , θ 3 ) ∈ H | θ 3 < π -θ 0 and (θ 3 < ψ 0 (θ 0 ) or ψ 1 (θ 0 ) < θ 3 )}.
This result is a generalization of [START_REF] Byrtus | G 1 Hermite interpolation by PH cubics revisited[END_REF] focusing on the polynomial curves, which corresponds to the case K α = 1 for α = 0 in our study. The ATPH feature provides therefore a wider domain of solutions through the variations induced by the α parameter. Remark 1. If ∆ ≥ 0 and G ε < 0, the curve exists with l 0 < 0 and therefore l 2 < 0. A length being positive, the notion of algebraic measure would then be more appropriate to cover this case. As seen in Figure 8, this means geometrically that the orientations of vectors P 1 -P 0 and t 0 are opposite, as the orientations of P 3 -P 2 and t 3 are. If the constraints of the Hermite problem concern only the direction of the tangents and not their orientation, these solutions are convenient. The only condition that θ 0 and θ 3 must then verify is ∆ ≥ 0, ie (θ 0 , θ 3 ) ∈ D + .

Sign of roots

As L ε = l 2 /l 0 , its value is positive. However, equation ( 8) can have negative roots. We will show that they are also solutions of the Hermite problem. Relation [START_REF] Cattiaux | Normalized C-Basis and C-Curves[END_REF] gives two expressions for β 1 . The second one leads to slightly different expressions of equations ( 8) and ( 9):

sin θ 0 -L K α sin θ 0 + θ 3 2 + L 2 sin θ 3 = 0, ( 17
)
cos θ 0 -L K α cos θ 0 + θ 3 2 + L 2 cos θ 3 = ∥P 3 -P 0 ∥ l 0 . (18)
Expression [START_REF] Romani | Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation[END_REF] being the value of G ε , we notice that L ε is a solution of (8) if and only if -L ε is a solution of [START_REF] Romani | Algebraic-Trigonometric Pythagorean-Hodograph Space Curves[END_REF]. Therefore, the solution -L ε of ( 17) is identical to the expression given by ( 9) for L ε . So the value of l 0 is the same in both cases.

Proposition 2. When L ε < 0, we still have l 0 = ∥P 3 -P 0 ∥ G ε and l 2 = L 2 ε l 0 . In this case,

β 1 = π + θ 0 + θ 3 2
and the polygon (P 0 , P 1 , P 2 , P 3 ) is then crossed (see Figure 9).

P 0 P 3 P 1 P 2 Figure 8: Improper solution when G ε is negative, for α = 2π 3 , θ 0 = - π 4
and θ 0 = 3π 5 .

P 0 P 3 P 1 P 2 P 1 P 2 L + >0 L -<0
Figure 9: Example of a double solution with L + > 0 and L -< 0.

A crossed polygon often gives a looped curve, which is rarely suitable in CAGD. As a consequence, it is important to determine the sign of L ε when G ε > 0 and ∆ ≥ 0. Some considerations about the roots of equation ( 8) lead to the following result. Proposition 3. When the roots of equation (8) exist and verify G ε > 0, we have L + > 0 for all the values of θ 0 and θ 3 , and L -> 0 if and only if θ 3 > π. The polygon is then crossed if and only if ε = -1 and θ 3 < π.

4.4 Particular case: θ 3 = 0 and θ 3 = π As said in Section 3, equation ( 8) is of degree 1 when θ 3 = 0 or θ 3 = π.

If θ 3 = 0, its solution is then L = L 0 = -2 √ K α cos θ 0 2 which is always negative. So β 1 = π+ θ 0 + θ 3 2
(according to Proposition 2) and its polygon is Z-shaped. Denoting G 0 the equivalent of G ε for L 0 , we obtain from (18

C  0  3 P 0 P 3 P 1 P 2 (a) Case 1: 0 ≤ θ 3 < θ 0 + π. C  0  3 P 1 P 3 P 0 P 2 (b) Case 2: θ 0 + π < θ 3 ≤ π.
) G 0 = 1 + 2 K α + 2 1 + 1 K α cos θ 0 which is positive if and only if θ 0 > arccos K α + 2 2(K α + 1)
-π.

If θ 3 = π, the solution of equation ( 8) is

L = L π = -2 √ K α sin θ 0 2 which is positive. With the same convention, G π = 1 K α (2 cos θ 0 -2 + K α ) is positive if and only if θ 0 > -arccos 1 - K α 2 .
5 Evolution of the solution polygon with respect to α

The shape of the control polygon determines the curve and may sometimes involve the presence of inflection points, cusps or loops. As said previously, these elements are undesirable for most CAD, CAD/CAM, or computer graphics applications. A sufficient condition to eliminate undesired elements according to the variation diminishing property is to consider convex polygons.

We then focus the rest of this paper on this case. Obviously, this requirement is equivalent to θ 3 < π and ε = 1. The corresponding two cases are illustrated in Figure 10.

The ATPH solution of the Hermite problem depends on α which gives an additional degree of freedom, the polynomial solution being a particular case obtained for α = 0. This free parameter can be used to solve various additional problems (velocity, interpolation point, . . . ), as will be shown in Section 6. It is therefore important to analyze the evolution of the curve with respect to α. For this purpose, we will study a geometrical version of our problem through the different configurations of the control polygon.

Considering two points A and B, we denote by AB the algebraic measure from A to B. Let D 0 (resp. D 3 ) be the line passing through P 0 (resp. P 3 ) with t 0 (resp. t 3 ) as orientation vector. Firstly, we suppose the existence of an intersection point of D 0 and D 3 , denoted by C.

According to the initial constraints of the Hermite problem, the points P 1 and P 2 must be chosen such that P 1 -P 0 and t 0 have the same orientation, and P 2 -P 3 and t 3 are opposite vectors. The measures l 0 and l 2 are here defined algebraically with l 0 = P 0 P 1 and l 2 = P 2 P 3 , which is consistent with the fact that these two values remain positive. Moreover, as θ 1 = θ 2 according to (4), the triangle CP 1 P 2 is isosceles with apex C. We therefore define the measures a = CP 0 and b = CP 3 . These two values are constant with respect to α. As CP 1 = -CP 2 , we have

l 0 + a = l 2 -b.
Moreover, the angles of CP 0 P 3 being fixed with respect to α, there exists a constant value c such that l 1 = ±c(l 0 + a). From relation (4), we deduce a new expression of K α as a function of l 0 , denoted by K α and written as

K α (l 0 ) = c 2 (l 0 + a) 2 l 0 (l 0 + a + b) .
The derivative of this function with respect to l 0 is:

d K α dl 0 = - c 2 (l 0 + a) l 2 0 (l 0 + a + b) 2 (l 0 (a -b) + a(a + b)) .
To study the sign of this expression, we make the assumption |a| > |b|. As defined in Section 2, the B-basis (Z 0 , Z 1 , Z 2 , Z 3 ) has the same property of symmetry as the Bernstein basis. So the case |a| < |b| arises from the former one by considering the symmetrical problem with θ ′ 0 = -θ 3 and θ ′ 3 = -θ 0 . The geometric constraints lead then to two distinct situations, as illustrated by Figure 10.

In the first case, we have 0 ≤ θ 3 < θ 0 + π, with a < 0 and b > 0. Since l 0 < |a|, l 0 + a < 0. Moreover, l 0 + a = l 2 -b with l 2 > 0, so l 0 + a + b > 0. Finally, l 0 > -(a + b) > 0 and a -b < a < 0 lead to l 0 (a -b) + a(a + b) < 0. The derivative is therefore negative. Similarly in the second case, when θ 0 + π < θ 3 ≤ π, we have a > 0, b < 0 and l 0 + a > 0. As a > -b, we have a + b > 0 and a -b > 0, so l 0 (a -b) + a(a + b) > 0. The derivative is negative again.

In the considered domain, the function K α is a decreasing and continuous bijection with respect to l 0 . Moreover, the K α form described in ( 3) is an increasing and continuous function of α (see Proposition 1). So l 0 = K -1 α (K α ) is continuous and decreasing with respect to α. The property is also true for l 2 , since l 2 = l 0 + a + b, the value a + b being constant relatively to K α . Proposition 4. When ε = 1, θ 3 < π and D 0 and D 3 intersect, the lengths l 0 and l 2 continuously decrease as α increases.

When D 0 and D 3 are parallel, we have θ 3 = θ 0 + π, as illustrated by Figure 11. Indeed, for θ 3 = θ 0 + 2π, ∆ is always negative and the interpolation problem has no solution. Let P ′ 0 denote the projection of P 3 onto D 0 , along the line (P 1 P 2 ) and we suppose that P 0 P 1 > P 2 P 3 (the other case being symmetrical, as before). Let a and c be the distances P ′ 0 P 0 and P ′ 0 P 3 respectively. Here again, these two values are independent of the choice of α. As θ 1 = θ 2 , the quadrilateral P ′ 0 P 1 P 2 P 3 is then a rectangle and we have l 2 = l 0 -a and l 1 = c. So according to equation (4), c 2 = K α l 0 (l 0 -a). Hence, we define

K α (l 0 ) = c 2 l 0 (l 0 -a)
.

Since l 0 > a, this function is obviously continuous and decreasing when l 0 ∈ R + . So, as for the intersecting case, we can establish the following result:

Proposition 5. When ε = 1, θ 3 < π and D 0 and D 3 are parallel, the lengths l 0 and l 2 continuously decrease as α increases.

Remark 2. For ε = 1 and θ 3 < π, the study points out that l 0 and l 2 also decrease as α increases. Though, for ε = -1 and θ 3 < π, it can be proved that l 0 and l 2 are increasing functions with respect to α.

Fitting method and unique interpolant

As a concrete example of use of the free parameter α in the Hermite interpolation, we propose an optimization process to determine its required value to satisfy an additional interpolation constraint. After determining the interval to which the α parameter should belong to get a solution, we infer the domain P where the extra interpolation point D has to be to ensure the existence of a solution. The fitting process is developed and illustrated via some examples.

The domain P

According to Section 4, the required conditions for the existence of a solution are ∆ > 0 and G ε > 0. These constraints modify the initial interval ]0, 2π[ of α. For ε = 1 and θ 3 < π, we can determine α min and α max , depending on θ 0 and θ 3 , such that the Hermite problem has a solution if and only if α ∈ ]α min , α max [. On the studied domain, it can be easily proved that ∆ > 0. Therefore, the existence of a solution depends on the sign of G + . Indeed, according to Proposition 3, we have L + > 0 if G + > 0 without any condition on α. Moreover, the domain Γ + is bounded by the condition

θ 3 < ψ 1 (θ 0 ). If θ 3 < θ 0 + 4π 3
this inequality is always true and α min = 0 (as

K α ∈ ]1, 4[ from Proposition 1). Otherwise, it is equivalent to K α > 2 + 2 cos(θ 3 -θ 0 ) and α min is the value for which K α = 2 + 2 cos(θ 3 -θ 0 ). (19)
This equation being transcendental, no closed-form solution exists and a numerical approximation is therefore necessary. As there is no upper bound for Γ + , we have evidently α max = 2π. The corresponding solution curve is then degenerated to the segment [P 0 P 3 ].

Properties 4 and 5 show that the lengths l 0 and l 2 continuously decrease as α increases. According to the variation diminishing property, the convex curve r α moves continuously from r α min to r αmax . The continuous curve r α then describes the area delimited by r α min and [P 0 P 3 ], which is the domain P we are looking for (see Figure 12). So, for any point D in P, there exists α * ∈]α min , α max [ and t D ∈ [0, α * ] such that r α * (t D ) = D. This transcendental equation needs once again an algorithmic solving. Remark 3. When θ 3 > θ 0 + 4π 3 and α min is a solution of (19), we have G + = 0 implying that the length l 0 = ∥P 3 -P 0 ∥ G + is theoretically infinite. Practically, even though the used value is an approximation, l 0 still increases dramatically with the accuracy of α min . Even if it can be seen as an advantage since it involves a very large P domain, it can lead to serious numerical instabilities when verifying the belonging of D to P and determining α * . It is rather preferable to choose a value slightly greater than the solution of (19) for α min .

The optimization process

The first question that must be answered is whether or not D belongs to P. It is done with a ray-tracing method [START_REF] Shimrat | Algorithm 112: Position of Point Relative to Polygon[END_REF][START_REF] Galetzka | A Simple and Correct Even-Odd Algorithm for the Point-in-Polygon Problem for Complex Polygons[END_REF]. Let us consider the polygon resulting from a discretization of the curve R α min , that we have closed by joining the points P 0 and P 3 . The algorithm consists in casting a ray from D and in counting the intersections of this ray with the edges of the polygon.

If the number of intersections between the ray and each edge is even, the point D is outside, otherwise D is inside. Once this step has been done, the fitting error defined by E(α) = ∥D -R α (t D )∥ 2 is minimized. The value t D ∈ [0, α] is the parameter related to the closest orthogonal projection of D onto R α (cf. Figure 13a). In other words, its value is obtained by determining the solution of

R ′ α (t).(D -R α (t)) = 0 ( 20
)
that minimizes the distance between D and R α (t). Equation [START_REF] Wu | Geometric Hermite interpolation by a family of spatial algebraic-trigonometric PH curves[END_REF] being transcendental, an iterative Newton-Raphson-based approach is done to get t D and then E(α). A dichotomy-based process is finally applied to minimize E(α) with respect to α (cf. Figure 13b). The reader can refer to Appendix A for further details about the algorithm behind the optimization process.

Qualitative comparisons between PH and ATPH curves

In order to illustrate the contribution of ATPH, several examples comparing cubic-like ATPH and cubic PH solutions are detailed here. In each case, we interpolate a part of a classic curve. The control points P 0 and P 3 , as well as the angles θ 0 and θ 3 , are chosen from the parameterization of the reference curve.

Circle arcs

Since circle arcs are of major interest to the CAGD, the example of the circle is essential.

In [START_REF] Cattiaux-Huillard | Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves[END_REF], we show that there is a cubic-like ATPH which represents it exactly, whatever the radius and the considered arc. Of course, this is not the case with PH curves. Not only does the PH Hermite interpolation only provides an approximation, but a solution also does not always exist. Indeed, to interpolate a circle arc, the initial condition is θ 3 = -θ 0 = 1 2 α where α is the arc measure. The ATPH R α fits then exactly the circle arc without optimisation process. Denoting by R the PH solution, it exists only if θ 3 < θ 0 + 4π 3 (see [START_REF] Byrtus | G 1 Hermite interpolation by PH cubics revisited[END_REF]), i.e if θ 3 < 2π 3 or α < 4π 3 . Moreover, when θ 3 approaches 2π 3 , l 0 tends to infinity and therefore the error diverges.

Let E(θ 3 ) = R 1 2 -S be the error, where S is the apex of the circle arc (see Figure 14a) defined by

S = 1 2 (P 0 + P 3 ) - ∥P 3 -P 0 ∥ 2 1 sin θ 3 - 1 tan θ 3 .j.
According to [START_REF] Meek | Geometric Hermite interpolation with Tschirnhausen cubics[END_REF] for θ 3 = -θ 0 , the PH solution of this Hermite problem verifies

l 0 = l 2 = ∥P 3 -P 0 ∥ 1 + 2 cos θ 3 . P 0 P 3 S R(1/2) E (a) Graphical interpretation of E(θ 3 ) π 6 π 3 π 2 2 π 3 θ 3 2 4 6 E(θ 3 ) (b) Graph of the error function E(θ 3 )
Figure 14: Error between ATPH circle arc (blue) and its PH approximation (orange). Then, we have

P 0 P 3 P 1 P 2 (a) α = 6π 7 P 0 P 3 P 1 P 2 P 1 P 2 (b) α = 4π 3 -0.2
R 1 2 = 1 2 (P 0 + P 3 ) - 3∥P 3 -P 0 ∥ 4(1 + 2 cos θ 3 ) sin θ 3 .j.
The error is

E(θ 3 ) = 1 ∥P 3 -P 0 ∥ R 1 2 -S = 3 sin θ 3 4(1 + 2 cos θ 3 ) - 1 2
1 sin θ 3 - 1 tan θ 3 .
Its behavior is illustrated by Figure 14b, and Figure 15 illustrates the difference between the PH and ATPH solutions for two values of α.

Clothoid

Let us consider the example of a clothoid, defined by the normalized Fresnel integrals

C 1 (t) =        x(t) = t 0 sin π 2 u 2 du, y(t) = t 0 cos π 2 u 2 du.
The control points P 0 and P 3 are chosen such as P 0 = C 1 (-2.2) and P 3 = C 1 (-1.6). The tangents at these points verify θ 0 ≈ -1.91 and θ 3 ≈ 1.67. By considering the interpolation point D = C 1 (-1.9), the difference between the clothoid and the obtained ATPH solution is not discernible with the naked eye, unlike the PH one (see Figure 16).

Lemniscate

Let us consider the lemniscate, defined by the parametric equations

C 2 (t) =      x(t) = a √ 2 cos t 1 + sin 2 t , y(t) = a √ 2 sin t cos t 1 + sin 2 t ,
with t ∈ [0, 2π], herein for a = 1. We perform two different interpolations. In Figure 17a, we consider a symmetric arc with P 0 = C 2 7π 6 , P 3 = C 2 7π 6 , θ 3 = -θ 3 ≈ 1.39. Figure 17b shows the fitting of a non-symmetric arc with P 0 = C 2 7π 12 , P 3 = C 2 13π 12 , θ 0 ≈ -1.89 and θ 3 ≈ 1.17. Once again, we clearly see that the cubic-like ATPH curve is closer to the reference curve than the PH one.

Bernoullian quartic

Let us consider the Bernoullian quartic, defined by the parametric equations

C 3 (t) =       
x(t) = (a 2 -b 2 ) sin t cos t √ a 2 sin 2 t + b 2 cos 2 t y(t) = ab √ a 2 sin 2 t + b 2 cos 2 t , with t ∈ [0, π], herein for a = 1 and b = 2. We have P 0 = C 3 5π 6 , P 3 = C 3 7π 15 , then θ 0 ≈ -2.07 and θ 3 ≈ 0.91. Figure 18 illustrates three cases putting forward the position of D on the shape of the interpolating curve. In Figure 18c, the quartic and ATPH curve are nearly overlapped.

Conclusion

In this paper, we have studied the construction process of a Hermite interpolation within the framework of the cubic-like ATPH. The domain Γ in which solutions exist has been rigorously described. Through the presence of the α parameter, the solution of the Hermite problem provided by ATPH curves is not unique. The analysis carried out between the parameter α and the control polygon allows us to understand its influence on the curve. This degree of freedom can be taken into account to solve additional problems. We have proposed an application through a fitting process illustrated by several examples.

3 Figure 2 : G 1

 321 Figure 2: G 1 Hermite interpolation data.

- 3 .Figure 4 :

 34 Figure 4: Domain D + for α = 0 (polynomial case), α = 7π 6 and α = 11π 6 .

- 3 .Figure 5 : 6 - 3 Figure 6 :Figure 7 :

 356367 Figure 5: Edges of D + , Γ + and Γ -domains for α = 7π 6 .

Figure 10 :

 10 Figure 10: Shape of the control polygon when the lines D 0 and D 3 instersect.

3 Figure 11 :

 311 Figure 11: Shape of the control polygon when the lines D 0 and D 3 are parallel.

2 Figure 12 :

 212 Figure 12: Domain P for θ 0 = -2π 3 , θ 3 = 3π 5 and α min = 0.

  The intermediate curves are in blue and the optimal solution R α * is in orange.

Figure 13 :

 13 Figure 13: Fitting process with an interpolation point D: orthogonal projection on the left and dichotomic fitting on the right.

Figure 15 :

 15 Figure 15: Reconstruction of a circle arc by ATPH (blue) and PH curves (orange).

3 P 1 P 2 P 1 P 2 Figure 16 :

 3121216 Figure 16: Interpolation of a clothoid arc with the PH (orange) and the ATPH curves (blue).

3 P 1 P 2 ( 3 P 1 P 2 (Figure 17 :

 31231217 Figure 17: Interpolation of a lemniscate arc with a PH (orange) and an ATPH curves (blue).

  Interpolation with D = C 3 11π 15

Figure 18 :

 18 Figure 18: Interpolation of a Bernoullian quartic arc with the PH (orange) and the ATPH curves (blue).

In the following, notations in bold upper case will refer to points, and those in bold lower case to their complex form.

Acknowledgments

The authors are grateful to the anonymous referees for their useful comments that helped to improve the clarity of the paper.

A Algorithm of the optimization process Algorithm 1: Optimization with respect to α Input: Point D to fit, computational accuracy parameter η Output: The optimized value of α


Charles Paillard

Laurent Bellaiche

Light, a new handle to control the structure of cesium lead iodide

CsPbI3 is of high interest for photovoltaic applications owing to its low bandgap, provided it could be stabilized in its γ perovskite phase. Instead, it energetically prefers to adopt a so-called yellow, δ phase, with much larger bandgap and reduced photovoltaic properties. Here, using an original constrained Density Functional Theory method, we mimic the effect of thermalized photoexcited carriers, and show that larger concentrations in photo-excited carriers (i.e. larger optical pump fluences) effectively reduce the energy difference between the δ non-perovskite ground state and the γ perovskite phase. Even further, the stability of the phases could be potentially reversed and therefore the γ phase stabilized under strong illumination. We also report large photostriction, i.e., large photo-induced strain for all phases in this material, making CsPbI3 suitable for other applications such as photo-driven relays and photoactuators.

I. INTRODUCTION

Halide perovskites represent a new hope to achieve efficient and affordable solar power conversion. They have quickly grown to achieve more than 25% power conversion efficiency [START_REF] Jeong | Pseudo-halide anion engineering for α-FAPbI3 per-392 ovskite solar cells[END_REF]. Their domain of application extends far beyond the mere solar cell, as their good photo-or electroluminescent properties [START_REF] Protesescu | Nanocrystals of Cesium Lead Halide 396 Perovskites[END_REF]3] make them prime candidates for laser or light-emitting diode applications [4,[START_REF] Chouhan | Synthesis, optoelectronic 410 properties and applications of halide perovskites[END_REF]. However, many of those halide perovskites suffer from severe stability issues with respect to humidity or temperature, in particular when they are made of a hybrid of organic -inorganic cations [START_REF] Senocrate | Ther-413 mochemical Stability of Hybrid Halide Perovskites[END_REF]. As a result, the fully inorganic lead halide perovskites, CsPbX 3 (X=I, Br, Cl) have been considered most promising owing to their larger stability. [START_REF] Xiang | A review on the sta-416 bility of inorganic metal halide perovskites: challenges 417 and opportunities for stable solar cells[END_REF]. Among those, CsPbBr 3 significantly starts absorbing at wavelengths below 550 nm (green), while CsPbCl 3 absorbs in the near UV spectral domain.

[2] As a result, those two compounds are not particularly promising to convert efficiently the solar spectrum into useful energy.

On the other hand, the perovskite phase of CsPbI 3 absorbs wavelengths as high as 700 nm [START_REF] Protesescu | Nanocrystals of Cesium Lead Halide 396 Perovskites[END_REF]. Unfortunately, lead cesium iodide is only metastable in a perovskite phase; the more stable δ-phase has much worse optical absorption properties [START_REF] Valastro | Optical behaviour of γ-black CsPbI3 phases formed 422 by quenching from 80 •C and 325 •C[END_REF]. The instability of the perovskite phase has recently been elucidated to stem from rattling of the Cs ion in its coordination cage, based on X-ray diffraction measurements on single crystals from 100 K to 295 K [START_REF] Straus | Understanding the Instability of the Halide Per-426 ovskite CsPbI3 through Temperature-Dependent Struc-427 tural Analysis[END_REF]. The perovskite phase can nonetheless be stabilized by substituting some I -with Br -at the expense of degraded absorption properties [START_REF] Protesescu | Nanocrystals of Cesium Lead Halide 396 Perovskites[END_REF], or metastabilized through cooling from the high-temperature phase [START_REF] Steele | Thermal unequilib-435 rium of strained black CsPbI 3 thin films[END_REF]. Yet, temperature or pressure are slow, cumbersome and energy-inefficient * charles.paillard@centralesupelec.fr the γ phase is dynamically stable in dark conditions [START_REF] Jain | The Materials Project: A materials 440 genome approach to accelerating materials innovation[END_REF], 60 our results may provide a pathway to trigger a transi-61 tion to the perovskite phase without having to realize 62 heavy and slow thermal cycles [START_REF] Steele | Thermal unequilib-435 rium of strained black CsPbI 3 thin films[END_REF]. Furthermore, a large 63 photo-induced strain is also calculated in all phases and 64 in particular in the stable δ-phase, which is of high im-65 portance to use CsPbI 3 in photostrictive devices [START_REF] Kundys | Photostrictive materials[END_REF].

66

II. METHODS

67

In order to study the possibility of triggering a phase 68 transition towards a perovskite phase in CsPbI 3 , we used 69 the constrained-DFT scheme developed in Ref. [START_REF] Paillard | Photoinduced Phase Transitions in Ferroelectrics[END_REF] as 70 implemented in Abinit [START_REF] Gonze | [END_REF][START_REF] Romero | Abinit: Overview, and focus on selected 474 capabilities[END_REF][16][START_REF] Torrent | Implementation of the projector augmented[END_REF][START_REF] Bottin | Large-497 scale ab initio calculations based on three levels of par-498 allelization[END_REF], with a plane wave cut- [START_REF] Jollet | Generation of 501 Projector Augmented-Wave atomic data: A 71 element 502 validated table in the XML format[END_REF]20] and the AtomPAW software [START_REF] Holzwarth | A Projec-506 tor Augmented Wave (PAW) code for electronic struc-507 ture calculations, Part I: atompaw for generating atom-508 centered functions[END_REF]. The discretization of the Brillouin zone was performed using a Γ-centered mesh having 7×10 -3 Bohr -1 sampling density along each reciprocal lattice vector direction. Self-consistent field cycles were considered converged when the difference in forces between two iterations was smaller than 10 -8 Ha.Bohr -1 ; structural relaxation was performed until the maximum calculated force laid below 5 × 10 -7 Ha.Bohr -1 .

Fermi-Dirac distributions with a smearing temperature of 0.004 Ha characterize the occupation of electronic states.

The lattice constants obtained in the electronic ground state are in good agreement with lattice constants measured experimentally or calculated using the Local Density Approximation (LDA) [START_REF] Marronnier | Anhar-513 monicity and Disorder in the Black Phases of Cesium 514 Lead Iodide Used for Stable Inorganic Perovskite Solar 515 Cells[END_REF][START_REF] Liu | 517 Structural Properties and Stability of Inorganic CsPbI 3 518 Perovskites[END_REF] (see Supplementary Information [24]). The reciprocal space paths used for the band structures in Figure 1 are detailed in Ref. [START_REF] Setyawan | High-throughput elec-524 tronic band structure calculations: Challenges and 525 tools[END_REF] In order to simulate photo-excitation, we constrained a finite number n ph of electrons (resp. holes) to lie in the conduction (resp. valence) bands with a Fermi-Dirac distribution characterized by two separate quasi-Fermi levels for photo-excited electrons and holes.

III. PHOTO-INDUCED PHASE TRANSITIONS A. Structural and electronic properties in dark

We start by describing the electronic and energetic properties of CsPbI 3 in "dark" conditions (n ph = 0). The band structures of the δ and γ phases are depicted in Fig- ures 1a,d. The γ phase (Figure 1a) has a direct bandgap of 0.63 eV located at the Γ point. The electronic bandgap of the δ phase is indirect and 1.79 eV large. The top of the valence band is located on the Y Γ segment, while the bottom of the conduction band is close to X. The direct bandgap, depicted with black empty circles, is located near X in reciprocal space, and is 1.80 eV large.

In Table I, we summarize the values of bandgaps for the γ and δ phases, and compare with experimental and theoretical works. The predicted bandgaps calculated with SOC underestimate both experimental and theoretical values obtained without SOC, as consistent with the litterature [START_REF] Even | 528 Importance of Spin-Orbit Coupling in Hybrid Or-529 ganicInorganic Perovskites for Photovoltaic Applica-530 tions[END_REF][START_REF] Brgoch | 533 Ab Initio Calculations of Band Gaps and Absolute Band 534 Positions of Polymorphs of RbPbI 3 and[END_REF][START_REF] Tung | Strain Induced 554 Topological Insulator Phase in CsPbBrxI3-x (x = 0, 1, 2, 555 and 3) Perovskite: A Theoretical Study[END_REF].

The energy of the four phases without photoexcitation is depicted in Figure 2 (see points at n ph = 0). The lowest-energy structure is the δ-phase, which lies below the γ-phase by roughly 80 meV/f.u. This qualitatively differs from Ref. [START_REF] Marronnier | Anhar-513 monicity and Disorder in the Black Phases of Cesium 514 Lead Iodide Used for Stable Inorganic Perovskite Solar 515 Cells[END_REF], but agrees well with experimental observations of the δ-phase being the most stable state.

The difference with Ref. [START_REF] Marronnier | Anhar-513 monicity and Disorder in the Black Phases of Cesium 514 Lead Iodide Used for Stable Inorganic Perovskite Solar 515 Cells[END_REF] might be due to a different exchange-correlation functional used. We note, furthermore, that the relative energies of the phases δ, γ, β and α agree well with the observed phase transitions upon cooling. [START_REF] Valastro | Optical behaviour of γ-black CsPbI3 phases formed 422 by quenching from 80 •C and 325 •C[END_REF][START_REF] Steele | Thermal unequilib-435 rium of strained black CsPbI 3 thin films[END_REF][START_REF] Liu | 517 Structural Properties and Stability of Inorganic CsPbI 3 518 Perovskites[END_REF][START_REF] Sutton | Cubic or 559 Orthorhombic? Revealing the Crystal Structure of 560 Metastable Black-Phase CsPbI 3 by Theory and Exper-561 iment[END_REF][START_REF] Yao | Research and 563 progress of black metastable phase CsPbI 3 solar cells[END_REF] showed that photo-excited carriers had a strong impact 152 on phonon instabilities. [START_REF] Paillard | Photoinduced Phase Transitions in Ferroelectrics[END_REF][START_REF] Peng | Topological phonons in oxide perovskites con-572 trolled by light[END_REF] In particular, in cubic 153 lead titanate and barium titanate, it had been found that 154 phonon instabilities related to the development of electri-155 cal polarization were suppressed under illumination [START_REF] Paillard | Photoinduced Phase Transitions in Ferroelectrics[END_REF].

δ-phase γ-phase E g,indirect E g,direct E g,direct
156

Here, the phonon spectrum of the cubic α phase (see Sup- erate semiconductors [START_REF] Pasynkov | On some problems of the phenomenolog-575 ical theory of ferroelectric-semiconductors[END_REF] or to explain photostriction in 173 ferroelectric oxides [START_REF] Paillard | Ab initio[END_REF], one may simply say that the dif-174 ference of energy between the two phases is:

E γ (n ph ) -E δ (n ph ) = ∆E 0 + δE g (n ph ).n ph (1)
In the above equation, ∆E 0 is the energy difference 176 between the δ and γ phase in "dark" conditions (roughly 177 80 meV/f.u.); δE g (n ph ) = E g,γ (n ph ) -E g,δ (n ph ) denotes 178 the difference of bandgaps between the γ and δ phases,

IV. PHOTOSTRICTION

We also looked at the photostrictive response of CsPbI 3 . It is given in Table II, in which estimates use the value of the photo-induced strain at n ph = 0.025 e/f.u. in Figure 3. We chose such value because it is small enough to be in a linear regime for most phases in CsPbI 3 . Consequently, we plot, in Figure 3, the photo-induced strain for all four phases of CsPbI 3 considered in this work. [START_REF] Paillard | Ab initio[END_REF]. Note also that we report in Table II, photo-induced strains from ∆SCF calculations [START_REF] Paillard | Ab initio[END_REF]44,46] corresponding to indirect transitions to match as closely as possible the results that would be obtained using the constrained-DFT method employed here.

We must also stress that at large concentrations in photo-excited carriers (n ph > 0.1 e/f.u.), the photoinduced response can become highly non-linear, especially in the β and α phases. While large photo-induced strain is observed across all phases in the large n ph regime, Figure 3 and Table II show that the δ phase possesses the largest photomechanical sensitivity under low illumination conditions.

A recent study has attributed the large photostriction in cubic cesium lead halide perovskite to the strong antibonding nature of the bottom of the conduction band. (

FIG. 3. Photo-induced strain in lead cesium iodide, along the principal crystallographic axes of each phase (γ), (β), (δ) and (α). Dashed open symbols in the δ and γ phases show calculations with SOC.

Material

Phase δa/a δb/b δc/c % / (e/f.u

.)] CsPbI3 α -0.02 - - β 0.5 - -0.8 γ 0.2 1.4 1.5 δ, -9.9 15.3 -5.

BiFeO3 (perovskite [44])

R3c -1.5 --BaTiO3 (perovskite [START_REF] Paillard | Ab initio[END_REF] )

R3m -0.2 - - PbTiO3 (perovskite [38]) P 4mm -1.0 - -3.1 SrCoO2.5 (brownmillerite [45]) Ima2 - - +0.4 SnS (2D ferroelectric [46])
-56.9 14.2-28.4 -TABLE II. Estimates of the linear photo-induced strain in lead cesium iodide calculated using the constrained-DFT method (no SOC/SOC values reported). Other reported compounds were calculated using the ∆SCF (Self-Consistent Field) method.

V. CONCLUSION 286

The constrained-DFT approach employed in the We estimate the photo-excited carrier concentration.

Considering the power balance sketched in Figure 4, the total absorbed energy from a laser of fluence I 0 impinging on a sample of cross-section S, thickness d, having absorption and reflection coefficient α and R is

E tot = (1 -R)I 0 (1 -e -αd )S (A1)
As a result, the total number of photons absorbed is

N photons = E tot hω = (1 -R)I 0 (1 -e -αd )S hω , (A2)
with hω the average photon energy within the incident beam. Assuming that each absorbed photon yields an electron -hole pair, we would thus have a photo-excited carrier density, expressed in e/f.u. of

n ph = N photons Sd × V δ 4 , (A3)
where V δ is the volume of the δ phase unit cell. The factor 1/4 comes from the fact that there are 4 formula units in the δ phase unit cell.

Assuming, for the sake of simplicity, that photons in the laser beam carry an energy hω = 3 eV (413 nm), we take for the δ phase a value of α ≈ 2 × 10 5 cm -1 [START_REF] Valastro | Optical behaviour of γ-black CsPbI3 phases formed 422 by quenching from 80 •C and 325 •C[END_REF]. We

  control parameters. Methods allowing to easily and effi-42 ciently control the appearance of the perovskite phase in 43 CsPbI 3 are thus highly desirable. 44 In the present work, we study the relative stability 45 of the main four phases of CsPbI 3 : the δ-phase, which 46 is the non-perovskite ground state; the γ-phase which 47 is the low-energy orthorhombic perovskite phase; the β-48 phase is an intermediate tetragonal perovskite structure, 49 while the α-phase is the cubic prototypical and high-50 temperature perovskite. All four phases are depicted in 51 the Supplementary Information.

52

  Density Functional Theory (DFT) calculations are em-53 ployed in an original scheme which constrains excited 54 electrons (resp. holes) in the conduction (resp. valence) 55 bands during the self-consistent cycle to mimic thermal-56 ized photo-excited carriers. Our results indicate that 57 above bandgap light may help tune the relative stabil-58 ity of the γ and δ phases. Even further, knowing that 59

71

  off of 40 Ha. The Perdew-Burke-Ernzerhof (PBE) ex-72 change correlation functional revised for solids (named 73 PBESol) was employed. All calculations reported here 74 include spin-orbit coupling (SOC). Plane Augmented 75 Wave (PAW) pseudopotentials were regenerated for 76 PBESol using the PBE pseudopotential parameters used in the Jollet-Torrent-Holzwarth (v1.1) table

FIG. 1 .

 1 FIG. 1. Band structures obtained in the γ (a-b) and δ (c-d) phases for n ph = 0.00 e/f.u. (electronic ground state; a and c) and n ph = 0.6 e/f.u. (b and d) respectively; orange plain circles indicate the location of the electronic bandgap, and black empty circles depict the direct bandgap.

  157plementary Information) shows that photo-excited carri-158 ers hardly modify the unstable phonon branches. This is 159 consistent with the fact that no other perovskite-phase 160 becomes more stable than the γ phase in Figure2. There 161 does not seem to be any other perovskite phase that could 162 develop under illumination besides the γ phase, which is 163 itself dynamically stable[START_REF] Jain | The Materials Project: A materials 440 genome approach to accelerating materials innovation[END_REF]. 164 C. Toy model 165 As a result, the case could be made that the proposed 166 phase transition origin lies in the difference of electronic 167 bandgaps of the δ and γ phases. In a two-level system 168 (e.g. limiting ourselves to the bottom of the valence band 169 and top of the conduction band), a simple toy model can 170 be designed to further prove this point. Akin to the ex-171 tension of Landau's theory to the description of degen-172

179 and n

  photo-excited carriers are distributed across more than 189

  a, b and c refer to the lattice constants of the principal axes sorted by increasing length. In the case of the α cubic phase for which a = b = c, we report only a, while for the β tetragonal phase, a = b = c, and we report only a and c. Interestingly, and as also summarized in

[ 47 ]

 47 Our results indicate that the strong photo-induced strain permeates through all phases of CsPbI 3 , at least in the large n ph regime. The large photo-induced strain in CsPbI 3 , and in general in hybrid perovskites [48], may find applications in improving the photovoltaic power conversion efficiency [49]. Interestingly, to achieve photostriction comparable to the 0.125% expansion reported in MAPbI 3 films [48], one could use 413 nm wavelength laser pulses of 8.5 mJ.cm -2 .

287

  photo-induced phase transition towards the γ perovskite 295

  Appendix A: Photo-induced carrier estimate

  FIG. 4. Sketch of what happens to a ray of light of intensity I0 impinging on a material.

342

  This allows us to obtain a crude estimate of the value 343 of n ph depending on the pump fluence, which we plot 344 in Figure5. Note that in Figure5, we assumed a sam-345 ple uniformly illuminated of thickness 323 nm, area size 346 S = 0.0625 cm 2 , which is submitted to a laser pulse car-347 rying an energy up to 4 mJ. The geometrical parameters 348 involved were taken from photostriction experiments car-349 ried out on thin films of MAPbI 3 [35]. One observes that 350 reasonable pulse energy can lead to the desired concen-351 tration of photo-excited carriers in thin films. 352 Note that in the previous simple derivations, we as-353 sumed that every electron-hole pair relaxes into thermal-354 ized carriers in their respective bands (valence bands for 355 holes; conduction bands for electrons). We also assume 356 that the recombination time of excited carriers is large 357 enough compared to the pulse duration. Typical relax-358 ation times in CsPbI 3 are of the order of a few nanosec-359 onds [52, 53] (depending on the sample quality). Picosec-360 ond laser pulses could therefore be suitable this model. 361 A more complex model would solve a balance equation or 362 the real time Bethe-Salpether equation to obtain a more 363 accurate relation between the pump fluence and the av-364 erage number of photoexcited carriers.

FIG. 5 .

 5 FIG.5. Estimated photo-induced carrier concentration induced by a 3 eV laser pulse. Dotted blue and red line correspond to the concentration necessary to achieve the δ -→ γ transition with and without SOC, according to Figure2of the main manuscript.