Exact integration for products of power of barycentric coordinates over $d$-simplexes in $R^n$

HAL Id: hal-01816347

https://hal-univ-paris13.archives-ouvertes.fr/hal-01816347

Preprint submitted on 15 Jun 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Exact integration for products of power of barycentric coordinates over d-simplexes in R

Francois Cuvelier

To cite this version:

Francois Cuvelier. Exact integration for products of power of barycentric coordinates overd-simplexes inRⁿ. 2018. �hal-01816347�

Exact integration for products of power of barycentric coordinates over d -simplexes in R

François Cuvelier^∗ 2018/06/15

Abstract

Exact integral computation over a d-simplex in Rⁿ for products of powers of its barycentric coordinates is done in [9] by using mathematical induction and coordinate mappings. In this note we give a new proof using Laplace transformations without mathematical induction.

Contents

1 Notations and denitions 2

2 Some results on the unit d-simplex 3

2.1 unitd-simplex volume . . . . 3

2.2 Magic formula . . . . 5

3 Some results on a d-simplex in Rⁿ 6 3.1 Gradients of Barycentric coordinates on ad-simplex . . . . 6

3.2 Integration over ad-simplex . . . . 8

3.3 Volume of ad-simplex . . . 11

3.4 Magic formula . . . 11

∗Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS UMR 7539, 99 Avenue J-B Clé- ment, F-93430 Villetaneuse, France, [email protected]

This work was supported in part by the ANR project DEDALES under grant ANR-14- CE23-0005.

Local shape functions of a large variety of nite element on a d-simplex KĂRⁿcan be expressed in function of the barycentric coordinatestλ₀, . . . , λ_du ofK and their derivatives (see [1] for examples).

In [9], the authors give a proof of the magic formula: letννν “ pν0, . . . , νdq P N^d`1,then

ż

K d

ź

i“0

λ^ν_iⁱpqqqqdqqq“d!|K|

d

ś

i“0

ν_i! pd`

d

ř

l“0

νiq!

(1)

where |K| is the volume ofK. In their proof, mathematical induction and co- ordinate mappings are mainly used. In this note we give a new proof of this formula using Laplace transformations without mathematical induction.

Firstly we recall denitions of ad-simplex in Rⁿ and of its barycentric co- ordinates. Therafter we introduce Laplace transforms to compute the volume of the unit d-simplex Kˆ ĂR^d and the magic formula (1) over K.ˆ In the last section, we propose to compute the gradients of the barycentric coordinates by solving linear systems. We also present the mapping of an integral over a d-simplex inRⁿ to the reference unit d-simplex, allowing to proove (1).

1 Notations and denitions

LetnPN^˚ be the space dimension anddP v0, nw.We recall the denition of a d-simplex inRⁿ as well as its barycentric coordinates.

Denition 1 (d-simplex) A d-simplexK ĂRⁿ is the convex hull of pd`1q pointsqqqqqqqqq⁰, . . . ,qqqqqqqqq^dof Rⁿ which form the vertices of K.

K“

# q

qqPRⁿ |qqq“

d

ÿ

i“0

θ_iqqqqqqqqqⁱ, with@iP v0,dw, θ_iě0, and

d

ÿ

i“0

θ_i“1 +

. (2) For example, a 2-simplex is a triangle and a3-simplex is a tetrahedron. It will be always assumed that ad-simplex is not degenerated, i.e., the set of vectors tqqqqqqqqqⁱ´qqqqqqqqq⁰u^d_i“1is linearly independent.

Denition 2 (Barycentric coordinates) Let K Ă Rⁿ be a non-degenerate d-simplex and tqqqqqqqqqⁱu^d_i“0 its vertices. The parametrization of K with a convex combination of the vertices reads as follows

K“

# q

qqPRⁿ |qqq“

d

ÿ

i“0

λ_ipqqqqqqqqqqqqqⁱ, with @iP v0,dw, λ_ipqqqq ě0, and

d

ÿ

i“0

λ_ipqqqq “1 +

.

The coecients λ0pqqqq, . . . , λdpqqqqare called the barycentric coordinates onK(3)of q

q q.

As immediat property, the barycentric coordinates on Ksatisfy

λ_ipqqqqqqqqq^jq “δ_i,j, @pi, jq P v0,dw. (4)

2 Some results on the unit d-simplex

The unit d-simplexKˆ^dĂR^d is dened by thed`1vertices ˆq

qqˆ q qqˆ

qqq⁰,qqqqqqˆˆqˆqq¹,¨ ¨ ¨ ,qqqˆqˆqqqqqˆ^d(

“ 000,ˆeeeˆˆ¹,¨ ¨ ¨,eeˆˆˆe^d( where ˆeeeˆˆ¹, . . . ,eˆˆeeˆ^d(

is the standard basis ofR^d.We have Kˆ^d“

# ˆ q q

qPR^d|qqˆq“

d

ÿ

i“0

λˆ_ipˆqqqqˆqqqqˆqqqqqˆⁱ, withλˆ_ipˆqqqq ě0, and

d

ÿ

i“0

ˆλ_ipˆqqqq “1 +

. (5) As immediat property, the barycentric coordinatespˆλiq^d_i“0onKˆ^dsatisfy

ˆλipˆqqqqqqˆqˆqq^jq “δi,j, @pi, jq P v0,dw. (6) and are explicitly given withqˆqq“ px1,¨ ¨ ¨ , xdq^tPKˆ^dby

λˆ0pˆqqqq “1´

d

ÿ

i“1

xi and@iP v1,dw, ˆλipˆqqqq “xi. (7) Indeed, asqqqˆqˆqqˆqqq⁰“000,we have

ˆ q q q“

d

ÿ

i“0

ˆλ_ipˆqqqqˆqqqqqqˆqˆqqⁱ“

d

ÿ

i“1

ˆ qqqqqqqqˆˆqⁱˆλ_ipˆqqqq Fromqqqˆqˆqqqqqˆⁱ“ˆeˆeeˆⁱ, @iP v1,dw, we obtain

d

ÿ

i“1

ˆ q q qˆ qqqˆ

qqqⁱλˆipˆqqqq “

¨

˝ qˆqqˆqqqqqˆq¹ ¨ ¨ ¨ qˆqqqqqqqˆˆq^d

˛

‚

¨

˚

˝ λˆ1pqqˆqq

... λˆ_dpˆqqqq

˛

‹

‚“Id

¨

˚

˝ ˆλ1pqˆqqq

... λˆ_dpˆqqqq

˛

‹

‚“

¨

˚

˝ λˆ1pqqˆqq

... λˆ_dpˆqqqq

˛

‹

‚

and thus

ˆ qqq“

¨

˚

˝ x₁

... xd

˛

‹

‚“

¨

˚

˝ λˆ₁pˆqqqq

... ˆλdpqqˆqq

˛

‹

‚.

From (5), we have

d

ÿ

i“0

ˆλipˆqqqq “1 and thus

ˆλ0pˆqqqq “1´

d

ÿ

i“1

λˆipˆqqqq “1´

d

ÿ

i“1

xi.

2.1 unit d-simplex volume

There are several ways to compute the volume |K|ˆ of the d-simplex Kˆ Ă R^d which is given by the following integral:

|K| “ˆ ż

Kˆ

1dˆqqq“ ż1

0

ż1´x₁ 0

ż1´x₁´x₂ 0

. . .

ż1´px₁`...`x_d´1q 0

1dxd. . . dx3dx2dx1.

An elegant way to perform this integration is explained in [6], section 18.10, and uses a Laplace transform. To use this method, we note that

Kˆ “R^d`X t1´ px1`. . .`xdq ě0u. (8) So we also have

|K| “ˆ ż

R^d<sub>`</sub>Xt1´px<sub>1</sub>`...`x_dqě0u

1dxd. . . dx1.

By using a dirac function and extending the integration domain to R<sup>d`1</sup>` , we also have

|K| “ˆ ż

R<sup>d`1</sup>`

δpx1`. . .`xd`xd`1´1qdxd`1dxd. . . dx1

To use the Laplace transform theory, we dene the function f by fptq “

ż

R<sup>d`1</sup><sub>`</sub>

δpx1`. . .`xd`xd`1´tqdxd`1dxd. . . dx1

so that|K| “ˆ fp1q.The Laplace transform off is given by Lpfqpsq “

ż8 0

fptqe^´stdt

“ ż

R<sup>d`1</sup><sub>`</sub>

ˆż₈

0

δpx₁`. . .`x_d`x<sub>d`1</sub>´tqe^´stdt

˙

dx_d`1dx_d. . . dx₁

“ ż

R<sup>d`1</sup><sub>`</sub>

expp´s

d`1

ÿ

i“1

x_iqdx_d`1dx_d. . . dx₁

“

d`1

ź

i“1

ż8 0

expp´sx_iqdx_i

“ 1 s^d`1.

By using the inverse Laplace transform table (see [8] for example), we have L^-1psÞÑ d!

s^d`1qptq “t^d.

Asf “L^-1˝Lpfqand by linearity of the inverse Laplace transform we obtain fptq “t^d

d!. So the volume of the unitd-simplex is

|K| “ˆ 1

d! (9)

2.2 Magic formula

Letννν“ pν0, . . . , νdq PN^d`1.The magic formula is given by ż

Kˆ d

ź

i“0

λˆ^ν_iⁱpˆqqqqdˆqqq“ śd

i“0νi! pd`řd

i“0ν_iq! (10)

This formula is often used in P¹-Lagrange nite element methods because P¹- Lagrange basis functions on a d-simplex are the associated barycentic coordi- nates. For example, one can refer to [7] (section 8.2.1, page 179, formulap8.3q), [9], [4] section 7.3.3 page 126, [3] for d P v1,3w, [2] as exercise for d “ 2 and d“3. In this section, we propose a proof of this formula using Laplace trans- form theory. LetIpνqˆ denote the integral of (10). The barycentic coordinates λˆi are given in (7) and so withqqˆq“ px1, . . . , xdqand using (8) we obtain

Ipνq “ˆ ż

Kˆ

p1´

d

ÿ

i“1

x_iq^ν0

d

ź

i“1

x^ν_iⁱdx_d. . . dx₁

“ ż

R^d<sub>`</sub>Xt1´px<sub>1</sub>`...`x_dqě0u

p1´

d

ÿ

i“1

x_iq^ν0

d

ź

i“1

x^ν_iⁱdx_d. . . dx₁

From section 2.1, by using a dirac function and by extending the integration domain to R<sup>d`1</sup>` we obtain withνd`1“ν0

Ipνˆ q “ ż

R<sup>d`1</sup>`

δpx1`. . .`xd`xd`1´1qx^ν_d`1⁰

d

ź

i“1

x^ν_iⁱdxd`1dxd. . . dx1

“ ż

R<sup>d`1</sup><sub>`</sub>

δpx1`. . .`xd`xd`1´1q

d`1

ź

i“1

x^ν_iⁱdxd`1dxd. . . dx1

To use the Laplace transform theory, we dene the function fννν by f_νννptq “

ż

R<sup>d`1</sup><sub>`</sub>

δpx₁`. . .`x_d`x<sub>d`1</sub>´tq

d`1

ź

i“1

x^ν_iⁱdx_d`1dx_d. . . dx₁

so thatIpνˆ q “fνννp1q. The Laplace transform offννν is given by Lpfνννqpsq “

ż8 0

fνννptqe^´stdt

“ ż

R<sup>d`1</sup>`

ˆż8 0

δpx1`. . .`xd`xd`1´tqe^´stdt

˙d`1

ź

i“1

x^ν_iⁱ dxd`1dxd. . . dx1

“ ż

R<sup>d`1</sup>`

expp´s

d`1

ÿ

i“1

xiq

d`1

ź

i“1

x^ν_iⁱ dxd`1dxd. . . dx1

“

d`1

ź

i“1

ż8 0

x^ν_iⁱexpp´sxiqdxi

“

d`1

ź

i“1

LptÞÑt^νiqpsq

In a classical Laplace transform table (see [8] for example), we have LptÞÑ t^k

k!qpsq “ 1 s^k`1 and by linearity of the Laplace transform

LptÞÑt^kqpsq “ k!

s^k`1. So we obtain

Lpfνννqpsq “

d`1

ź

i“1

νi! s^νi`1 “

śd`1 i“1 νi! s<sup>d`1`řd`1i“1νi</sup>

By using the inverse Laplace transform table, we have L^-1psÞÑ 1

s^kqptq “ t^k´1 k´1.

With the linearity of the inverse Laplace transform we obtain fνννptq “L^-1pLpfνννqpsqqptq

“

śd`1 i“1 νi! pd`řd`1

i“1ν_iq!t^{d`řd`1i“1νi}. AsIpνq “ˆ fνννp1qandνd`1“ν0,the equation (10) is proved.

3 Some results on a d-simplex in Rⁿ

3.1 Gradients of Barycentric coordinates on a d-simplex

Lemma 3 Let K Ă Rⁿ be a non-degenerate d-simplex and and tqqqqqqqqqⁱu^d_i“0 its vertices. The barycentric coordinatespλipqqqqq^d_i“0are solution of the linear system

¨

˚

˚

˚

˝

1 1 ¨ ¨ ¨ 1 0

... A^t_KAK

0

˛

‹

‹

‹

‚

¨

˚

˚

˚

˚

˝ λ₀pqqqq λ₁pqqqq

... λ_dpqqqq

˛

‹

‹

‹

‹

‚

“

¨

˚

˚

˝ 1 AKpqqq´qqqqqqqqq⁰q

˛

‹

‹

‚ (11) whereA_KPMn,dpRqis dened by

A_K“

¨

˝ qqqqqqqqq¹´qqqqqqqqq⁰ ¨ ¨ ¨ qqqqqqqqq^d´qqqqqqqqq⁰

˛

‚ (12)

The barycentric coordinates are multivariate polynomials of rst degree and their gradients are given by

` ∇∇∇λ1pqqqq ¨ ¨ ¨ ∇∇∇λdpqqqq ˘

“AKpA^t_KAKq^-1 (13) and

∇∇∇λ0pqqqq “ ´

d

ÿ

i“1

∇

∇∇λipqqqq. (14)

Proof: Asřd

i“0λ_ipqqqq “1,we have qqq“

d

ÿ

i“0

λipqqqqqqqqqqqqqⁱùñqqq´qqqqqqqqq⁰“

d

ÿ

i“1

pqqqqqqqqqⁱ´qqqqqqqqq⁰qλipqqqq “AK

¨

˚

˝ λ₁pqqqq

... λ_dpqqqq

˛

‹

‚

Due to linear independence oftqqqqqqqqqⁱ´qqqqqqqqq⁰u^d_i“1,

HK^def“A^t_KAK PMd,dpRq (15) is a regular matrix and the barycentric coordinates are solution of the linear system

A^t_KA_K

¨

˚

˝ λ₁pqqqq

... λdpqqqq

˛

‹

‚“A^t_Kpqqq´qqqqqqqqq⁰q and

d

ÿ

i“0

λ_ipqqqq “1.

In matrix form these equations can be written as (11) and we deduce that the barycentric coordinatesλ_iare multivariate polynomials of rst degree. So their gradients are constants onK.

The ane map/transformation FK from the unit d-simplex Kˆ Ă R^d to KĂRⁿ is given by

q q

q“AKqˆqq`qqqqqqqqq⁰“FKpˆqqqq. (16) So we have

A^t_Kpqqq´qqqqqqqqq⁰q “A^t_KAKqˆqq“HKˆqqq and thusF_K^-1:KĂRⁿÝÑKˆ ĂR^d is dened by

ˆ q q

q“H^-1_KA^t_Kpqqq´qqqqqqqqq⁰q “F_K^-1pqqqq. (17) So we have

λ_ipqqqq “ pˆλ_i˝F_K^-1qpqqqq and λˆ_ipˆqqqq “ pλ_i˝F_Kqpˆqqqq (18) One can remark that ifd“nthenAK is a regular square matrix andH^-1_KA^t_K “ A^-1_K.

Now, we may compute partial derivative ofλi and @i P v0,dw, @j P v1, nw, we obtain withqqˆq“ pˆx1, . . . ,xˆdqandqqq“ px1, . . . , xnq

Bλi

Bx_jpqqqq “

d

ÿ

l“1

Bλˆi

Bˆx_jpF_K^-1pqqqqqBF_K,l^-1 Bx_j pqqqq

From (17), denoting B_K “H^-1_KA^t_K P M_d,mpRq gives ^BF_Bx^K,l-1_j pqqqq “ pB_Kql,j. The barycentric coordinates are polynomials of rst degree, so their gradients are constants and we obtain

∇∇∇λi“B^t_K∇∇∇ˆλˆi

(in fact B_K is the Jacobian matrix of F_K^-1). The matrix H_K is regular and symmetric, soB^t_K “AKH^-1_K and we obtain

∇∇∇λi“AKH^-1_K∇∇∇ˆλˆi. (19)

From (7), we deduced that

´ ∇∇∇ˆˆλ₁ ¨ ¨ ¨ ∇∇∇ˆλˆ_d

¯

“I_d (20)

and thus

` ∇∇∇λ1pqqqq ¨ ¨ ¨ ∇∇∇λdpqqqq ˘

“AKH^-1_K

´ ∇∇∇ˆλˆ1 ¨ ¨ ¨ ∇∇∇ˆˆλd

¯

“A_KH^-1_K. Asřd

i“0λipqqqq “1,we immediately have

∇∇∇λ0pqqqq “ ´

d

ÿ

i“1

∇

∇∇λipqqqq.

˝ From (13) and (14), we immediatly have:

Remark 4 The gradients of the barycentric coordinates are linear combinations of tqqqqqqqqq¹´qqqqqqqqq⁰, . . . ,qqqqqqqqq^d´qqqqqqqqq⁰u.

3.2 Integration over a d-simplex

If K is a non-degenerated d-simplex inR^d, from (16) we have J_FKpˆqqqq “AK. ThenAK is a regular square matrix and we have the classical formula:

ż

K

fpqqqqdqqq“ |detpA_Kq|

ż

Kˆ

f˝F_Kpˆqqqqdˆqqq (21) The following theorem extend this result tod-simplex inRⁿ,with1ďdďn.

Theorem 5 Let KĂRⁿ be a non-degeneratedd-simplex andf :KÝÑR. ż

K

fpqqqqdqqq“ˇ

ˇdetpA^t_KA_Kqˇ ˇ

1{2ż

Kˆ

f˝F_Kpˆqqqqdˆqqq (22) whereK is the unit d-simplex inRⁿ, AKPMd,npRqis dened by

AK “

´

qqqqqqqqq¹´qqqqqqqqq⁰ qqqqqqqqq²´qqqqqqqqq⁰ ¨ ¨ ¨ qqqqqqqqq^d´qqqqqqqqq⁰

¯ (23)

andFK : ˆKÝÑK is given by

F_Kpˆqqqq “A_Kqqˆq`qqqqqqqqq⁰ (24) Proof: The set vvv¹, . . . , vvv^d(

is linearly independent so we can extend it to a basis vvv¹, . . . , vvvⁿ(

. We denote by AP Mn,npRqthe matrix such that the i-th column is the vectorvvvⁱ for alliP v1, nw.So we have

A“

´

AK vvv^d`1 ¨ ¨ ¨ vvvⁿ

¯ (25)

By the QR-factorization theorem apply to the matrix A P M_npRq, there is an orthogonal matrix Q PMnpRq and a regular upper triangular matrixR P MnpRqsuch that

A“QR

So we have

Q^tA“R

and we dene the matrixA¯PMn,dpRqto be the rstdcolumns ofR: A¯“Q^tAK.

We can also note that A¯“

´

¯ q qq¯

qqqqqq¯¹´qqq¯q¯qq¯qqq⁰ qqq¯q¯qqqqq¯²´qqq¯q¯qq¯qqq⁰ ¨ ¨ ¨ qqq¯q¯qq¯qqq^d´q¯qqqqqqq¯¯q⁰

¯

“

´

¯ q q q¯ qqq¯

qqq¹ qqq¯q¯qq¯qqq² ¨ ¨ ¨ qqq¯q¯qqqqq¯^d

¯ . LetF¯:Rⁿ ÝÑRⁿ be the bijective function dened by

Fpx¯ xxq “Q^tpxxx´qqqqqqqqq⁰q “xxx¯ (26) and

q¯qqqqqqq¯¯qⁱ“Fp¯qqqqqqqqqⁱq “Q^tpqqqqqqqqqⁱ´qqqqqqqqq⁰q, @iP v0,dw.

By constructionq¯qqqqqqq¯¯q⁰ “ 000 and, @i P v1, dw, qqqqqq¯¯q¯qqⁱ is the i-th column of the upper triangular matrix R.So we have

@iP v0,dw, qqq¯q¯qq¯qqqⁱPVectpeee¹, . . . , eee^dq where eee¹, . . . , eeeⁿ(

is the standard basis of Rⁿ. The set q¯qqqqqqq¯¯q⁰, . . . ,q¯qq¯qqqqq¯q^d(

are the vertices of thed-simplex K¯ “F¯pKqand we deduce

K¯ ĂVectpeee¹, . . . , eee^dq. (27) By change of variables, we obtain

ż

K

fpqqqqdqqq“ ż

K¯

f ˝F¯^-1p¯qqqq|detpJF¯^-1p¯qqqqq|d¯qqq

whereJF¯^-1 is the Jacobian matrix ofF¯^-1.From (26), we haveJF¯^-1p¯qqqq “Qand as Qis an orthogonal matrix, detpJF¯^-1pqqqqq “¯ 1.So we obtain

ż

K

fpqqqqdqqq“ ż

K¯

f ˝F¯^-1p¯qqqqd¯qqq. (28) LetPPMd,npRqdened by

P“`

Id Od,n´d

˘ and

@iP v0,dw, qqq¯¯q¯¯qq¯¯qqqⁱ“P¯qqq¯qqqqq¯qⁱ PR^d. From (27), we deduce

@iP v0,dw, ¯qqqqqq¯q¯qqⁱ“P^tq¯¯qq¯¯qqqqq¯¯qⁱ “ ˆ q¯¯qqqqqqq¯¯¯¯qⁱ

000

˙ .

Letg¯“f˝F¯^-1 andK¯¯ be thed-simplex in R^d with verticesqqq¯¯q¯¯qq¯¯qqqⁱ, iP v0,dw. We denote byP : ¯K¯ ĂR^dÝÑK¯ ĂRⁿ the application dened byPpqq¯¯qq “P^tq¯¯qq.We denote byg¯¯: ¯K¯ ÝÑRthe application dened by

¯¯

gp¯¯qqqq “¯g˝Ppq¯¯qqq “¯g ˆ qq¯¯q

000

˙ .