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Regularized Kelvinlets: Sculpting Brushes based on Fundamental Solutions of Elasticity Supplemental Material FERNANDO DE GOES, Pixar Animation Studios DOUG L. JAMES, Pixar Animation Studios, Stanford University ACM Reference format: Fernando de Goes and Doug L. James. 2017. Regularized Kelvinlets: Sculpt- ing Brushes based on Fundamental Solutions of Elasticity Supplemental Material . ACM Trans. Graph. 36, 4, Article 1 (July 2017), 3 pages. DOI: http://dx.doi.org/10.1145/3072959.3073595 A Derivatives of 3D Regularized Kelvinlets We rst provide the expressions for the derivatives of regularized Kelvinlets and locally ane regularized Kelvinlets in 3D. Grab brush: By direct dierentiation of Eq. (6) in the main text, we obtain the gradient of the 3D displacement eld u ε ( r ): u ε ( r ) = 1 r 3 ε b ( ( f t r ) I + rf t + fr t ) - a fr t - 3 r 5 ε h a 2 ε 2 f + b ( f t r ) r i r t . (1) Observe that, at x 0 , we have u ε (0) = 0, thus indicating that the brush center moves rigidily. Locally Aine brush: By dierentiating Eq. (12) of the main text, we now obtain the gradient of the 3D displacement eld e u ε ( r ): e u ε ( r ) = 1 r 3 ε b ( F t + F + tr(F )I ) - aF - 3 r 5 ε h a 2 ε 2 F + b ( r t Fr )I + b rr t ( F + F t ) i - 3 r 5 ε b ( F t + tr(F )I + F ) - aF rr t + 15 r 7 ε h a 2 ε 2 F + b ( r t F r )I i rr t . (2) At the brush center x 0 , this expression simplies to: e u ε (0) = 1 ε 3 b ( F t + F + tr(F )I ) - 5 2 a F . (3) Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for prot or commercial advantage and that copies bear this notice and the full citation on the rst page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). © 2017 Copyright held by the owner/author(s). 0730-0301/2017/7-ART1 $15.00 DOI: http://dx.doi.org/10.1145/3072959.3073595 Twist brush: In this case, we have a skew-symmetric force matrix F , which implies F + F t = 0 and tr(F ) = 0. This matrix can then be written as a cross product matrix [ q] × of a vector q, and the gradient at x 0 simplies to t ε (0) = - 5 2 a ε 3 [ q] × . (4) Note that this gradient matrix is skew-symmetric, thus indicating that the deformation at the brush center is a (innitesimal) rotation and can be controlled by the three DoFs provided by the vector q. Scale brush: This case has a force matrix F = s I and then s ε (0) = s (2 b - a) 5 2 1 ε 3 I . (5) Therefore, the brush center x 0 is locally deformed by a uniform scaling, with a single DoF. Pinch brush: The case of a pinch brush has a symmetric and trace- less force matrix, i.e., F + F t = 0 and tr(F ) = 0. This implies that the gradient x 0 is of the form p ε (0) = 1 2ε 3 (4 b - 5a ) F . (6) As a consequence, the deformation at x 0 is also symmetric with zero trace, in a total of ve DoFs. B Derivatives of 2D Regularized Kelvinlets We can also compute the derivatives for regularized Kelvinlets and locally ane regularized Kelvinlets in 2D, following the dierentia- tion of Eqs. (23) and (24) in the main text. The resulting expressions and their properties are very similar to the 3D case. Grab brush: u ε (r) = 2 r 2 ε b ( ( f t r) I + r f t + f r t ) - a f r t - 2 r 5 ε 2 f + 2 b ( f t r)r r t . (7) At x 0 , we have u ε (0) = 0, thus indicating that the brush center is locally under a rigid deformation. ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.

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Page 1: Regularized Kelvinlets: Sculpting Brushes based on Fundamental ... - Pixargraphics.pixar.com/library/Kelvinlets/gradDerivation.pdf · Regularized Kelvinlets: Sculpting Brushes based

Regularized Kelvinlets:Sculpting Brushes based on Fundamental Solutions of ElasticitySupplemental Material

FERNANDO DE GOES, Pixar Animation Studios

DOUG L. JAMES, Pixar Animation Studios, Stanford University

ACM Reference format:

Fernando de Goes and Doug L. James. 2017. Regularized Kelvinlets: Sculpt-ing Brushes based on Fundamental Solutions of Elasticity SupplementalMaterial . ACM Trans. Graph. 36, 4, Article 1 (July 2017), 3 pages.

DOI: http://dx.doi.org/10.1145/3072959.3073595

A Derivatives of 3D Regularized Kelvinlets

We �rst provide the expressions for the derivatives of regularizedKelvinlets and locally a�ne regularized Kelvinlets in 3D.

Grab brush: By direct di�erentiation of Eq. (6) in the main text, weobtain the gradient of the 3D displacement �eld uε (r ):

∇uε (r ) =1r3ε

[b

((f tr ) I + r f t + f r t

)− a f r t

]−

3r5ε

[a2 ε

2 f + b (f tr )r]r t .

(1)

Observe that, at x0, we have ∇uε (0) = 0, thus indicating that thebrush center moves rigidily.

Locally A�ine brush: By di�erentiating Eq. (12) of the main text,we now obtain the gradient of the 3D displacement �eld uε (r ):

∇uε (r )=1r3ε

[b

(F t + F + tr(F )I

)− aF

]−3r5ε

[a2 ε

2F + b (r t Fr )I + b rr t(F + F t

) ]−3r5ε

[b

(F t + tr(F )I + F

)− aF

]rr t

+15r7ε

[a2 ε

2F + b(r t Fr )I]rr t .

(2)

At the brush center x0, this expression simpli�es to:

∇uε (0) =1ε3

[b

(F t + F + tr(F )I

)−52a F

]. (3)

Permission to make digital or hard copies of part or all of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor pro�t or commercial advantage and that copies bear this notice and the full citationon the �rst page. Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).© 2017 Copyright held by the owner/author(s). 0730-0301/2017/7-ART1 $15.00DOI: http://dx.doi.org/10.1145/3072959.3073595

Twist brush: In this case, we have a skew-symmetric force matrixF , which implies F +F t = 0 and tr(F )= 0. This matrix can then bewritten as a cross product matrix [q]× of a vectorq, and the gradientat x0 simpli�es to

∇tε (0) = −52a

ε3[q]× . (4)

Note that this gradient matrix is skew-symmetric, thus indicatingthat the deformation at the brush center is a (in�nitesimal) rotationand can be controlled by the three DoFs provided by the vector q.

Scale brush: This case has a force matrix F = sI and then

∇sε (0) = s (2b − a)521ε3I . (5)

Therefore, the brush center x0 is locally deformed by a uniformscaling, with a single DoF.

Pinch brush: The case of a pinch brush has a symmetric and trace-less force matrix, i.e., F + F t = 0 and tr(F ) = 0. This implies thatthe gradient x0 is of the form

∇pε (0) =12ε3(4b − 5a ) F . (6)

As a consequence, the deformation at x0 is also symmetric with zerotrace, in a total of �ve DoFs.

B Derivatives of 2D Regularized Kelvinlets

We can also compute the derivatives for regularized Kelvinlets andlocally a�ne regularized Kelvinlets in 2D, following the di�erentia-tion of Eqs. (23) and (24) in the main text. The resulting expressionsand their properties are very similar to the 3D case.

Grab brush:

∇uε (r) =2r2ε

[b

((f t r) I + r f t + f rt

)− a f rt

]−

2r5ε

[aε2 f + 2b(f t r)r

]rt .

(7)

At x0, we have uε (0) = 0, thus indicating that the brush center islocally under a rigid deformation.

ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.

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1:2 • F. de Goes and D. L. James

Locally A�ine brush:

∇uε (r) =2r2ε

[b

(F t + F + tr(F )I

)− aF

]−2r4ε

[aε2F + 2b (rt F r)I + 2b rrt

(F + F t

) ]−4r4ε

[b

(F t + tr(F )I + F

)− aF

]rrt

+8r6ε

[aε2F + 2b (r t Fr )I

]rrt .

(8)

At a brush center x0, this expression simpli�es to:

∇uε (0) =2ε2

[b

(F t + F + tr(F )I

)− 2a F

]. (9)

Twist brush: A 2D twist brush is described by a skew-symmetricforce matrix F = t J , where t indicates a scalar and

J =

[0 −11 0

].

By replacing this matrix to Eq. (26) in the main text, we obtain

tε (r) = −2a t(1r2ε+ε2

r4ε

)J , (10)

and its gradient at x0 is of the form

∇tε (0) = −t4aε2

J . (11)

This indicates that the deformation at the brush center is a rotationand is parameterized by a single DoF, representing the vorticity atx0. Note that the vorticity is a scalar in 2D, but a vector in 3D.

Scale brush: Similar to the 3D case, the scale brush has a forcematrix F = s I and is written as

s(r) = 2(b − a)(1r2ε+ε2

r4ε

)(s r) . (12)

Its gradient at x0 is

∇sε (0) = 4 s (2b − a) 1ε2I . (13)

Therefore, the brush center x0 is locally deformed by a uniformscaling, with a single DoF.

Pinch brush: The case of a pinch brush has a symmetric and trace-less force matrix F , and its displacement �eld is given by

pε (r)=−2a(1r2ε+ε2

r4ε

)F r + 4b

[1r2εF−

1r4ε

(rt F r

)I

]r. (14)

The gradient of a pinch brush at x0 is expressed by

∇pε (0) =4ε2(b − a ) F . (15)

Due to the properties of F , it is easy to check that the deformationat x0 is symmetric with zero trace, in a total of two DoFs.

C Symmetrized brushes

Without loss of generality, consider a plane centered at the originand with normal vector n. This plane de�nes a re�ection matrix ofthe form M =I−2nnt . Then the image of a point x is given by Mx .A symmetrized displacement �eld with force vector f and brushcenter x0 is de�ned by

u(x) = [Kε (x − x0) +Kε (x −Mx0)M ] f . (16)

Proposition: Any symmetrized deformation generated by (16) satis-�es the identity

u(Mx) = Mu(x), ∀x . (17)

Proof:

First, we introduce some additional notation to make our derivationmore concise. We denote д =x−x0 and h =x−Mx0, and indicatetheir norms by д and h, respectively. Also, we rewrite the Green’sfunction Kε in Eq. (6) of the main text as

Kε (r ) = αr I + βrrrt , (18)

where αr and βr are functions of the distance r and include thebrush parameters (ε, µ,ν ). Since MM =I , we have ‖Mx ‖ = ‖x ‖ forany x . Moreover we can show that:

Mx − x0 = M (x −Mx0) = Mh,

Mx −Mx0 = M (x − x0) = Mд.(19)

Now replacing (16), (18), and (19) to (17), we obtain:

u(Mx) = [Kε (Mh) +Kε (Mд)M] f

=[αhI + βhMhhtM + αдM + βдMддtMM

]f

=[αhMM + βhMhhtM + αдM + βдMддt

]f

=[M

(αhI + βhhh

t ) M +M (αдI + βдддt

) ]f

= M [Kε (h)M +Kε (д)] f

= Mu(x),

thus verifying the proposition. �

A similar result is also valid for the symmetrized version of locallya�ne regularized Kelvinlets and for their counterparts in 2D.

D Strain-Stress Formulation

In Section 3 of the main text, we recapped the formulation of thepotential energy for linear elastostatics using displacement �eldsu. Next we review an alternative yet equivalent formulation of theelastic potential energy in terms of strain and stress tensors. Bothexpressions are known results in the theory of linear elasticity, see,e.g., [Slaughter 2002].

ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.

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Regularized Kelvinlets • 1:3

In linear elasticity, the strain tensor ϵ and the stress tensor σ arede�ned in terms of the displacement �eld u by:

ϵ = 12(∇u + (∇u)t

),

σ = µ(∇u + (∇u)t

)+

2µν(1 − 2ν ) (∇ · u) I ,

(20)

where µ and ν are the elastic shear modulus and the Poisson ra-tio, respectively. Note that both tensors are symmetric. The elasticpotential energy (Eq. (1) of the main text) is then expressed by:

E(u) = 12 〈ϵ,σ 〉F − 〈b,u〉, (21)

where 〈·, ·〉F indicates the Frobenius inner product for 2-tensor�elds integrated over the in�nite volume, and b indicates the exter-nal body forces. Using the divergence theorem, one can show thatthe integrated Frobenius inner product can be converted into anintegrated inner product of vector �elds, i.e.:

〈ϵ,σ 〉F = − 〈u,∇ · σ 〉. (22)Further expanding the divergence of σ yields to:

∇ · σ = µ∆u +µ

(1 − 2ν )∇(∇ · u). (23)

Finally, by substituting (22) and (23) into (21) and using divergencetheorem once more, we reproduce Eq. (1) of the main text.

References

W. S. Slaughter. 2002. The Linearized Theory of Elasticity. Birkhäuser Basel.

ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.