Geometric Modelingpanozzo/gp/Assignment3.pdfCSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo...

CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo

Geometric Modeling Assignment 3: Discrete Differential Quantities

Acknowledgements: Julian Panetta, Olga Diamanti


Assignment 3 (Optional)• Topic: Discrete Differential Quantities with libigl

• Vertex Normals, Curvature, Smoothing (Tutorials 201-205)

per-face

per-vertex, uniform

per-vertex, quadratic fit


Vertex Normals

uniform average area-weighted average mean-curvature based

quadratic fitting on k-nearest neighborsPCA on k-nearest neighbors

libigl tutorial #201, igl::principal_curvature


Curvature

min max

mean gauss

libigl tutorials #202, #203


• Mean Curvature

• Gaussian Curvature

• Min Curvature • Max Curvature

H(vi) = 0.5kLc(vi)k

Curvature

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)

(cot↵j + cot�j) (f(vj)� f(v))

�Sp = �2Hn

H(vi) = 0.5kLc(vik

C0 C1 C2

@f

@t= ��f

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)

1(vi) = H(vi) +pH2(vi)�G(vi)

2(vi) = H(vi)�pH2(vi)�G(vi)

C0 C1 C2

@f

@t= ��f

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

1 = H +pH2 �G

1 = H �pH2 �G

G = 12

2H = 1 + 2


Data Smoothing: Grid Case• Image/2D data smoothing:

Filter out noise/rapid oscillations

• Solve the heat equation over some time period:

• Intuition: move toward the average of neighboring values

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

@f

@t= ��f = �

✓@2f

@x2+

@2f

@y2

◆


Data Smoothing: Grid Case

• Approach: discretize laplacian with finite differences

• Now we have an ordinary differential equation (ODE)

• Integrate (time-step) the ODE, e.g., with forward/backwards Euler

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

Diffusion Flow

10

∂f

∂t= λ∆f

diffusion constant

Laplace operator

@f

@t= ��f = �

✓@2f

@x2+

@2f

@y2

◆


Data Smoothing: Surface Case• Laplace-Beltrami operator lets us smooth functions on surfaces, S

• E.g., with forward Euler (explicit smoothing) on a triangle mesh, M:

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

@f

@t= ��Sf

discrete Laplacian weightsvertices adjacent vi

f t+dt(vi) = f t(vi) + � dt �Mf t(vi)

= f t(vi) + � dt

0

@X

vj2N(vi)[{vi}

wijft(vj)

1

A


Mesh Smoothing• We can view the vertex positions as a (vector-valued) function to smooth!

• Here, first-order explicit time stepping looks like:

• Note: Laplacian weights generally change when mesh changes!

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



C0 C1 C2

@f

@t= ��f

f(vi) =

2

4xi

yizi

3

5

f t+dt(vi) = f t(vi) + � dt �Mf t(vi) =)vnewi = vi + � dt (�Mv)i

= vi + � dt

0

@X

vj2N(vi)[{vi}

wijvj

1

A


Explicit Smoothing• In matrix form:

• Intuition: vertices moving toward neighbor average.

Diffusion Flow on Meshes

• Continuous

• Discretization

21

0 Iterations 10 Iterations 100 Iterations

@p

@t= ��p

pi pi + �t��pi


• Continuous

• Discretization

21


@p

@t= ��p



• Continuous

• Discretization

21


@p

@t= ��p


RANDOM EQUATIONS FOR SLIDES 3

Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...

wN1 wN2 · · · wNN

3

7775= {wij}

wij =

8<

:

0 i 6= j, @ edge (i, j)1 i 6= j, 9 edge (i, j)�|N(vi)| i = j

wij =

8>><

>>:

0 i 6= j, @ edge (i, j)cot↵j+cot�j

2A(vi)i 6= j, 9 edge (i, j)

�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

(I � �dtL)vn+1 = vn

vn+1 = v

n + �dtLvn+1

vn+1 � v

n = �dtLvn

vn+1 = (I + �dtL)vn

@v

@t= �L(v)

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:


wij =

8>><

>>:



�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775


vn+1 = v

n + �dtLvn+1

vn+1 � v

n = �dtLvn


@v

@t= �L(v)

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:


wij =

8>><

>>:



�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775


vn+1 = v

n + �dtLvn+1

vn+1 � v

n = �dtLvn


@v

@t= �L(v)

@f

@t= ��f


Implicit Smoothing• “Backward Euler” is more stable

• Take larger time steps without mesh “blowing up” (becoming jagged)

original explicit, 1000 iterations,λ = 0.01

implicit, 1 iteration, λ = 20


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:


wij =

8>><

>>:



�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775


vn+1 = v

n + �dtLvn+1

vn+1 � v

n = �dtLvn


@v

@t= �L(v)

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:


wij =

8>><

>>:



�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775


vn+1 = v

n + �dtLvn+1

vn+1 � v

n = �dtLvn


@v

@t= �L(v)

@f

@t= ��f

libigl tutorial #205

Laplacian at next (unknown) step! Must solve a linear system:


2D Rectangular Grid Laplacian

Measures difference from neighbor’s average (if we divide by 4/h^2):

11 -4 1

1

RANDOM EQUATIONS FOR SLIDES

OLGA DIAMANTI

1. Laplace-operator

f : R3 ! R

rf =

2

66664

@f@x

@f@y

@f@z

3

77775

�f : R3 ! R

�f = divrf = r ·rf

rf =h

@f@x

@f@y

@f@z

i

2

66664

@f@x

@f@y

@f@z

3

77775=

@2f

@x2+

@2f

@y2+

@2f

@z2

f : S ! R

�f = divSrSf

�f(xi) =X

j2N(i)

f(xj)� 4f(xi)

�f(v) =X

vj2N(v)

f(vj)� f(v)

=X

vj2N(v)

f(vj)� kf(v), k = |N(v)|

1


OLGA DIAMANTI

1. Laplace-operator

f : R3 ! R

rf =

2

66664

@f@x

@f@y

@f@z

3

77775

�f : R3 ! R


rf =h

@f@x

@f@y

@f@z

i

2

66664

@f@x

@f@y

@f@z

3

77775=

@2f

@x2+

@2f

@y2+

@2f

@z2

f : S ! R

�f = divSrSf

�f(xi) =X

j2N(i)

f(xj)� 4f(xi)

�f(v) =X

vj2N(v)

f(vj)� f(v)

=X

vj2N(v)

f(vj)� kf(v), k = |N(v)|

1

Weights can be represented by a “stencil:”

h

h �hf(xi) :=1

h2

2

4

0

@X

xj2N(xi)

f(xj)

1

A� 4f(xi)

3

5


2D Rectangular Grid Laplacian1

1 -4 11


OLGA DIAMANTI

1. Laplace-operator

f : R3 ! R

rf =

2

66664

@f@x

@f@y

@f@z

3

77775

�f : R3 ! R


rf =h

@f@x

@f@y

@f@z

i

2

66664

@f@x

@f@y

@f@z

3

77775=

@2f

@x2+

@2f

@y2+

@2f

@z2

f : S ! R

�f = divSrSf

�f(xi) =X

j2N(i)

f(xj)� 4f(xi)

�f(v) =X

vj2N(v)

f(vj)� f(v)

=X

vj2N(v)

f(vj)� kf(v), k = |N(v)|

1


OLGA DIAMANTI

1. Laplace-operator

f : R3 ! R

rf =

2

66664

@f@x

@f@y

@f@z

3

77775

�f : R3 ! R


rf =h

@f@x

@f@y

@f@z

i

2

66664

@f@x

@f@y

@f@z

3

77775=

@2f

@x2+

@2f

@y2+

@2f

@z2

f : S ! R

�f = divSrSf

�f(xi) =X

j2N(i)

f(xj)� 4f(xi)

�f(v) =X

vj2N(v)

f(vj)� f(v)

=X

vj2N(v)

f(vj)� kf(v), k = |N(v)|

1

Derivation:

Compute second derivatives by finite-differencing the forward difference approximation to first derivative:

Sum second partial derivative approximations for each dimension:

@2f

@x2(x, y) ⇡ 1

h

✓f(x+ h, y)� f(x, y)

h� f(x, y)� f(x� h, y)

h

◆

=f(x+ h, y) + f(x� h, y)� 2f(x, y)

h2=)

�f(x, y) =@2f

@x2(x, y) +

@2f

@y2(x, y)

⇡ f(x+ h, y) + f(x� h, y) + f(x, y + h) + f(x, y � h)� 4f(x, y)

h2

�hf(xi) :=1

h2

2

4

0

@X

xj2N(xi)

f(xj)

1

A� 4f(xi)

3

5


Irregular Grid (Mesh) LaplacianDiscrete Laplace-Beltrami

k First approach: Assume uniformity as in the grid case

CGL slideset 8

0

( ) ( ) = limh

f f x h f xx ho

w � �w

1( ( ) ( ),.., ( ) ( ))kf f v f v f v f v� � �

• First approach: “umbrella operator” • assume all edges are unit length • depends only on connectivity, not vertex locations

• Notice, when edge length h = 1 the 2D grid laplacian is:

• This version easily generalizes to an arbitrary mesh, M:

�uniformM f(v) =

X

vj2N(v)

✓f(vj)� f(v)

◆

=

0

@X

vj2N(v)

f(vj)

1

A� |N(v)|f(v)

�hf(xi) =1

h2

2

4

0

@X

xj2N(xi)

f(xj)

1

A� 4f(xi)

3

5 =X

xj2N(xi)

✓f(xj)� f(xi)

◆

#neighbors


The “Umbrella” (Uniform) Laplacian

In Matrix Form:

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



a(x, y, z) = x

b(x, y, z) = y

b(x, y, z) = z

f(x, y, z) =

2

4a(x, y, z)b(x, y, z)x(x, y, z)

3

5

vsmoothed = vi + ��Sf(vi)

= vi + �

X

vj2N(vi)

wivj

C0 C1 C2

Y = LFRANDOM EQUATIONS FOR SLIDES 3

Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:

�|N(vi)| i = j

0 i 6= j, @ edge (i, j)1 i 6= j, 9 edge (i, j)

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:

�|N(vi)| i = j


F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:

�|N(vi)| i = j


F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:


wij =

8>><

>>:



�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f

�uniformM f(v) =

0

@X

vj2N(v)

f(vj)

1

A� |N(v)|f(v)


The Cotangent Laplacian

•Accounts for edges’ differing lengths. •Converges to the continuous Laplace-Beltrami operatorwith mesh refinement

•Derivation in next recitation lecture.

Discrete Laplace-Beltrami

k Second Approach: Cotangent Formula

CGL slideset 10



CGL slideset 10



CGL slideset 10

�cotanM f(v) =

1

2A(v)

X

vi2N(v)

(cot(↵i) + cot(�i))

✓f(vi)� f(v)

◆

�S


The Cotangent Laplacian

In Matrix Form:

2 OLGA DIAMANTI

�Sf(v) =1

2A(v)

X

vj2N(v)


�Sp = �2Hn

H(vi) = 0.5kLc(vik

G(vi) =

2⇡ �Pj✓j

A(vi)



a(x, y, z) = x

b(x, y, z) = y

b(x, y, z) = z

f(x, y, z) =

2

4a(x, y, z)b(x, y, z)x(x, y, z)

3

5

vsmoothed = vi + ��Sf(vi)

= vi + �

X

vj2N(vi)

wivj

C0 C1 C2

Y = LFRANDOM EQUATIONS FOR SLIDES 3

Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:

�|N(vi)| i = j


F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:

�|N(vi)| i = j


F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:

�|N(vi)| i = j


F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f


Y =

2

66664

�f(v1)�f(v2)�f(v3)

· · ·�f(vN )

3

77775

L =

2

6664

w11 w12 · · · w1N

w21 w22 · · · w2N...

... · · ·...


3

7775= {wij}

wij =

8<

:


wij =

8>><

>>:



�P

vk2N(vi)wik i = j

F =

2

66664

f(v1)f(v2)f(v3)· · ·

f(vN )

3

77775

@f

@t= ��f

�cotanM f(v) =

1

2A(v)

X

vi2N(v)


✓f(vi)� f(v)

◆


Eigen Sparse Matrix• Full-sized Laplacian can be huge for large meshes

• but most elements are zero!

• Instead, only store non-zero elements: Sparse Matrix

#include <Eigen/Sparse>

Eigen::SparseMatrix<double> Laplacian;


Eigen Sparse Matrix• How to construct a sparse matrix in Eigen://declare size of matrix L = SparseMatrix<double> (V.rows(), V.rows());

//declare list of non-zero elements (row, column, value) std::vector<Eigen::Triplet<double> > tripletList;

//insert element to the list //if multiple triplets exist with the same row and column, values will be *added* tripletList.push_back(Eigen::Triplet<double>(source,dest,C(i,e)));

//construct matrix from the list L.setFromTriplets(tripletList.begin(), tripletList.end());

see igl::cotmatrix (V,F)


How to build cotangent matrix

• Iterating over vertices is slow

• Instead, iterate over faces and add cot terms to the vertices incident each face edge

for i = 1:number_of_faces for j = 1:face_valence source_vertex = faces(i,j); destination_vertex = faces(i,(j+1) % face_valence); weight = ....; //laplacian weight for edge (source_vertex, destination_vertex)(cotan or uniform) Laplacian(source_vertex, destination_vertex) += weight; Laplacian(destination_vertex, source_vertex) += weight; Laplacian(destination_vertex, destination_vertex) -= weight; Laplacian(source_vertex, source_vertex) -= weight;

end end



CGL slideset 10

�cotanM f(v) =

1

2A(v)

X

vi2N(v)


✓f(vi)� f(v)

◆


Averaged or Summed?• We’ll see next time that the cotan laplacian we introduced:computes the average Laplacian over some area “A” around v (hence the division by A)

• This gives an asymmetric matrix. But if we instead compute the “summed” laplacian (i.e., don’t divide by A), we end up with a symmetric matrix.

• This “summed,” symmetric Laplacian is more standard in the finite element community, and is related to our Laplacian by a “lumped (diagonal) mass matrix” holding the areas:

M =

2

6664

A(0) 0 · · · 00 A(1) · · · 0...

... · · ·...

0 0 · · · A(N)

3

7775

lumped mass matrix (averaging region areas)

�cotanM f(v) =

1

2A(v)

X

vi2N(v)


✓f(vi)� f(v)

◆

Laveraged = M�1Lsummed

[Lsummed]ij =

8>>><

>>>:

12

✓cot(↵ij) + cot(�ij)

◆if j 2 N(i)

�P

j2N(i)[Lsummed]ij if i = j

0 otherwise

(Laveraged corresponds to formula above)


Averaged or Summed?• Suppose we have a linear system: i.e. using the asymmetric cotangent Laplacian we introduced earlier.

• Right hand side vector “b”represents a scalar field, storing the desired Laplacian value (not the summed value!) at each vertex.

• Solving symmetric sparse linear systems matrices is much more efficient, so we prefer to solve with the summed quantities:

Laveragedx = b

MLaveragedx = Lsummedx = Mb


How to compute A(v)• Barycentric area

• Connect edge midpoints and triangle barycenters

• Each of the incident triangles contributes 1/3 of its area to all its vertices, regardless of the placement

+ Simple to compute + Always positive weights - Heavily connectivity-dependent - Changes if edges are flipped


How to compute A(v)• Voronoi area

• Connect edge midpoints and triangle circumcenters

• The 3 resulting quadrilaterals are not equareal

• Sum contributions from incident triangles

+ Only depends on vertex positioning - More complicated computations


How to compute A(v)• Voronoi area

• Connect edge midpoints and triangle circumcenters

• The 3 resulting quadrilaterals are not equiareal


• For obtuse triangles, circumcenter is outside the triangle -> negative areas!

+ Only depends on vertex positioning - More complicated computation - May introduce negative weights (obtuse triangles)


How to compute A(v)• Voronoi area - compromise

• Connect edge midpoints with:

• triangle circumcenters , for non-obtuse triangles

• midpoint of opposite edge, for obtuse ones


+ Only depends on vertex positioning - More complicated computations


Linear Systems with Eigen

• Cholesky factorization works only for symmetric, positive definite A! • Other solvers available, e.g. SparseLU:

• http://eigen.tuxfamily.org/dox/TutorialSparse.html

#include <Eigen/SparseCholesky> // ... SparseMatrix<double> A; // fill A VectorXd b, x; // fill b // solve Ax = b SimplicialLDLt<SparseMatrix<double> > solver; solver.compute(A); if(solver.info()!=Success) return; // decomposition failed x = solver.solve(b); if(solver.info()!=Success) return; // solving failed // solve for another right hand side: x1 = solver.solve(b1);

http://eigen.tuxfamily.org/dox/TutorialSparse.html


Questions?

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