mesh deformation based on discrete differential geometry l.
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Mesh Deformation Based on Discrete Differential Geometry. Reporter: Zhongping Ji. 4.4.2007. Leonhard Euler 1707.4.15-1783.9.18 . Papers. (A) Linear Rotation-invariant Coordinates for Meshes [SIG 2005] Yaron Lipman Olga Sorkine David Levin Daniel Cohen-Or Tel Aviv University

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  • (A) Linear Rotation-invariant Coordinates for Meshes [SIG 2005]
  • Yaron Lipman Olga Sorkine David Levin Daniel Cohen-Or
  • Tel Aviv University
  • (B) Volume and Shape Preservation via Moving Frame Manipulation [TOG 2007]
  • Yaron Lipman Daniel Cohen-Or Gal Ran David Levin
  • Tel Aviv University
about authors
About authors
  • A Ph.D. student at The School of Mathematical Sciences, Tel-Aviv University.His supervisors are Prof. David Levin and Prof. Danny Cohen-Or.
a rotation invariant
( A ) Rotation-Invariant
  • A rotation-invariant representation for triangular mesh based on two discrete forms;
  • The first form locally discribes the tangential component of the surface;
  • The second form locally discribes the normal component of the surface.
classical differential geometry
Classical differential geometry
  • The first and second fundamental forms;
  • Given two fundamental forms I and II, and they hold some compatibility conditions, Gauss-Codazzi-Mainardi equations;
  • Given an initial right-hand frame, then solve a first-order linear PDE.
discrete forms
Discrete Forms
  • Notations
first form
First form

The length and the signed angle


First form:


second form
Second form

Second form:

Height function of a vertex above the tangent plane


  • Given the discrete form coefficients and the orientation bits at vertex i, the 1-ring neighborhood of i is defined up to a rigid transformation.
  • Fixing the position of vertex i, and fixing the direction of one edge emanating from vertex i and the normal, uniquely defines all the rest of the 1-ring vertex positions.
discrete surface equations
Discrete surface equations
  • Define a discrete frame at vertex i,

Discrete surface equations:


theorem 1
Theorem 1
  • The coefficients of the discrete surface equations can be expressed by the discrete forms.
theorem 2
Theorem 2
  • Given an initial discrete frame at an arbitrary vertex i0, and given that the first and second discrete forms with the orientation bits are taken from an existing mesh, there exists a unique solution to the set of discrete surface equations.
theorem 3
Theorem 3
  • Given the discrete frame at each vertex, there exists a unique embedding of the vertices of the mesh in R3 up to a translation, such that the resulting mesh has the given discrete forms and discrete frames.

More constraints can be added to this equation.

Geometry difference equations:

mesh representation
Mesh representation
  • The first discrete form is represented by the lengths of the projected edges and angles between adjacent projected edges.
  • The second discrete form represents the orthogonal component.
  • This sums to 3di scalars, and the scalars are rigid motion invariant.
To reconstruct the mesh given its discrete form coefficients, an initial discrete frame and a position of one vertex in space.
  • Construct the discrete surface equations;
  • Add the initial discrete frame as an additional equation;
  • Solve the augmented linear system in the least-squares sense to obtain the discrete frames;
  • Construct the geometry difference equations;
  • Add the given initial vertex position as an additional equation;
  • Solve the augmented linear system in the least-squares sense to get the positions of the vertices.
mesh editing
Mesh editing
  • Two constraints and two sparse linear systems;
  • Transformation constraints on the local frames and translation constraints on the vertex positions;
  • Discrete surface equations and geometry difference equations.
future work
Future work
  • Directly manipulate the local frames;
  • Improve the co-existence of constraints on the local frames and the vertex positions;
  • Use this representation for detail editing, such as detail enhancement.
b moving frame
( B ) Moving Frame
  • Isometric shape-preserving
  • Aiming at minimizing distortion of the shape.
  • Local volume preservation
  • Control the local volume by scaling the moving frames.
  • Look for a deformation which maintains the first fundamental form intact (isometry), and minimizes changes in the second fundamental form.
  • Based on the Stokes’ theorem, using a carefully designed differential form which establishes a connetion between local volum and surface properties.
shape preserving
  • Isometric deformation
  • Defining a rigid motion-invariant geometric distance(GD) between two isometric surfaces;
  • Given a surface and a set of rotational constraints, look for an isometric deformation minimizing the GD to the original surface under the constraints.
shape operator
Shape operator

Orthonormal frame:

Tangent plane:

Shape operator:

Second fundamental form:

geometric distance
Geometric distance

Point to point:

Surface to surface:

meaning: measures to what extent the surface M and deformed surface are rigid motion of each other.

Geometric deformation problem: Given a surface M, the goal is to deform it into another surface, subject to some prescribed constraints such that the geometric distance between them is minimal.

solve the problem
Solve the problem

Dirichlet-type integral:

An energy functional on the rotation’s field.

The minimizer of the intergral, subject to constraints, is a harmonic function defined on a Lie group.

shape preserving method
Shape-preserving method
  • Compute a rotation field R such that the enerty is minimized subject to constraints;
  • Apply rotations over the moving frames, and reconstruct the isometry f.
parametrization of so 3
Parametrization of SO(3)
  • Dim(SO(n)) = n(n-1)/2
  • Two parametrization:
    • orthogonal parametrization;
    • conformal parametrization.
volume preservation
Volume preservation
  • Motivation,Strokes’ theorem:
  • Designing a two-form:

is a volumetric form and reflects the local volume underlying the pointpon surfaceM.





derive two form
Derive two-form
  • Let (b1,b2,b3) denote the standard basis in R3,
calculate scaling factor
Calculate scaling factor

Scale the moving frame, scaling factor:

calculate h t
Calculate h(t)
  • Defining the depth function
  • Solve the ODE
  • Select the set of Constraints: static and handle vertices;
  • Create and factorize the Laplace-Beltrami matrix of ROI;
  • Calculate the local depth field L of the mesh;
  • Calculate the rotation field R: MSO(3) and apply to the moving frames at each vertex;
  • Reconstruct the mesh in a least-squares sense;
  • Calculate the scaling factor for volume correction;
  • Scale the moving frames and reconstruct the mesh.

Without volume correction


Without volume correction


With volume correction





future work60
Future work
  • It cannot preserve simultaneously the surface area and volume of an object. An exciting avenue for future research is to investigate orher complete local surface descriptors, whose preservation yields a shape preservation of both surface and volume.