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Surface Metric Tensor

The basic idea of the indirect triangulation technique consists in the meshing of the planar parametric space and in mapping that mesh to the surface. In order to ensure that the elements mapped back to the surface are isotropic (of the aspect ratio close to one), nicely shaped (close to the equilateral triangle), and well respecting the prescribed element spacing, it is necessary to capture properly the distortion and stretching induced by the mapping function given by Eq. (1). Both can be conveniently described by the surface metric tensor in its principal coordinate system, which defines the stretches in the principal directions of the surface.

The stretch induced by the mapping given by Eq. (1) can be expressed as a relation between the lengths of a differential in the parametric space and the corresponding differential in the real space of a curve passing through point (Fig. 2). The squared length of the differential with components and is given by

(15)


The differential in the real space is defined by

(16)


where and are the components of the gradient of the surface at point defined by Eqs (9) and (10). Then the square of the length of the differential comes from

(17)
   

which is known as the first fundamental form of the surface. The non-diagonal terms = of the surface metric tensor represent the angle (more precisely its cosine multiplied by and ) between parametric curves at point . Assuming that there are orthogonal directions and in the parametric space (Fig. 2), for which the term vanishes, the squared length of the differential becomes

(18)


while for the differential holds

(19)


It is therefore clear that the roots of diagonal terms of the metric tensor , expressed in the principal coordinate system and as

(20)


directly correspond to the stretches in and directions, respectively. The problem thus reduces to finding the eigenvalues and of the surface metric tensor . According to the well known relationships for the tensor spectral decomposition, the declination between the and directions is obtained from

(21)


The inverse stretching for the elements in the parametric space in the directions  and , respectively, can be then evaluated as

(22)


where and are given by

(23)
(24)

Taking into account the required local element size at point , the final formula for the element size in the parametric space in directions and yield

(25)


Thus the anisotropic triangulation of the parametric space is controlled at each point by three values - orientation of the principal directions and element size and in the principal directions. These quantities can be visualized in the parametric space by an ellipse (ellipse of stretches) aligned with the principal directions and with half-axes of size in the direction and in the direction .

From the mathematical point of view, however, it is more convenient to describe the distortion and stretching of the mapping by an anisotropic metric in which the length of every optimal edge (optimal in terms of the prescribed mesh density) is evaluated to one.1 Obviously, such a metric, expressed in the local coordinate system corresponding to the principal directions, is given by

(26)


After the transformation back to the original (global) coordinate system and , the metric can be written as

(27)
   

where is the standard transformation matrix of the form

(28)


The anisotropic meshing of the parametric space is thus controlled by the surface metric tensor divided by the square of the locally prescribed element size. The length of vector in the parametric space evaluated in this metric is then expressed by

(29)


which can be further simplified for (locally) constant metric to

(30)




Footnotes

... one.1
A mesh with all edges having the length in this metric equal to one is called the unit mesh. However, such a mesh is generally hard to achieve. It is usually enough if an edge conforms to   [79].


Next: Parametric Space Meshing Up: Indirect Triangulation of 3D Surfaces Previous: Surface Representation

Daniel Rypl
2005-12-07