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#

Figures

**Figure 1:**
Rational Bezier entities: a) curve, b) surface.

**Figure 2:**
Principal directions at point

on a parametric surface.

**Figure 3:**
Location of ideal point

on the ellipse of stretches.

**Figure 4:**
Location of ideal point

for

equal to a)

, b)

, c)

, d)

.

**Figure 5:**
Location of ideal point

for angle of edge

equal to:
a)

, b)

, c)

, d)

, e)

, f)

.

**Figure 6:**
Degenerated surfaces: a) an octant of a sphere, b) a rectangle.

**Figure 7:**
Evaluation of the ellipse of stretches at a singular point

.

**Figure 8:**
Triangulation of a surface with collapsed side.

**Figure 9:**
Weighting factors for the smoothing in the parametric space.

**Figure 10:**
Model (left) and mesh (right) of a hemisphere modeled by
a single patch with two singularities.

**Figure 11:**
Model (left) and mesh (right) of a hemisphere modeled by
a single patch without stretch singularity.

**Figure 12:**
Model (left) and mesh (right) of a hemisphere modeled by
four patches with a singularity.

**Figure 13:**
Principal stretches (left) and anisotropic
mesh (right) in the parametric space of a hemisphere modeled by
a single patch with two singularities.

**Figure 14:**
Principal stretches (left) and anisotropic
mesh (right) in the parametric space a hemisphere modeled by
a single patch without stretch singularity.

**Figure 15:**
Principal stretches (left) and anisotropic
mesh (right) of one of the patches in the parametric space of
a hemisphere modeled by four patches with a singularity.

**Figure 16:**
Anisotropic mesh of hyperbolic paraboloid in the parametric space.

**Figure 17:**
Anisotropic mesh of twisted plane in the parametric space.

**Figure 18:**
Model (left) and mesh (right) of a hyperbolic paraboloid.

**Figure 19:**
Model (left) and mesh (right) of a twisted plane with a hole.

**Figure 20:**
Octree hierarchy (2D case).

**Figure 21:**
Enforcement of the octree one-level difference (2D case).

**Figure 22:**
Octree built around a mechanical part (left) and corresponding element spacing contours (right).

**Figure 23:**
Octree built around a double bended curve.

**Figure 24:**
Projection of point

: a) to a curve, b) to a surface.

**Figure 25:**
Discretization of a curve.

**Figure 26:**
Location of the ``ideal'' point

of a tentative triangle.

**Figure 27:**
Correction of the ``ideal'' point

due to the surface curvature.

**Figure 28:**
Correction of the ``ideal'' size due to the element size gradation.

**Figure 29:**
Neighbourhood of point

(2D case).

**Figure 30:**
Selection of point

(2D case).

**Figure 31:**
Uniform mesh of a circle with (left) and without (right) the application of the
ellipsoidal neighbourhood

.

**Figure 32:**
Intersection check on a poorly parameterized surface.

**Figure 33:**
Diagonal edge swapping.

**Figure 34:**
Model (left) and mesh (right) of a hemisphere modeled by
a single patch with two singularities.

**Figure 35:**
Model (left) and mesh (right) of a hemisphere modeled by
a single patch without stretch singularity.

**Figure 36:**
Model (left) and mesh (right) of a hemisphere modeled by
four patches with a singularity.

**Figure 37:**
Model (left) and mesh (right) of a rotational hyperboloid.

**Figure 38:**
Model (left) and mesh (right) of a chair.

**Figure 39:**
Chair - performance diagram.

**Figure 40:**
Chair - distribution of the element quality.

**Figure 41:**
Chair - distribution of the dihedral angle.

**Figure 42:**
Model and mesh of Utah teapot.

**Figure 43:**
Two levels of hierarchical refinement of a triangular element.

**Figure 44:**
Two stages of the refinement - splitting and averaging.

**Figure 45:**
Averaging mask of the Loop's scheme (after splitting).

**Figure 46:**
Averaging mask of the butterfly scheme (before splitting).

**Figure 47:**
Loop's scheme applied to a star-shaped polyhedron.

**Figure 48:**
Averaging mask of the modified butterfly scheme (n = 7).

**Figure 49:**
Averaging mask of curve interpolating scheme: a) 4-point curve mask, b) 3-point curve mask.

**Figure 50:**
Modified butterfly scheme applied to a star-shaped polyhedron.

**Figure 51:**
Projection to a discrete surface: a) target triangle localization, b) progressive
refinement of the target triangle.

**Figure 52:**
Localization of a point into a target triangle: a) initial
configuration, b) first refinement, c) second refinement, d) final
(limit) configuration.

**Figure 53:**
Correction of the sub-triangle selection: a) backtracking
process, b) sub-triangle numbering, c) neighbour numbering.

**Figure 54:**
Unit hemisphere - performance diagram (EP).

**Figure 55:**
Unit hemisphere - performance diagram (AP).

**Figure 56:**
Unit hemisphere - deviation from the ideal shape.

**Figure 57:**
Mesh of a concrete dam with a spillway.

**Figure 58:**
Concrete dam - performance diagram.

**Figure 59:**
Stanford bunny - performance diagram.

**Figure 60:**
Mesh of Stanford bunny.

**Figure 61:**
Location of the ``ideal'' points

and

of
a tentative quadrilateral.

**Figure 62:**
Selection of the base edge for the generation of the second triangle.

**Figure 63:**
Mesh schemes for the determination of smoothing weights.

**Figure 64:**
Topological operations:
a) merge operations (left), b) swap operations (middle),
c) refine operation (right top), d) coarse operation (right middle),
e) split operation (right bottom).

**Figure 65:**
Transform operations.

**Figure 66:**
Mesh generation process on a planar domain with two openings.

**Figure 67:**
Mixed (left) and all-quadrilateral (right) graded mesh of a complex planar domain.

**Figure 68:**
Mixed (left) and all-quadrilateral (right) uniform mesh of a torus.

**Figure 69:**
Mixed (left) and all-quadrilateral (right) graded mesh of a chair.

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*Daniel Rypl *

2005-12-07