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Polygonal TechniquesPolygonal TechniquesPolygonal TechniquesPolygonal Techniques

2002. 5. 15 2002. 5. 15

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Introduction

• This chapter– Discuss a variety of problems that are

encountered within polygonal data sets

• The goal for polygonal representation– Accuracy and speed

• The three topic– Tessellation, consolidation, simplification

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Term definition

• Tessellation– Polygonal of the different forms have to

split into more tractable primitives• Displayed correctly

• Consolidation– The process which encompasses merging

and linking polygonal data• Improve data display

• Simplification– Attempting to remove unnecessary or

insignificant features• More speed

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Three-Dimensional Data - 1

• Deferent representation for deferent types of objects– Solid-base modelers

• Voxels (fig – a)• Constructive solid geometry (fig – b)

Fig - a

Fig - b

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Three-Dimensional Data - 2

– Surface-base modelers• Mesh (fig – b)• Subdivision (fig – c)• Parametric surface (fig – a)• Implicit surface

– f(x, y, z) = 0

Fig - b Fig - cFig - a

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Tessellation - 1

• Tessellation– The process of splitting a polygon into a set of

polygons

• Reasons– Triangle : optimization for Many graphics APIs and

hardware (triangulation)– Renderer : handling convex polygons (convex

partitioning)– In order to catch shadows and reflected light

The different types of tessellation

(b. convex partition, c. triangulation, d. uniformly meshed)

a b c d

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Tessellation - 2

• Process of tessellation– How best to project a three-dimensional

polygon into two dimensions• Best plane

– Selection the largest projected area among the xy, the yz, the zx

– Using the greatest magnitude in the polygonal’s normal

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Tessellation - 3

– Problems• Polygon facets by creating the modelers can be

badly warped – produce an hourglass or bowtie quadrilateral

(fig-a)– Using line / line intersection test (section 10.12)

• Polygons are not always made of a single outline– Using the join edge (fig-b)

Fig - aFig - b

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Shading problems

• Split the quadrilateral into the triangles – Provide proper input to the renderer– Avoid bowtie problems– Notice

• Select the good diagonal edge for renderer

• The different ways – Using the minimized differences (between color)

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Shading problems

• The different ways– Using the diagonal that gives the largest angle

between the two triangles

Left : each quadrilateral split along the same diagonal

Right : the split is made using the diagonal whose corners are

closer in height

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Shading problems

• Problem of warped quadrilateral– Texture is applied – When tessellation apply, distort texture – Requirement

• Fine meshing of surface• Use texture-coordinate interpolation scheme

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Edge cracking and T-Vertices

• Producing the NURBS by modeler – Happen the edge cracking at the shared edge by the

two surface

– Solution• Edge stitching

– When joining flat surface is that of T-vertices, happen to related problem

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Consolidation

• Consolidation– To find and adjust the links between polygons for

rendering

• Advantage – Improve efficiency

• The display polygon meshes• The area of backface culling

• Quality-related problem– Face orientation

• Use a right-handed orientation (counter-clockwise direction)

Problem : When model has the only surface normal.

Left : don’t have the normal per vertex

Right : have the normal per vertex

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Consolidation

Solution for forming polygonal meshes 1. Form edge-face structure 2. Find groups of polygons (determine solidity)3. For each group, flip faces to gain consistency4. Determine what the inside of the mesh is 5. Find smoothing groups and compute vertex

normals6. Find boundaries7. Create polygonal meshes

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Consolidation

– Form edge-face structure• Form a list of all edges in the entire set of polygons• Sorting by vertex coordinates • Ex) v1(-3,5,2), v2(-3,6,-8) v1 => v2 sorting

– Find groups of polygons (determine solidity)• A group consist of polygons which are connected by

edges• Ex) two group of the teapot : the pot and the lib

– For each group, flip faces to gain consistency• Ensure orientation consistency (counter-clockwise

outlines)• Select an arbitrary starting polygon• Traversal neighbor polygons• No consistency, apply flipping

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Simplification - 1

• Simplification– The process of taking a complex model and

reducing complexity – Data reduction or decimation

• The main advantage – Be quicker to render

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Simplification - 2

• The operation of simplification– Edge collapse

• The strategy for new vertex placement– Select one vertex or the other vertex [Melax]

• Faster, Lower-quality – Check the midpoint on the edge [Hoppe]– Check both vertices on the edge [Garland and

Heckbert]• Higher-quality, Extra processing, code and momory

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Simplification - 3

• Problem of edge collapse– Happen to edge crossing– Solution

• Using of the best cost function

– Happen to ambiguous edge collapse

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• The extension of the cost function– by Garland and Heckbert– Be to weight the value by the areas of the triangles

associated with each plane– Be to use a cost function based on maintaining other

surface features – Be to preserve the material changes, texture map

edges, color-per-vertex changes

Simplification - 4

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Simplification - 5

• The simplification of LOD– Problem

• Happen to “Popping effect”

– Solution• Be to use alpha blending to cross-fade between

model representation• Be to use geomorphs to increase or decrease the

level of detail