Modeling and representation 1 – comparative review and polygon mesh models 2.1 Introduction 2.2 Polygonal representation of three-dimensional objects 2.3.

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Modeling and representation 1 – comparative review and polygon mesh models 2.1 Introduction 2.2 Polygonal representation of three-dimensional objects 2.3 High-level methods – constructive solid geometry 2.4 High-level creation using modellers/ediotrs

2.1 Introduction  Modelling and representation is a general phrase which can be applied to any or all of the following aspects of objects: Creation of a three-dimensional computer graphics representation The technique or method or data structure used to represent the object Manipulation of the representation – in particular changing the shape existing model.

2.1 Introduction  The representation of an object is very much an unsolved problem in computer graphics  Representation for user or for renderer Polygon mesh representation :both Bi-cubic parametric patches and constructive solid geometry (CSG): for user, it may be converted into polygon meshes for rendering

2.1 Introduction  Different representational methods have their advantages and disadvantages.  There is no universal solution to the many problems.  Particular modelling methods have evolved for particular contexts.

2.1 Introduction  Mainstream models used in computer graphics Polygonal Bi-cubic parametric patches CSG (constructive solid geometry) Spatial subdivision techniques Implicit representation

2.1 Introduction  Polygonal Objects are approximated by a net or mesh of planar polygonal facets. With this form we can represent, to an accuracy that we chose, an object of an shape.

2.1 Introduction  Example

2.1 Introduction  Bi-cubic parametric pathes These are “ curved quadrilaterals ”. Each patch is specified by a mathematical formula that gives the position of the patch in 3D space and its shape. A significant advantage of the representation is its ecnomy.

2.1 Introduction  Example Polygon mesh (2048 elements) Patch (32 patches)

2.1 Introduction  CSG (constructive solid geometry) This in an exact representation to within certain rigid shape limits. The CSG method is a volumetric representation – shape is represented by elementary volumes or primitives

2.1 Introduction  Example

2.1 Introduction  Spatial subdivision techniques This simply means dividing the object space into elementary cubes, known as voxels. Labelling each voxel as empty or as containing part of an object.

2.1 Introduction  Implicit representation Surfaces defined by underlying mathematical formula An implicit function is, for example: (which is definition for a sphere)

2.1 Introduction  Example

2.2 polygonal representation of three- dimensional objects 2.2.1 creating polygonal objects 2.2.2 manual modelling of polygonal objects 2.2.3 automatic generation of polygonal objects 2.2.4 interactive/mathematical generation of polygon objects

2.2 polygonal representation of three- dimensional objects  This is the classic representational form in 3D graphics  Advantage creating polygonal objects is straightforward visually effective algorithms exist highly efficient renderer simpler elements  Disadvantage complex and high creation cost do not allow simple shape manipulation hard to do exact collision detection accuracy

2.2 polygonal representation of three- dimensional objects  Hierarchical structure vertex-based boundary model  representing faces in terms of a sequences of vertices edge-based boundary model  representing faces in terms of a closing sequence of edges

2.2 polygonal representation of three- dimensional objects  Conceptual hierarchy

2.2 polygonal representation of three- dimensional objects  Topological representation

2.2 polygonal representation of three- dimensional objects  Practical data structure

2.2 polygonal representation of three- dimensional objects  Attribute Polygon attribute Edge attribute Vertex attribute

2.2 polygonal representation of three- dimensional objects  Polygon attributes Triangular or not Area Normal to the plane containing the polygon Coefficients (A,B,C,D) of the plane containing the polygon where Ax + By + Cz + D=0 Whether convex or not Whether it contain holes or not

2.2 polygonal representation of three- dimensional objects  Edge attributes Length Whether it is an edge between two polygons or an edge between two surfaces Polygons on each side of the edge

2.2 polygonal representation of three- dimensional objects  Vertex attribute Polygons that contribute to the vertex Shading or vertex normal – the average of the normals of the polygons that contribute to the vertex Texture coordinates (u,v) specifying a mapping into a two-dimensional texture image

2.2 polygonal representation of three- dimensional objects  Another problems Scale problem: the needs to control the detail of objects Solution : maintain a hierarchy of models in different detail and use the one appropriate. Problem s to be solved:  Visual disturbances  How to generate the hierarchy  How many levels

2.2.1 Creating polygonal objects  Four common examples of polygon modelling methods: Using a three-dimensional digitiser or adopting an equivalent manual strategy Using an automatic device such as a laser ranger Generating an object from a mathematical description Generating an object by sweeping

2.2.2 manual modelling of polygonal objects  The easiest way to model a real object is manually using a three-dimensional digitiser.

2.2.3 Automatic generation of polygonal objects  A device that is capable of creating very accurate or high-resolution polygon mesh objects from real objects is a laser ranger.

2.2.4 Interactive/mathematical generation of polygon objects  Many polygonal objects are generated through an interface into which a user puts a model description in the form of a set of curves.  The most popular paradigm is that of sweeping a cross-section in a variety of different ways.

2.2.4 Interactive/mathematical generation of polygon objects  Examples

2.2.4 Interactive/mathematical generation of polygon objects  Synder ’ s rail curve product surfaces

2.2.4 Interactive/mathematical generation of polygon objects  Synder ’ s Affine Transformation Surface

2.2.4 Interactive/mathematical generation of polygon objects  Three problems The size of the polygonal primitives depends on the excursion of the spine curve. How do we orient the cross-section with respect to a varying spine. How do we prevent cross-sections self-intersecting

2.2.4 Interactive/mathematical generation of polygon objects  Three problems

2.2.4 Interactive/mathematical generation of polygon objects  Solve problem1 The more polygons occur when the curvature twists rapidly. The most direct way to do this is to use the curve subdivision algorithm and subdivide the curve until a linearity test is positive.

2.2.4 Interactive/mathematical generation of polygon objects  Solve problem2 (Frenet frame) The unit length tangent vector T: The derivative of the curve V: The principal normal N: The second derivative of the curve A: B:

2.2.4 Interactive/mathematical generation of polygon objects  Solve problem2 (Frenet frame)

2.2.4 Interactive/mathematical generation of polygon objects  Solve problem3

2.3 High-level methods – constructive solid geometry (CSG)  The CSG approach is a powerful high-level tool that is found in many modelling packages.  It does not manipulate polygons directly but produces polygon models after the modelling or design phase is complete.  The logic of the shape in this representation is in how the final shape can be made or represented as a combination of primitive shapes.

2.3 High-level methods – constructive solid geometry (CSG)  Motivation Facilitate an interactive mode for solid modelling  Object representation Primitives are combined using Boolean set operators and linear transformations An object representation is stored as an attributed tree.  Disadvantage High rendering cost Limited primitives and operators

2.3 High-level methods – constructive solid geometry (CSG)  Boolean operations possible between solids. Union Subtraction Intersection

2.3 High-level methods – constructive solid geometry (CSG)  CSG tree

2.3 High-level methods – constructive solid geometry (CSG)  Geometric complex objects

2.4 High-level creation using modellers/editors  The previous sections described modelling methods that are commonly embedded in modellers/editors.  Such software will generally contain many high-level facilities.

2.4 High-level creation using modellers/editors  Example

2.4 High-level creation using modellers/editors

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