Representing 3D objects

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Representing 3D objects

• 3D coordinate systems
• Polygons and curved patches
• Methods for generating 3D models

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Introduction

• previously have looked at representing 2D images
• bitmap and Vector representation
• how can we represent 3D objects?
• rare to use 3D equivalent of pixels
• voxels colour of each point in 3D space
• normally Vector representation
• can't display directly
• need to convert to a 2D image
• shows 3D model from a particular viewpoint

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Representing 3D objects
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Representation of 3D objects

• object representated by set of vertices (points
  in 3D space)
• apply colour or texture (image)
• each vertex maps into particular position on
  texture

tiger.x Microsoft DirectX samples
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Representing points in 3D space

• need the x, y and z coordinate of each point
• (x, y, z)
• we are familiar with the x, y coordinate system
  from 2D geometry
• z axis is perpendicular to the other two axes
• but which way does it go?
• there are two possibilities
• the right-handed and left-handed systems

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• Right-handed 3D coordinates
• the positive z axis comes out of the
  screen/paper

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• Left-handed 3D coordinates
• the positive z axis goes into the screen/paper

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Right and left handed systems

• grasp the z-axis with the relevant hand
• curl your fingers from the positive x-axis to the
  positive y-axis
• your thumb will point in the positive z-direction
• or, point your thumb along the positive x-axis
• point your index finger along the positive y-axis
• the rest of your fingers will point in the
  positive z-direction
• why does it matter?
• models created in the two systems are mirror
  images of each other
• and can never be superimposed no matter how they
  are rotated
• just like your hands
• and many biologically active molecules

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Polygonal mesh

• vertices connected by edges to make flat polygons
• best representation for rendering
• converting to 2D image
• triangles
• guaranteed to be flat
• or quadrilaterals
• less edges
• need to make sure vertices are coplanar

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Curved patches

• shape defined by control points
• 3-D equivalent of Bezier or other curve
• edges of patch defined by 2D curve
• smoother than polygon representation
• few points needed
• more intuitive to modify (sculpt)
• usually converted to polygons for rendering

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How can we define the vertices?

• generate from existing solid objects
• start with real 3D object, find and record
  suitable vertices and edges
• build up from 3D primitives
• build up from 2D curves

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Generating vertices from an existing solid object

• define by hand
• tedious
• difficult to visualise in 3D
• The Utah teapot
• a early and widely-used model
• 1975
• 110 vertices
• generated by Martin Newell using graph paper,
  pencil and a teapot
• edited control points for Bezier patches by hand
• see "A Brief History of the Utah Teapot"
• http//www.sjbaker.org/teapot/

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Generating vertices from an existing solid object

• 3D digitiser
• manually position and click
• can use more vertices where more detail is needs
• 3D laser ranger
• automatically scan at set resolution (slices)

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Geometric primitives

• can be specified mathematically
• need only to define a few attributes
• can calculate position of vertices
• to any desired resolution
• cube
• width
• width, height, depth of cuboid
• sphere
• radius
• x2 y2 z2 r2

cube
sphere
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Geometric primitives
torus
cylinder
cone
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Constructive solid geometry

• combine primitive objects using boolean
  operations
• union
• intersection
• difference

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Revolving curves to create a surface

• define curve in 2D
• Bezier curve or NURB
• rotate to get 3D shape

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Lofting and sweeping

• lofting
• define a series of 2D profile curves
• stack them up to define contours of surface
• sweeping
• sweep a 2D shape along a line
• to define cylinder
• shape determines cross section
• sweep along a curve to get a twisted cylinder

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3D modelling tools

• Maya
• http//usa.autodesk.com/adsk/servlet/index?siteID
  123112id7639525
• 3DS Max
• http//usa.autodesk.com/adsk/servlet/index?id5659
  302siteID123112
• Lightwave 3D
• http//www.newtek.com/lightwave/
• MilkShape3D
• http//www.swissquake.ch/chumbalum-soft/index.html
  

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Using a 3D modelling tool

• start with one of the previous techniques
• geometric primitive
• combine using boolean operations
• or revolving, lofting or sweeping 2D curves
• then edit to produce desired model
• scale, rotate, translate
• move vertex positions
• define extra vertices and polygons for more
  control
• duplicate vertices so they can be moved apart
• extrude faces
• sculpt NURBS models by moving control points

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Polygonal modelling with Maya
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Polygonal modelling with Maya
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NURBs modelling with Maya

• start with simple NURBs surface (patches)
• move control points to sculpt

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What makes a good model?

• more vertices finer detail
• but more space to store
• more time to render (important for games)
• some areas need more detail
• face
• near joints to allow movement
• other places can get away with less
• inside
• facing away from viewer
• less important objects

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Further work

• Kerlow
• Chapter 4 "Basic Modelling Techniques"
• Hearn and Baker
• Chapter 8 "Three-Dimensional Object
  Representations "
• Tutorial
• Begin polygonal modelling with Maya
• Temple example