shape representation 1

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shape representation 1

• polygons and polyhedra

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How do we represent the shape of an object?
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Eeeew

• We dont really need to represent all the details
  of an object
• Just approximate the shape
• Enough that it looks right when displayed

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What do we mean by shape?
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The interior of the shape?
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The boundary of the shape?
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Representation of shape

• In computer graphics, we usually use boundaries
• Curves in 2D
• Surfaces in 3D
• A curve or surface is an (infinite) set of points
• This is a problem, since our computers dont have
  infinite memory

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Approximating a curve
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Linear approximation
We can use a line as a (bad) approximation to a
curve
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Piecewise-linear approximation
We can use several lines to make a better
approximation
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Approximating a curve
By adding more lines, we can make it as accurate
as we like
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Approximating a curve
By adding more lines, we can make it as accurate
as we like
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Polylines
We end up with a bunch of line segments that
share endpoints
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Vertices and edges
The endpoints are called vertices and the line
segments edges
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Vertices and edges
We can change the shape by moving a vertex
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Closed curves and polygons
If the curve joins up with itself, we say its a
closed curve
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Closed curves and polygons
A closed curve made out of lines is called a
polygon
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Concave polygons
concavity
A polygon with an indentation is called a concave
polygon
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Convex polygons
A polygon with no concavities is called convex
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Representing surfaces

• To represent 3D objects, we need to represent
  their surfaces

Source http//www.d-vw.com/news/advanced/active_l
ighting.htm
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Planar approximation

• We approximated 2D shapes with groups of line
  segments(aka polygons)
• What would the 3D equivalent be?
• The 3D analog of a line is a plane
• So we approximate surfaces with groups of planar
  patches
• What would those patches look like?

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Planar approximation

• What would those patches look like?
• Planes intersect in lines
• So many planes intersect in many lines
• Gee, this sounds familiar

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Polygonal approximation

• Whenever 4 or more planar patches intersect, they
  form a polygon
• Shapes modeled with planar patches form groups of
  polygons

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Vertices, edges, and faces

• In 3D
• The corners of the polygons are still called
  vertices
• But the lines between them are called edges
• And the polygons themselves are called faces
• Every edge is part of two faces
• Every vertex is part of at least two lines

vertices
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Vertices, edges, and faces

• In 3D
• The corners of the polygons are still called
  vertices
• But the lines between them are called edges
• And the polygons themselves are called faces
• Every edge is part of two faces
• Every vertex is part of at least two lines

edges
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Vertices, edges, and faces

• In 3D
• The corners of the polygons are still called
  vertices
• But the lines between them are called edges
• And the polygons themselves are called faces
• Every edge is part of two faces
• Every vertex is part of at least two lines

faces
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Polygonal meshes

• When large groups of polygons form a surface, we
  call it a mesh
• Meshes polygons share a common set of vertices
• Polygon meshes are the (current) standard
  technology for surface modeling

Source http//www.d-vw.com/news/advanced/active_l
ighting.htm
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Polygonal meshes
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Meshes have a long tradition
Perspective Study of a Chalice Paolo Uccelloin
1430-1440.
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Triangle meshes

• 3D modeling programs usually represent surfaces
  as arbitrary polygons
• Most graphics cards
• Provide hardware for drawing triangles
• Draw other polygons by making them out of
  multiple triangles
• Polygon meshes are usually converted to triangle
  meshes for final rendering
• So many artists model directly using the triangle
  mesh so they have as much control as possible

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Resolution and polygon count

• Graphics card speed is often measured in
  triangles per second
• A 1M tri/sec card can render
• A 10,000 triangle scene at 100fps
• A 100,000 triangle scene at 10 fps
• So models trade off speed for accuracy
• More polys more realistic still frames
• More polys less realistic animation
• Many artists in the game industry specialize in
  low poly count modeling

Source http//cache-www.intel.com/cd/00/00/01/45/
14535_continuous_detail_levels_f2.jpg
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10000 triangles
Source http//www.cs.utah.edu/wyman/classes/wave
lets/proj4/
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9000 triangles
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8000 triangles
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7000 triangles
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6000 triangles
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4000 triangles
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3000 triangles
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Polygon modeling techniques

• 3D Scanning
• Sorry, we cant do that
• Make a primitive shape
• Most modelers can make spheres, cones, cubes,
  etc. automatically
• Deform an existing shape
• Rotate, translate, scale
• Stretch (scale in one direction)
• Move the vertices around
• Add/remove vertices(Modeling tool will
  split/merge polygons accordingly)

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Solids of revolution

• Solids of revolution are common in the real world
• Lathes are machines specifically design to
  manufacture them

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Turning an object on a lathe
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An electronic latheMaking a wine glass in Maya

• Start by drawing the curve for the profile of the
  object

Source http//www.3dvf.com/modules/publish/Articl
es-Dossiers_1429_2.html
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Making a wine glass

• Start by drawing the curve for the profile of the
  object
• Select the revolve operation

Source http//www.3dvf.com/modules/publish/Articl
es-Dossiers_1429_2.html
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Making a wine glass

• Start by drawing the curve for the profile of the
  object
• Select the revolve operation
• Voila!

Source http//www.3dvf.com/modules/publish/Articl
es-Dossiers_1429_2.html
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Extruding an existing face

• Most modelers will let you pull a face out from
  the surface
• It will automatically create new polygons to
  rejoin the face to the surface

Source http//www.3d-station.com/pages/tutorials/
tutorial.php?id131page1
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Extruding an existing face

• Most modelers will let you pull a face out from
  the surface
• It will automatically create new polygons to
  rejoin the face to the surface

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Extruding an existing face
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Example Making a T-Shirt

• Start with a cube

Source http//zobal.free.fr/tuts/acm/en_acm2.htm
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Face extrusion

• Extrude the top face

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Face extrusion

• Pull the sides in to make the neck
• Do the same thing to make the arm

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Face deletion

• Right now, the arm is closed at the end
• Remove the face on the end to make an arm hole

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Removing one side

• Remove
• All the faces on the left side
• The bottom face
• And generally tweak the vertices until you like
  them

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Forming the whole thing

• Now clone the mesh
• Flip it over
• And join it to the old mesh

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Smoothing

• Most modeling tools have a smoothing function
• Warning this increases the number of polygons
  dramatically!
• After smoothing, tweak to taste

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Rendering

• The previous images were wireframes
• The just show the vertices and edges
• Rendering is the process of producing a final
  image from the polygon mesh

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Surface reflection
incident ray

• Surface reflectance is very complicated
• There are two main models of reflectance
• Specular (glossy) surfaces bounce it directly off
• Lambertian (matte) bounce it evenly in all
  directions

reflected rays
specular reflection (highlights)
incident ray
reflected rays
Lambertian reflection (matte/diffuse)
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Specular reflection

• Mirror-like reflection
• Mirrors are near-perfect specular reflectors
• Incident and reflected rays have at (almost) the
  same angle
• In practice, theres some scattering
• All wavelengths are (usually) reflected equally
• So reflection has the color of the light

incident ray
reflected rays
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Lambertian reflection

• Lambertian/diffuse/matte reflection
• Perfect non-glossy paint
• Light reflected equally in all directions
• Brightness depends on illumination angle
• When light hits and an angle, its spread out
  over a wider area (1/sin ? times wider)
• So the intensity of the light coming out is
  dimmed by sin ?
• Not all wavelengths are reflected equally
• The reflection has the color of the surface (at
  least if the light is white)

?
d
?
d/sin ?
beam spreads out by 1/sin ?
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Surface normals

• (For reasons that will be clearer later)
• We usually measure the angle a little differently
• We use the angle between the light and
• A line sticking straight out from the surface
• This is called the surfaces normal
• This means the dimming factor is cos ?
• Because we measured ? differently

surface normal
?
d
?
d/cos ?
beam spreads out by 1/cos ?
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Specular and diffuse reflection
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Normals in 3D

• Each point on a surface has a normal
• Vector that sticks out perpendicularly to the
  surface
• For polygonal models, all points in a polygon
  have the same normal
• Although rendering sometimes uses fake normals
  to generate a better appearance

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Flat shading

• Since all points on a face have the same normal
• So they really look flat
• And the boundaries are very noticeable

Source http//www.cgl.uwaterloo.ca/mmwasile/cs68
8/project.html
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Normal interpolation

• By interpolating the normals, we get a much
  smoother look without using more polys
• Each pixel in a face recalculates its normal
• Using an average of the normals of each vertex
• Weighted by distance from the vertex
• Closer vertices exert more influence

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Phong shading

• Phong shading is a common lighting model based on
• Specular and diffuse components
• With normal interpolation
• Ambient component
• Added evenly to all pixels
• Acts like a fill light

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Phong shading
Source wikipedia
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Whats in a model?

• Geometry
• All the vertices of the mesh
• Connectivity information (which vertices form
  which edges/faces)

• Material information
• Diffuse color
• Specular color
• Specular hardness
• Ambient color
• Different polys can have different materials
• Each polygon can only have one material

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Texture mapping

• Individually coloring each polygon would be
  tedious
• It would also take a lot of triangles to make a
  realistic object

Source http//escience.anu.edu.au/lecture/cg/Text
ureJ3D/printNotes.en.html
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Texture mapping

• Graphics hardware has the ability to paint faces
  with sections of a bitmap image
• Called textures
• Vertices are tagged with texture coordinates
• Specify a pixel within the texture
• Pixels in the interior of the face are painted
  with the pixels between the pixels of the
  vertices
• Texture is often used to make objects look more
  complicated than they really are
• There are no actual depressions between the
  bricks in this image

Source http//escience.anu.edu.au/lecture/cg/Text
ureJ3D/printNotes.en.html
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The curse of topology

• Texture maps are flat
• Most 3D objects arent
• To map a flat texture to a curved 3D object we
  have to
• Stretch some parts
• Squish others
• Cut holds (sometimes)

Source http//ligwww.epfl.ch/silic/predocschool/
Project/projectreport.htm
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Mesh
Source http//www.randomstuff.org.uk/laura/Galle
ry_digital_art/silenus.html
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Texture maps
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Mapped mesh
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With and without texture
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Normal mapping

• Normal mapping is another way to make a low poly
  model look like a high poly model
• Provide a texture map for normals
• Lets the artist paint changing normals over a
  single poly
• Normals respond to lighting as if they were part
  of the real geometry
• Q Why wouldnt we just paint dark and bright
  regions on a normal texture map?

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famous examples over the ages
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The death star trench sequence
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Super Mario 64 (Nintendo 1996)
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Super Mario Strikers (Nintendo 2005)
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Tomb Raider 1 (1996)
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Tomb Raider 7 (2006)
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High poly count modelingFinal Fantasy Advent
Children (2005)
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High poly count modelingFinal Fantasy Advent
Children (2005)