CS 352: Computer Graphics

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CS 352 Computer Graphics
Chapter 4 Geometric Objects andTransformations
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Overview

• Scalars and Vectors
• Coordinates and frames
• Homogeneous coordinates
• Rotation, translation, and scaling
• Concatenating transformations
• Transformations in OpenGL
• A virtual trackball

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Perspective

• How is it that mathematics can model the (ideal)
  world so well?

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Where we're headed

• Virtual trackball program
• Similar to primer demo
• Read in a 3D object and display it.
• Handle virtual trackball rotations
• Later add lighting and shading

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Background linear algebra

• Quick review of important concepts
• Point location (x, y, z)
• Vector direction and magnitudeltx, y, zgt

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Vectors

• Magnitude of a vector v
• Direction of a vector, unit vector v
• Affine sum P a Q (1-a) R

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Dot Product

• Def u v ux vx uy vy uz vz
• u v u v cos ?
• Uses
• Angle between two vectors?
• Are two vectors perpendicular?
• Do two vectors formacute or obtuse angle?
• Is a face visible?(backface culling)

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Cross Product

• u ? v ltuyvz - uzvy, uzvx - uxvz, uxvy - uyvxgt
• Direction normal to plane containing u, v (using
  right-hand rule in right-handed coordinate
  system)
• Magnitude uv sin ?
• Uses
• Face outward normal?
• Angle between vectors?
• Do two line segments intersect?

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Face outward normals

• How to find the outward normal of a face?
• Assume that vertices are listed in a standard
  order when viewed from the outside --
  counter-clockwise
• Cross product of the first two edges is outward
  normal vector
• Note that first corner must be convex

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Coordinate systems and frames

• Hierarchical modeling
• May deal with many coordinate systems viewer,
  model, world, viewport
• Frame origin basis vectors (axes)
• Need to transform between frames
• E.g. reading in a world description with several
  objects

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Transformations

• Changes in coordinate systems usually involve
• Translation
• Rotation
• Scale
• Rotation and scale can be represented as 3x3
  matrices, but not translation
• We're also interested in a perspective
  transformation
• We use 4D "Homogeneous coordinates"

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Homogeneous Coordinates

• A point (x, y, z, w) where w is a "scale factor"
• Converting a 3D point to homogeneous coordinates
  (x, y, z) ? (x, y, z, 1)
• Transforming back to 3-space divide by w
• (x, y, z, w) ? (x/w, y/w, z/w)
• (3, 2, 1) same as (3, 2, 1, 1) (6, 4, 2,
  2) (1.5, 1, 0.5, 0.5)
• Where is the point (3, 2, 1, 0)?
• Point at infinity or "pure direction."
• Used for vectors (vs. points)

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Homogeneous transformations

• Most important reason for using homogeneous
  coordinates
• All affine transformations (line-preserving
  translation, rotation, scale, perspective, skew)
  can be represented as a matrix multiplication
• You can concatenate several such transformations
  by multiplying the matrices together. Just as
  fast as a single transform!
• Modern graphics cards implement homogeneous
  transformations in hardware

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Translation
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Scaling

• Note that the scaling fixed point is the origin
• See transformation.exe

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Example reading in an object

• Object description with vertices and faces How
  to transform to unit cube?
• vertices 4
• 10 10 10
• 40 20 20
• 30 40 10
• 30 10 -10
• faces 4
• 1 2 3
• 2 4 3
• 1 4 2
• 1 3 4

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Point normalization

• Find min and max values of X, Y, Z
• Find center point, Xc, Yc, Zc
• Translate center to origin (T1)
• Scale (S)
• Translate center to (½, ½, ½) (T2)
• P' T2 S T1 P
• Modeling matrix M T2 S T1
• Note apparent reversed order of matrices

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Rotation

• General rotation about anaxis v by angle u with
  fixed point p
• With origin as fixed point, about x, y, or
  z-axis

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Rotating about another point

• How can I rotate around another fixed point, e.g.
  1, 2, 3?
• Translate 1, 2, 3 -gt 0, 0, 0 (T)
• Rotate (R)
• Translate back (T-1)
• T-1 R T P P'

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Rotating about another axis

• How can I rotate about an arbitrary axis?
• Can combine rotationsabout z, y, and xRx Ry Rz
  P P'
• Note that ordermatters and anglescan be hard to
  find

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Example spinning the object

• How to make the object rotate by 5 degrees about
  the z axis?
• Multiply all the points by the appropriate
  rotation matrix
• How to animate a continuous rotation?
• Every few ms, rotate and redisplay
• How to animate rotation and flying around?
• First apply rotation matrix, then translation
  matrix
• Better, multiply the two matrices together M
  T R (note apparent reversed order)
• P' M P
• Always work from original points, to avoid error
  accumulation

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Orthographic projection

• Zeroing the z coordinate with a matrix
  multiplication is easy

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Perspective projection

• Perspective projection can also be done with a
  single matrix multiply using homogeneous
  coordinates
• In the default camera settings, it's easy
• What happens when you divide through by w?

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Transformations in OpenGL

• Three coordinate frames in OpenGL
• World coordinates
• Eye (camera) coordinates
• Window coordinates
• Two transformations convert between them
• Modeling xform world coords -gt eye coords
• Projection xform eye coords -gt window coords

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OpenGL Transformations

• GL has a current transformation matrix (CTM) or
  Matrix Mode
• glMatrixMode(GL_MODELVIEW)
• glLoadIdentity()
• glTranslatef(4.0, 5.0, 6.0)
• glRotatef(45.0, 1.0, 2.0, 3.0)
• glTranslatef(-4.0, -5.0, -6.0)
• Vertices are transformed as they flow through the
  pipeline

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OpenGL Transformations 2

• If p is a vertex, pipeline produces Cp
  (post-multiplication only)
• Can concatenate transforms in CTM
• C ? I
• C ? T(4, 5, 6)
• C ? R(45, 1, 2, 3)
• C ? T(-4, -5, -6)
• C T-1 S T
• Note that last transform defined is first applied
  see transformation.exe

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Setting up Viewing Transform

• To set up an orthographic projection
  transformation for the viewing transform
• glMatrixMode(GL_PROJECTION)
• glLoadIdentity()
• glOrtho(0.0, width, 0.0, height, -1, 1)

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Manipulating matrices

• Loading a matrix
• GLfloat myarray16
• glLoadMatrixf(myarray)
• Pushing and Popping
• glPushMatrix()
• glTranslatef()
• glRotatef()
• draw something
• glPopMatrix()

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Putting it all together

• Load vertices and faces of object.
• Normalize put them in (-1, 1, -1, 1, -1, 1) cube
• Put primitives into a display list
• Set up viewing transform to display that cube
• Set modeling transform to identity
• To spin the object, every 1/60 second do
• Add 5 degrees to current rotation amount
• Set up modeling transform to rotate by current
  amount

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Virtual Trackball

• Imagine a trackball embedded in the screen
• If I click on the screen, what point on the
  trackball am I touching?

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Trackball Rotation Axis

• If I move the mouse from p1 to p2, what rotation
  does that correspond to?

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Virtual Trackball Rotations

• Rotation about the axis n p1 x p2
• Angle of rotation use
• p1 p2 p1 p2 cos ?
• Fixed point if you use the -1, 1 cube, it is
  the origin

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Data structures needed

• An array of vertices, old_verticesn3
• An array of normalized vertices, verticesn3,
  in the -1, 1 cube
• An array of faces containing vertex indexes, int
  facesnmax_sides. Use facesn0 to store
  the number of sides. Set max_sides to 12 or so.

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Virtual Trackball Program

• Stage 1
• Read in, normalize object
• Put into display list (with edges)
• Display with rotation
• Stage 2
• Virtual trackball rotations
• Stage 3
• Perspective, zoom
• Lighting/shading