VIRTUAL ENVIRONMENT

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VIRTUAL ENVIRONMENT
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VIRTUAL ENVIRONMENT

• The Dynamics of Numbers
• Numerical Interpolation
• Linear Interpolation
• Non-Linear Interpolation
• The Animation of Objects
• Linear Translation
• Non-Linear Translation
• Shapes and Objects in between

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The Dynamics of Numbers

• Virtual Environment A Complex Numerical DB
• It changes one number to another
• Important Numerical Envelope of the change
• Example In animating the bouncing of a ball,
  its centroid is used to guide its motion
• Realistic motions are simulated by numerical
  interpolation
• Numerical Interpolation Linear and Non-Linear

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Numerical Interpolation

• Interpolation produces a function that matches
  the given data exactly.
• The function then can be utilized to approximate
  the data values at intermediate points
• It is also used to produce a function for which
  values are known only at discrete points, either
  from measurements or calculations.

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Numerical Interpolation

• Given data points
• Obtain a function, P(x)
• P(x) goes through the data points
• Use P(x)
• To estimate values at intermediate points
• For example, given data points
• At x0 2, y0 3 and at x1 5, y1 8
• Find the following
• At x 4, y ?

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Numerical Interpolation
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Linear Interpolation

• Linear Interpolation - Simplest interpolation
  method
• Apply LI(a sequence of points) ? A polygonal line
  where each straight line segment connects two
  consecutive points of the sequence
• Therefore, for every segment (P,Q)
• P(x) (1 - x)P xQ where x ? 0,1
• By varying x from 0 to 1, we get all the
  intermediate points between P and Q
• P(x) P for x 0 and P(x) Q for x 1.
• We get points on the line defined by P, Q,
  when 0 x 1

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Non-Linear Interpolation

• A linear interpolation ensures that equal steps
  in the parameter x give rise to equal steps in
  the interpolated values
• It is often required that equal steps in x give
  rise to unequal steps in the interpolated values
• We can achieve this using a variety of
  mathematical techniques
• For example, we could use trigonometric functions
  or polynomials to achieve this

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Trigonometric interpolation

• sin2(x) cos2(x) 1
• If x varies between 0 and p/2
• cos2(x) varies between 1 and 0, sin2(x) varies
  between 0 and 1
• which can be used to modify the two interpolated
  values n1 and n2 as follows
• nn1cos2(t)n2sin2(t) for 0 t p/2

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Trigonometric interpolation
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Trigonometric interpolation

• Let n1 1 and n2 3

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Cubic interpolation
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Cubic interpolation
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Cubic interpolation
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The Animation of Objects

• Newtons Laws of Motion provide a useful
  framework to predict an objects behavior under
  dynamic conditions
• This scenario can be simulated within a Virtual
  Environment
• It can be achieved through a simple linear
  translation of objects

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Uses of Translation

• Modeling transformations
• build complex models by positioning simple
  components
• transform from object coordinates to world
  coordinates
• Viewing transformations
• placing the virtual camera in the world
• i.e. specifying transformation from world
  coordinates to camera coordinates
• Animation
• vary transformations over time to create motion

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Linear Translation

• Consider an object located at VEs origin and
  assigned a speed S0 across the XY plane
• To simulate its sliding movement, the x z
  coordinates of the object must be modified
  (tnow tprev)V0
• The objects new velocity can be computed after
  it is bouncing off the boundary.

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Linear Translation
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Non-Linear Translation

• Consider an object moving along the x-axis in 1s,
  pause momentarily and then returns to its
  original position in 2s
• The non-linear movement of the object can be
  simulated by computing the x-translation as a
  function of time
• At time t1, the translation begins. At time t2,
  i.e.,(t11) it pauses momentarily and at time t3,
  i.e., (t13), it comes to rest
• t T t1 while t1 T t2 T
  Current time
• t (T t1 1)/2 while t2 T t3 t- Control
  parameter

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Euler Angles

• Orientation of an object in computer graphics is
  often described using Euler Angles
• Orientation of an object in computer graphics is
  often described using Euler Angles
• RxRyRz
• Any axis order will work and could be used.
• Yaw, Pitch Roll

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Problem with Euler Angles

• Rotations not uniquely defined
• Ex (z, x, y) roll, yaw, pitch (90, 45, 45)
  (45, 0, -45)takes positive x-axis to (1, 1, 1)
• Cartesian coordinates are independent of one
  another, but Euler angles are not
• Remember, the axes stay in the same place during
  rotations
• Gimbal Lock
• Term derived from mechanical problem that arises
  in gimbal mechanism that supports a compass or a
  gyro
• Second and third rotations have effect of
  transforming earlier rotations, we lose a degree
  of freedom
• ex Rot x, Rot y, Rot z
• If Rot y 90 degrees, Rot z is equivalent to
  -Rot x
• With Euler Angles, knowing the transformations
  for the entire rotation does not help us with
  the intermediate rotations. Each is a unique set
  of three rotations about the x,y and z axes

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Quaternions

• Invented by Sir William Hamilton (1843)
• Do not suffer from Gimbal Lock
• Provide a natural way to interpolate intermediate
  steps in a rotation about an arbitrary axis
• Are used in many position tracking systems and VR
  software support systems

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Quaternions

• You can think of quaternions as an extension of
  complex numbers where there are three different
  square roots of -1.
• q w i x j y k z where
• i ?(-1), j ?(-1), k ?(-1)
• ij k, jk i, ki j
• ji -k, kj -i, ik -j
• You could also think of q as a value in
  four-dimensional space, q w, x, y, z
• Sometimes written as q w, v where w is a
  scalar and v is a vector in 3-space

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Shape and Object Inbetweening

• Shape Inbetweening
• It is a technique in the cartoon industry to
  speed up the process of creating art works
• The inbetween objects are derived from two key
  images drawn by a skilled animator
• The images are called as Key Frames
• The key frames shows an object in two different
  positions
• The number and position of inbetweenig images
  determines the dynamics of final animation

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Shape Inbetweening
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Object Inbetweening

• By inbetweening the z-coordinate, the above
  technique can be applied to 3D contours
• We cannot expect to interpolate between two
  different complex objects and geometrically
  consistent
• Given suitable geometric definitions, it is
  possible to transform one object into another and
  create subtle animation