Advanced Game Engineering RealTime Rendering

1
Advanced Game EngineeringReal-Time Rendering
Transforms

• ? ?
• 2006/03/22
• Dept. of CSE, Univ. of Incheon
• pucktan_at_incheon.ac.kr

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Contents

• Transforms
• Basic Transforms
• Special Matrix Transforms and Operations
• Quaternions
• Vertex Blending
• Projection

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Quaternions

• ? ?
• Quaternions wer invented back in 1843 by Sir
  William Rowan Hamilton.
• An extension to the complex numbers
• A powerful tool for constructing transforms
• They are superior to both Euler angles and
  matrices.
• Especially when it comes to rotations and
  orientations.
• ex) Quaternions are represented very
  compactly. Thay can be used for the stable
  interpolation of orientations
• A quaternion has four components.
• Expression

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Quaternions

• Mathematical Background
• Definition of Quaternions
• variable is called the real part of a
  quaternion,
• Imaginary part is , and and
  called imaginary units.
• For the imaginary part, , we can use all the
  normal vector operations,such as addition,
  scaling, dot product, cross product, and more.

A quaternion can be defined in the following
ways, all equivalent.
(3.28)
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Quaternions

• The multiplication operation between two
  quaternions, and , is derived as shown
  below.
• The multiplication of the imaginary units is
  noncommutative.

Multiplication
(3.29)
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Quaternions

• The definitions of addition, conjugate, norm, and
  an identity
• When is simplified, the
  imaginary parts cancel out and only a real part
  remains.
• Expression of Norm

Addition Conjugate Norm Identity
(3.30)
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Quaternions

• We derive a formula from Definition of the Norm
• Multiplicative inverse

(3.31)
(3.32)
(3.33)
?
(3.34)
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Quaternions

• The formula for the inverse uses scalar
  multiplication
• ? (3.29) ?? ??? ?????? ??!!
• 
  
  , which means that scalar multiplication is
  commutative
• ?

Conjugate rules Norm rules Laws of
Multiplication Linearity Associativity
(3.35)
(3.36)
(3.37)
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Quaternions

• A unit quaternion, is such
  that .
• For complex numbers, a two-dimensional unit
  vector
• The equivalent for quaternions is

(3.38)

(3.40)
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Quaternions

• Quaternion Transforms
• Unit quaternions is that they can represent any
  three-dimensional rotation
• Unit quaternions is that this representation is
  extremely compact and simple
• Example, put the four coordinates of a point or
  vectorinto the components of a quaternion ,
  and assume that we have a unit quaternion
  . Then
• rotates around the axis by an
  angle

(3.42)
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Quaternions

• This rotation, which clearly can be used to
  rotate around any axis.

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Quaternions

• Negating the axis, , and the real part, ,
  creates a quaternion which rotates exactly as the
  original quaternion does.
• Given two unit quaternions, and , the
  concatenation of first applying and then
  to a quaternion, ,is given by Equation
• is the unit quaternion representing
  the concatenation of the unit quaternions and

(3.43)
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Quaternions

• Matrix Conversion
• Some systems have matrix multiplication
  implemented in hardware.
• A quaternion, , can be converted into a
  matrix , as expressed in Equation

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Quaternions

• Ths scalar is .
• Once the quaternion is constructed, no
  trigonometric functions need to be computed, so
  the conversion porcess is efficient in practice.

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Quaternions

• The reverse conversion, from an orthogonal
  matrix.
• Differences made from matrix in Equation
• The trace, tr(M), of a matrix, M, is simply the
  sum of the diagonal elements of the matrix.

(3.46)

(3.47)
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Quaternions

• This result yields the following conversion for a
  unit quaternion
• To have a numerically stable routine. Divisions
  by small numbers should be avoided. Set

(3.48)

(3.49)
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Quaternions

• Which in turn implies that the largest of
  and u determine which of
  is largest. If is largest..
• Used to compute the largest of
• Equation (3.46) is used to calculate the
  remaining components of

(3.50)
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Quaternions

• Spherical Linear Interpolation
• operation that given two unit quaternions,
  and , and a parameter , computes
  an interpolated quaternion.
• This is useful for animating object, for example
  
• Expression by the composite quaternion, ,
  below
• For software implementations
• To compute , which is needed in the equation
  above,, the following fact can be used

(3.51)

(3.52)
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Quaternions

• Example
• The slerp function is perfectly suited for
  interpolating between two orientations.
• This procedure is not as simple with the Euler
  transform, because when several Euler angles need
  to be interpolated, gimbal lock can occur.

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Quaternions

• When more than two orientations, say
  are available, and we want to
  interpolate..? slerp could be used in a
  straightforward fashion.
• A better way to interpolate is to use some sort
  of spline.
• Expression of spline
• The squad function is constructed from repeated
  spherical interpolation using slerp. (ref.
  Section 12.1.1 ??? ?? ?? ????)

(3.53)
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Quaternions

• Rotation from One Vector to Another
• A common opertation is transforming from one
  direction s to another direction t.
• Normalize s and t
• Compute the unit rotation axis, called u
• and
  , where is the angle between s
  and t
• The quaternion that represents the rotation from
  s to t is then . .

(3.55)
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Quaternions

• Stability problems appear for both methods when s
  and t point in opposite directions, as a division
  by zero occurs.
• ? any axis of rotation perpendicular to s can be
  used to rotate to t

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Vertex Blending

• Imagine that an arm of a digital character is
  animated using two parts, a forearm and an upper
  arm.
• Then the joint between these two parts will not
  resemble a real elbow.? because two separate
  objects are used.

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Vertex Blending

• Vertex blending is one possible solution to this
  problem.
• Several other names have been This technique.
• Skinning
• Enveloping
• Skeleton-subspace deformation
• The forearm and the upper arm are animated
  separately as before, but at the joint the two
  parts are connected through an elastic skin.
• ? called stitching

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Vertex Blending

• One step further
• On can allow a single vertex to be transformed by
  several different matrices, with the results
  weighted and blended together.

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Vertex Blending

• The entire arm may be elastic, all vertices may
  be affected by more than one matrix, the entire
  mesh is often called a skin.

Figure 3.11. Areal example of vertex blending.
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Vertex Blending

• The vertex blending equation is
• p is the original vertex.
• u(t) is the transformed vertex whose position
  depends on the time t.
• is the ith bones world transform that
  changes with time to animate the object.
• The matrix transforms from the initial
  bones coordinate system to world coordinates.

(3.58)
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Vertex Blending

• Vertex blending can be done by the host CPU.
• If the application is the bottleneck, then it is
  worth doing this operation on the graphics
  hardware, if possible.
• DirectX 8 skinning method
• through a choice of either of two pipelines, the
  fixed function path
• This method supports a 256 matrix palette for
  vertex blending
• Each vertex can use a maximum of four of these
  matrices.
• through the vertex shader (Section 6.5)
• This matrix palette system cannot be used in
  conjunction with the vertex shader, as it is part
  of an entirely different pipeline.

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Vertex Blending

• Problem of Vertex blending
• Unwanted folding, twisting, and self-intersection
  can occur.

Figure 3.12. Problems with vertex blending
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