Interactive Computer Graphics Transformations

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Interactive Computer GraphicsTransformations

• James Gain and Edwin Blake
• Department of Computer ScienceUniversity of Cape
  Town
• July 2002jgain_at_cs.uct.ac.za

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Map of the Lecture

• Vector Geometry
• Vector, Affine and Euclidean Spaces
• Coordinate Systems
• Euclidean, Polar, Homogenous Coordinate Systems
• Transformations
• Coordinate System, Object, Viewing Transformations

Transformations
Viewing
Shading
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Vector Geometry

• Enables fundamental operations with points and
  vectors
• Independent of any particular co-ordinate system
• Robust, simple and efficient algorithms
• Operations
• Closest point, inside/outside and intersection
  tests
• Entities
• Vectors, points, lines, planes and polygons

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Points

• Discrete positions
• Represented by co-ordinate triples
  relative to co-ordinate axes
• Class Point double x,y,z
• But addition, subtraction, multiplication and
  division of points are not defined

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Illegal Operation on Points

• The results of arithmetic operations on points
• depend on the co-ordinate system
• have no geometric meaning
• A contradiction finding the midpoint
  is legal.

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Vectors

• Pointers indicating direction and magnitude
• Also represented by co-ordinate triples
  but with no fixed position
• Class Vector double i,j,k
• Clearly distinct from points

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Vector Space

• Domain in which vectors live
• Consists of
• A set of vectors
• Two closed operations addition and scalar
  multiplication
• Special zero vector, , for which
  and

Can be of any dimension, . Typically 2D or
3D Conceptually similar to an object-oriented
class (data vector, methods operations)
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Implementing a Vector Space

• class Vector
• double i,j,k
• inline void scale(double c)
• i i c j j c k k c
• inline Vector add(Vector a, Vector b)
• i a.i b.i
• j a.j b.j
• k a.k b.k

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Affine Space

• Domain in which points live
• Consists of
• A set of points
• An associated vector space
• Two extra operations
• (a) subtraction of two points to form
  a vector
• (b) addition of a point and vector to
  produce a new point
• Unlike the zero vector there is no distinguished
  point

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Implementing an Affine Space

• class Point
• double x,y,z
• inline void sub(Point q, Vector v)
• v-gti q.x - p.x v-gtj q.y - p.y
• v-gtk q.z - p.z
• inline void plus(Point p, Vector v)
• x p.x v.i y p.y v.j
• z p.z v.k

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Addition and Multiplication of Points

• undefined
• vector
• BUT midpoint calculation point
• Recast in terms of legal operations
• point
• point iff

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Euclidean Space

• An affine space with the additional concept of
  distance
• Consists of
• An affine space
• Two new operations, dot product and
  cross product
• Distance between two points length
  of the vector between them

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Length of a Vector

• From Pythagoras the length of a vector is
• In order to normalize a vector it is scaled by
  the reciprocal of its length

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Dot Product

• Variously known as inner, dot, or scalar product.
• Implementation
• double dot(Vector v, Vector w)
  return (a.i b.i a.j b.j a.k b.k)
• where is the angle between
  and fitted tail to tail.

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Cross Product

• Variously known as the outer, cross or vector
  product.
• and
• Implementation
• Vectorcross(Vector a, Vector b) i
  (a.j b.k) - (a.k b.j) j (a.k
  b.i) - (a.i b.k) k (a.i b.j) - (a.j
  b.i)
• Produces a vector orthogonal to the
  arguments in the same sense as the
  co-ordinate axes.

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Lines

• From two points, and , a spanning line
  segment or interpolating line can be found.
• Geometric entities usually have two forms
• Implicit
• Parametric
• Line equations (a point on the line
  satisfies)
• Implicit
• Parametric ( parameter)

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Planes

• From three points, , and , an
  interpolating plane can be found.
• Plane equations (a point on the plane
  satisfies)
• Implicit
  ( normal, point
  on the plane)
• Parametric
  ( parameters)
• The implicit equation can be rewritten in the
  more familiar form where

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Derivation of the Implicit Plane

• Given the three points, , and .
• Find a vector normal to the plane
• This assumes that the points are not collinear
  (all in a straight line).
• Set

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Exercise Backface Culling

• Given
• A left-handed 3D co-ordinate system
• A triangle with vertices, , and
  , which form a clockwise ordering when
  viewed from the front
• A viewpoint
• Design an algorithm which determined whether the
  back or front of is visible from

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Solution Backface Culling

• IF THEN RETURN undefined
• IF THEN RETURN side-on
• IF THEN RETURN back-facing
• ELSE RETURN front-facing

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Cartesian Co-ordinate Systems

• Consists of a reference point (the origin) and
  three mutually perpendicular lines passing
  through this point (the axes) labelled
  
• A labelling of the axes has either a left-handed
  or right-handed orientation
• Right-hand rule a co-ordinate system is
  right-handed if, whenever the thumb is aligned
  with ve and the forefinger with ve ,
  then the index finger points in the direction of
  ve . Otherwise the orientation is left-handed
• If unit vectors are aligned with the different
  axes then together they form an orthonormal
  (orthogonal and normal) basis for the co-ordinate
  system

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Polar Co-ordinate Systems

• Alternative system using rotation angles
• Orient a ray by rotating in the plane (
  ) and then elevating it towards the -axis (
  ). A point is then located at a distance ( )
  along the ray
• A point is defined by the polar
  co-ordinates
• Conversion from polar to Cartesian
  co-ordinates

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Transformations

• Objects typically consist of a collection of
  interrelated points, e.g. a polygon-mesh
• A predefined object can be modified to an
  arbitrary location, orientation and size by
  applying the same transformation to all its
  constituent points
• BUT, this only work if the transformation is
  affine
• Preserve straight lines
• Need only transform the endpoints of a line
  rather than every single point along it
• Transformations are extremely useful in both
  modelling and rendering

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Basic 2D Transformations

• Scale
• About the origin.
• By factors and .
• If then balanced,
  otherwise unbalanced.
• Translate
• Along the vector .
• Reflect
• In one or both of the co-ordinate
  axes.

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Further 2D Transformations

• Shear
• Parallel to an axis.
• By an angle .
• Rotate
• About the origin.
• Counter-clockwise by an angle .

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Derivation of Rotation

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The Matrix Representation of 2D Transformations

• A transformation can be encoded by
  pre-multiplying a transformation matrix,
  with a column vector of co-ordinates,
• Be aware that some texts post-multiply a
  co-ordinate row vector with a transformation
  matrix
• The transformation matrices are transposed with
  respect to each other.

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Transformation Matrices

• Scale (by factors and ) Shear
  (parallel to by )
• Reflect (in ) Rotate (by angle )

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Homogenous Co-ordinates

• 2D Translations cannot be encoded using
  matrix multiplication because points at the
  origin never move.
• Instead use homogenous co-ordinates
• These are projections of 3D points along a
  ray from the origin onto the plane

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Homogenous Projections

• An infinite number of homogenous co-ordinates map
  to every 2D point.
  are all equivalent to .
• Points at infinity have and cannot be
  projected onto the plane because this
  would involve division by zero
• These points can be used to represent vectors
• Conversion

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Matrices for Homogenous Co-ordinates

• where is a standard
  transformation
• Translation

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Concatenating Transformations

• It is frequently necessary to perform a sequence
  of transformations on the same object.
• These can be merged into a single transformation
  by matrix multiplication.
• Example shearing then reflecting

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Order is Important

• Matrix multiplication is not commutative
  
• The order in which transformations are combined
  affects the outcome.

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Non-Standard Transformations

• Centred at a point other than the origin
• Or involving a line which is not parallel to any
  co-ordinate axis.
• Example A rotation by about the point
• (1) Translate to the origin,
• (2) Rotate by about the origin,
• (3) Translate the origin back to ,
• The combined transformation is

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Matrices for Off-Origin Rotation
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Exercise Composite Reflection

• Write down but do not multiply out the composite
  transformation that reflects a point in an
  arbitrary line passing through point and
  , where
• Hint you will need to use translation, rotation
  and reflection

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Solution Composite Reflection

• Translate to the origin,
• Rotate the vector onto the -axis,
  , by the angle between and
• Reflect in the -axis,
• Reverse the rotation,
• Reverse the translation,

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Composite Reflection Matrices

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3D Transformations

• Employ 3D homogenous co-ordinates
• Transformation matrices
• translation scaling
  reflection

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3D Rotations

• Rotation (counterclockwise around the positive
  axis)
• about about about

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Transformation Pipeline
Modelling Transform
Object in object co-ordinates
Object in world co-ordinates
Viewing Transform
Object in viewing co-ordinates
Object in 2D screen co-ordinates
Perspective Projection
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Modelling Transforms

• Transformation of an object from a local
  co-ordinate frame to the world co-ordinate frame.
• The local system ( ) is expressed relative to
  the world system ( ) as three vectors ,
  and at an origin
• From to

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Viewing Transforms

• Need to transform an arbitrary co-ordinate system
  to the default viewing co-ordinate system (with
  the eye at the origin and the screen
  perpendicular to ).
• The camera is specified in world co-ordinates as
• A camera position at
  .
• A look point (screen centre) at
  .
• An up vector (which orients the camera).
• Transformations
• Translate the camera to the origin.
• Scale by so that the camera to screen
  distance is .
• Align the vector through rotation with
  the axis.
• Rotate around so that is aligned with
  the axis.

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Rotation about an Arbitrary Vector

• Task
• Rotate around the vector from to by
  an angle
• Solution
• Translate to the origin
• Rotate around by so that is
  aligned with the plane
• Rotate around by to align with
  the axis
• Rotate around by
• Reverse rotations 2,3
• Reverse translation 1

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OpenGL Transformations

• Current Transformation Matrix
• The transformation state of the system
• A concatenated matrix applied to all subsequent
  vertices
• Transformations in OpenGL
• In OpenGL CTM is a product of the Projection
  (GL_PROJECTION) and Model-View (GL_MODELVIEW)
  matrices
• Functions to change matrix state, set and
  concatenate matrices
• Functions to rotate, translate and scale

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Example OpenGL Transformations

• Task halve the x-length of an object centred at
  and rotate it by
  about the vector
• glMatrixMode(GL_MODELVIEW)
• glLoadIdentity()
• glTranslatef(4.0, 5.0, 6.0)
• glRotatef(45.0, 1.0, 2.0, 3.0)
• glScalef(0.5, 1.0, 1.0)
• glTranslatef(-4.0, -5.0, -6.0)

• Access the Model View matrix
• Load the identity transformation
• OpenGL tansformations added in reverse order
• Translate the origin to
• Rotate by about
• Scale by
• Translate to the origin

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Controlling OpenGL Matrices

• Order of Transformations
• The transformation specified most recently is the
  one applied first
• Conceptually analogous to pushing matrices onto a
  stack matrices are applied in the reverse of
  their specification
• Matrices can be loaded or concatenated directly
• glLoadMatrix(tflat) set current state to T
• glMultMatrix(tflat) postmult T with current
  state
• Glfloat tflat16 is an expansion of T44
  arranged in columns ( i.e tflat2 tmat02 )
  
• The state of a matrix can be saved
• glPushMatrix() saves the current matrix
• glPopMatrix() restores the last saved state

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Euler Angles (Azimuth, Elevation, Roll)

• Historically popular but flawed
  parametrization of orientation
• A general rotation is constructed as a
  sequence of rotations about 3 mutually
  orthogonal axes rolls about , and
• The order of the rolls is significant
• Arbitrarily choose the ordering
  A roll about by ,
  followed with a roll about
  by , and finally a roll about by
• Euler angle interpolation is not very stable.
  Especially with a small rotation in one axis and
  large rotation in another

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Gimbal Lock

• A gimbal mechanism consists of three concentric
  rings on pivots which support a compass or
  gyroscope.
• Gimbal lock occurs when two of the rings are
  accidentally aligned.
• Euler angles are also susceptible. If one axis is
  rolled into alignment with another then a degree
  of rotation freedom is lost.
• Unlike translations, rotations relative to
  separate axes are not independent.
• Example a -roll of rotates the
  -axis onto the -axis so that rolls about
  and are indistinguishable.

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Lerping Euler Angles

• Two Euler rotations
  and can be linearly
  interpolated to obtain inbetween rotations
• Problems
• This does not produce a natural steady rotation
  about a single vector but may instead cause weird
  oscillations. Reason the rolls about
  are not independent.
• The resulting interpolation differs depending on
  the ordering of the Euler angles. Reason the
  rolls about are not commutative.

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Example Poor Euler Angle Interpolation

• The interpolations
  and
  have the same start and end orientations but
  different intermediate orientations.

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Quaternions

• Aim to
• Guarantee a direct and steady rotation between
  any two key orientations
• Be independent of any particular co-ordinate
  system
• Quaternions were invented by Sir William
  Hamilton, after 10 years of work, on 16 October
  1843. In his elation he carved the formulae into
  the nearby Broome Bridge in Dublin
• They represent an orientation by a
  counter-clockwise rotation angle ( ) about an
  arbitrary vector ( )
• Advantages
• combining quaternions more efficient than matrix
  multiplication
• simple to convert between angle-axis, quaternion
  and transformation matrix representations of
  rotation

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Form of a Quaternion

• 
  
• such that
• are imaginary axes (
  are imaginary numbers) and
  is a real axis
• and are co-ordinates relative to
  these four axes
• In 3D a point
  s.t. with
  co-ordinates and axes
  lies on a sphere of radius
• Similarly, for quaternions the rotation with
  
  lies on a 4D hypersphere of radius

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Lerping vs. Slerping Quaternions

• Inbetweening key Quaternions and
  by linearly interpolating their
  components does not
• Produce equal changes in the quaternion
  for equal steps in . They speed up in
  the middle.
• Ensure vectors remain on the hypersphere.
• Rather use spherical linear interpolation
  (Slerping) which step through a constant
  angle.

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Evaluating Quaternions

• Advantages
• Flexible.
• No parametrization singularities.
• Smooth consistent interpolation of orientations.
• Simple and efficient composition of rotations.
• Disadvantages
• Each orientation is represented by two
  quaternions.
• Represent orientations not rotations (
  about is the same quaternion as about
  ).
• Complex!