1. Interactive Computer Graphics Transformations
James Gain and Edwin Blake
Department of Computer Science University of Cape Town
July 2002
Collaborative Visual Computing Laboratory

2. Interactive Computer Graphics Contents
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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5. 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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8. 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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10. Implementing an Affine Space
class Point
{ double x,y,z;
inline void sub(Point q, Vector * v)
{ v->i = q.x - p.x; v->j = q.y - p.y;
v->k = 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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11. 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:
Vector::cross(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)
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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18. 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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19. 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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20. 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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21. 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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22. 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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24. 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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25. Further 2D Transformations
Shear
Parallel to an axis.
By an angle .
Rotate
About the origin.
Counter-clockwise by an angle .
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26. Derivation of Rotation
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27. 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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28. Transformation Matrices
Scale (by factors and ) Shear (parallel to by )
Reflect (in ) Rotate (by angle )
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29. 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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30. 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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31. Matrices for Homogenous Co-ordinates
where is a standard transformation
Translation:
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32. 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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34. 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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35. Matrices for Off-Origin Rotation
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36. 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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37. 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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38. 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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41. Transformation Pipeline
Modelling Transform
Object in
object
co-ordinates
Object in
world
co-ordinates
Viewing Transform
Object in
2D screen
co-ordinates
Perspective Projection
Object in
viewing
co-ordinates
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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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44. 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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45. 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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46. 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, );
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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47. 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 tflat[16] is an expansion of T[4][4] arranged in columns ( i.e tflat[2] = tmat[0][2] )
The state of a matrix can be saved:
glPushMatrix() saves the current matrix
glPopMatrix() restores the last saved state
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48. 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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51. 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 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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54. 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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55. 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!
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