Rotation and Translation in Computer Graphics using 4 x 4
Homogenous Matrices and Quaternions
Lab. Eng., MSc Matti Pitkälä
Department of Automation
Laboratory, Lahti Polytechnic
Lahti, Finland
Email: [email protected]
Abstract
This paper introduces some of the key ideas in computer
graphics, for example how we use the principles of
mathematics using transformations such as rotation and
translation in 2D and 3D-viewing. What are possibilities to
achieve them in computer graphics.
1. Introduction
1.1 Overview of the problem
Computer graphics deals with the problem of image
synthesis. Given a model (usually mathematically based)
the problem of computer graphics is to produce realistic
image data which may be viewed on a graphics display
device. The process of producing the image data from the
scene model is called rendering. That images are
synthesized from mathematical models implies that
computer graphics is a mathematically based subject.
Image synthesis involves the physics of light, properties of
materials, etc. Animated imagery involves finite element
analysis, kinematics, sampling theory and other
mathematically based fields. The study of computer
graphics necessarily involves the study of many areas of
mathematics. An elementary view of some of these topics
are given in the following sections. Persons, who have an
interest in computer graphics should study so much
mathematics as possible.
The reverse problem of starting with image data and
recovering information is called image processing.
1.2 2D Viewing
We consider the problem of representing 2D graphics
images, which may be drawn as a sequence of connected
line segments. Such images may be represented as a matrix
of 2D points [xi yi].
For example pseudo-programming notation is used to
describe the viewing transformations and data object
representations.
square =:0 0 10 0 10 10 0 10 0 0
Figure 1: A square
The idea behind this representation is that the first point
represents the starting point of the first line segment drawn
while the second point represents the end of the first line
segment and the starting point of the second line segment.
The drawing of line segments continues in similar fashion
until all line segments have been drawn. A matrix having
n+1 points describes a figure consisting of n-line segments.
It is sometimes useful to think of each pair of consecutive
points in this matrix representation:
As a vector so that the square shown in Figure1 is the result
of drawing the vectors shown in Figure 2.
Figure 2: The Vectors in a square
1.2. 1 Rotation
Now we wish to rotate a figure around the origin of 2D
coordinate system. Figure 3 shows the point being rotated
degrees (by convention, counter clock-wise direction is
positive) about the origin.
Figure 3: Rotating a point about the origin
To rotate point (x1,y1) to point (x2,y2) we observe:
From the illustration we know that
sin(A+B) = y2/r
cos(A+B) = x2/r
sin(A) = y1/r
cos(A) = x1/r
And now, if we for example want to rotate our square 90degrees counter-clockwise (+), we get for the matrix:
cos90=0, sin90=1,-sin90=-1 and cos90=0 Æ 0 1
-1 0
1) (0, 0)* 0 1 Æ new point1 is (0, 0)
-1 0
2) (10, 0)* matrixÆ new point2 is (0, 10) and so on…..
Figure 4: The square, rotated 90 degrees.
1.2.2 Translation
Moving an object is called a translation. We translate the
point by moving to x and y coordinates, respectively,
the amount of the point should be shifted in the x and y
directions. We translate an object by translating each
vertex in the object. The translation equations may be
written as:
From the double angle formulas:
sin(A+B) = sin(A)*cos(B) + cos(A)*sin(B)
cos(A+B) = cos(A)*cos(B) – sin(A)*sin(B)
Substituting :
y2/r = (y1/r)*cos(B) + (x1/r)*sin(B)
Therefore: y2 = y1*cos(B) + x1*sin(B)
x2/r =(x1/r)*cos(B) - (y1/r)*sin(B)
And therefore: x2 = x1*cos(B) – y1*sin(B)
We have now:
x2 = x1*cos(B) - y1*sin(B)
y2 = x1*sin(B) + y1*cos(B)
We can write same as a single matrix equation. This
requires that we use a 2 by 2 matrix:
such that x*a+y*c = x+Tx and similarly x*b+y*d= y +Ty.
To rotate a line or polygon, we must rotate each of its
vertices.
If we consider the coordinates of the point (x, y) as o one
row two column matrix [x y] and the matrix
then, given the pseudo-definition
for matrix product, MP : = + / . or * , we can write
equations as the matrix-equation:
Figure 5 Translation
2. Rotation and translation using 4 x 4
homogenous matrices
2.1 Homogenous Coordinates
From the above text we now can see the impossibility of
representing a translation as 2 x 2 matrix. What is required
at this point is to change the setting (2D coordinate space)
in which there was the original problem. In geometry,
when one encounters difficulty when trying to solve a
problem in n-space, it is customary to attempt to re-phrase
and solve the problem in n+1 space. In our case this means
that we should look at our 2D problem in 3- dimensional
space. But how can we solve this problem? Consider that,
given a point (x, y) in 2-space, we map that point to (x, y,
1). That is, we inject each point in the 2D plane into the
corresponding point in 3 space in the plane z=1. If we are
able to solve our problem in this plane and find that the
solution lies in the plane z=1, then we may project this
solution back to 2 space by mapping each point (x, y, 1) to
(x, y).
To summarize, we inject the 2D plane into 3 space by the
mapping (x, y)Æ(x, y, 1).
The solution of our example square is now:
0 0 1
10 0 1
10 10 1
0 10 1
0 0 1
And we can rewrite the translation in form:
x’ = x + tx
y’ = y + ty
z‘=z
And using the 3 x 3 matrix:
Same for rotation:
x’ = x * cos(θ) – y * sin(θ)
y’ = x * sin(θ) + y * cos(θ)
z’ = z
And using the 3 x 3 matrix:
2.2 3D Rotation and Translation Using 4 x 4
Homogenous Matrices
Now we extend methods for translating and rotating in 3D
from 2D methods by including the z-axis (now it doesn’t
equal by value one).
Translation in a 3D homogeneous coordinate
representation; a point is translated from (x, y, z) to (x’, y’,
z’) using 4 x 4 matrix:
Tx, Ty and Tz specify the translation for the coordinates.
The above matrix is equivalent with this representation:
x’ = x + Tx
y’ = y + Ty
z’ = z + Tz
Example: We have an origin point (0, 0, 0) and let us move
it along z- and y-axis both 100 unit. We get:
x’
1 0 0 0
x
0
y’
0 1 0 100
y
100
z’
0 0 0 100
z
100
1
0 0 0 1
1
1
When Rotation will be specified in 3-D, we need to
designate a rotation axis (about which the object is to be
rotated), and the amount of angular rotation. In 2-D, we
always rotated about the xy-plane. The easiest rotations to
handle in 3-D are those that are parallel to a coordinate
axis. To simplify things, we will adopt the convention that
counterclockwise rotations about a coordinate axis are
produced with positive rotation angles, if we are looking
along the positive half of the axis toward the origin.
If we are thinking about 2-D rotations, then there are
always rotated about the z axis. So the standard 2-D
rotation equations give us what we need for rotations about
the
z-axis:
The above matrix is equivalent with this representation and
this is called Rz:
x’ = x * cos(θ) – y * sin(θ)
y’ = x * sin(θ) + y * cos(θ)
z’ = z
Transformations for rotations about the other two axes can
be obtained by cyclic permutation of the parameters in the
equations above. In other words, we replace x with y, y with
z and z with x. For the x-axis (in the yz plane) Rx:
y’ = y cos(θ) - z sin(θ)
z’ = y sin(θ) + z cos(θ)
x’ = x
For the y-axis (in the xz plane) Ry:
z’ = z cosθ - x sinθ
x’ = z sinθ + x cosθ
y’ = y
Here is the transformation matrices for Rx:
And here is the transformation matrices for Ry:
Even though quaternions remained a “solution in search of
a problem” for many years after their discovery,
applications in the fields of classical mechanics and the
theory of relativity were identified in the early twentieth
century. The capability of quaternions to represent threedimensional rotations about an arbitrary axis motivated
researchers to use quaternion algebra in rotational
kinematics equations. As a result, several new application
areas involving quaternion based algorithms, such as
robotics, orbital mechanics, aerospace technologies, and
inertial navigation systems have started to use.
3.2 Algebra for quaternions
As we know, the system of complex numbers is defined in
terms of i, a square root of -1 ( i*i = -1).
Although i is not a real number, we can write any complex
number in terms of real numbers, using a real and a
complex part. We can also define a conjugate and a
magnitude (or absolute value) for these numbers.
For example a point 1 unit along the z-axis rotated 90degrees about y:
Using the properties above, we can describe multiplication
for complex numbers:
A point 1 unit along the x-axis z rotates positively to
x about y:
Quaternions are an extension of complex numbers. Instead
of just i, we have three different numbers that are all square
roots of -1 labeled i, j, and k.
When we multiply two of these numbers together, they
behave similarly to cross products of the unit basis vectors:
ij = -ji = k
jk = -kj = i
ki = -ik = j
The conjugate and magnitude of a quaternion are found in
much the same way as complex conjugate and magnitude.
3. Rotation and translation using
quaternions
3.1 Introduction for quaternions
Quaternions are developed by Irish mathematician Rowan
Hamilton in mid nineteen century, when he was trying to
find a way to rotate a 3D vector by multiplying it by
another. After 15 years of hard work he solved the problem
by using a 4D notation. Quaternions provide useful
mechanism for describing rotations about some arbitrary
axis in 3D Graphics or Virtual Reality.
Unit quaternion is
The inverse of a quaternion refers to the multiplicative
inverse (or 1/q) and can be computed by q-1=q'/(q*q')
If a quaternion q has length 1, we say that q is a unit
quaternion. The inverse of a unit quaternion is its
conjugate,
q-1=q' .
We can represent a quaternion in several ways:
• as a linear combination of 1, i, j, and k,
• as a vector of the four coefficients in this linear
combination,
• as a scalar for the coefficient of 1 and a vector for
the coefficients of the imaginary terms.
We can write the product of two quaternions in terms of the
(s, v) representation using standard vector products in the
following way:
Representing rotations with quaternions :We will
compute a rotation about the unit vector, u by an angle
theta. The quaternion that computes this rotation is
so we can save a lot of time and preserve more numerical
accuracy with quaternions than with matrices.
Although let us look a matrix representation for
quaternion multiplication.
We can use the rules above to compute the product of two
quaternions and we get:
If we examine each term in this product, we can see that
each term depends linearly on the coefficients for q1. Also
each term depends linearly on the coefficients for q2. So,
we can write the product of two quaternions in terms of a
matrix multiplication.
When the matrix Lrow(q1) multiplies a row vector q2, the
result is a row vector representation for q1 * q2.
When the matrix Rrow(q2) multiplies a row vector q1, the
result is also a row vector representation for q1*q2.
⎛ cos(θ / 2 ) ⎞
q = ⎜⎜
⎟⎟
⎝ sin (θ / 2 ) u ⎠
We will represent a point p in space by the quaternion
P=(0,p) We compute the desired rotation of that point by
this formula:
Now we want to perform two rotations on an object. This
may come up in a manipulation interface where each
movement of the mouse adds another rotation to the current
object pose. This is very easy and numerically stable with a
quaternion representation.
Suppose q1 and q2 are unit quaternions representing two
rotations. We want to perform q1 first and then q2. To do
this, we apply q2 to the result of q1, regroup the product
using associativity, and find that the composite rotation is
represented by the quaternion q2*q1.
Therefore, the only time we need to compute the matrix is
when we want to transform the object. For other operations
we need only look at the quaternions. A matrix product
requires many more operations than a quaternion product
And, what we have, when computing rotation matrices
from quaternions. Now we have all the tools we need to
use quaternions to generate a rotation matrix for the given
rotation. We have a matrix form for left-multiplication by q
:
And a matrix form for right-multiplication by q’
The resulting rotation matrix is the product of these two
matrices.
q * P * q' = q * ( P * q') = q * (P Rrow(q')) = (P Rrow(q'))
Lrow(q) = P (Rrow(q') Lrow(q)) = P Qrow(q)
A matrix product requires many more operations than a
quaternion product so we can save a lot of time and
preserve more numerical accuracy with quaternions than
with matrices. Therefore, the only time we need to compute
the matrix is when we want to transform the object. For
other operations we need only look at the quaternions.
So using this matrix, we could compute Protated another
way: Protated = P * Qrow
There are applications in computer vision and robot vision
where quaternions could be effectively used in extracting
three dimensional orientation attitude information. The
importance of quaternions to graphics and games
programmers is also obvious, and also there are
possibilities to stimulate further research towards
extending/improving current applications.
4. Conclusion
Denavit and Hartenberg (1955) and later Fu et al. (1987)
and Foley et al. (1990) chose 4 x 4 homogeneous matrices
for transformations such as how to express object
translations, rotations and scaling. The 4 x 4 homogeneous
transformation matrices offer certain computational
advantages. First, they treat both rotations and translations
in a uniform way. Second, they can be compounded, such
that complex kinematics relationships can be modeled.
Finally, they are easily invertible.
Before 1980 there were many attempts to use also
quaternions for transformations in computer graphics, but
quaternions were not so useful. But after that and
nowadays there are various applications of quaternions,
particularly in the context of its increasing importance in
the fields of computer graphics, animation and virtual
reality. The fundamental properties of quaternions have
been outlined, and the mathematical preliminaries for using
quaternions as a rotation operator have been presented.
References
[1] R. Mukundan. Quaternions: From Classical Mechanics to Computer
Graphics, and Beyond. Proceedings of the 7th Asian Technology
Conference in Mathematics 2002 Pages 97-106.
[2] Leonard McMillan and Gary Bishop. Plenoptic Modeling: An ImageBased Rendering System. Proceedings of SIGGRAPGH 95, Los Angeles,
California, August 6-11,1995.
[3] Humberto Calderon and Stamatis Vassiliadis. Computer Graphics and
MOLEN paradigm: a survey.