Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
CE325: 3D Computer Graphics
3. Coordinate Geometry
Adrian F. Clark
alien@essex.ac.uk
VASE Laboratory, School of Computer Science and Electronic Engineering
University of Essex
2011–12
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Outline
1
Introduction
2
Trig & triangles
3
Transformations
4
Vectors
5
Matrices
6
Transformation matrices
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Introduction
It’s time for us to get to grips with the mathematical part of
CE325. We’ll start from basics and go through things fairly
slowly.
To read up on the mathematics, find a good maths book —
there are plenty in the library. You might find the
summaries and examples at
http://www.mathcentre.ac.uk/
useful too.
If you find you cannot handle the maths, do make an
appointment to see me — and bring along whatever
attempts you have made to understand things yourself.
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Right-Angled Triangles: Notation
Consider the right-angled
triangle with sides:
h: hypotenuse
a: adjacent
o: opposite
then Pythagoras’s
theorem tells us
h 2 = a2 + o 2 .
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Sines, cosines and tangents
We define the cosine and sine from the relationships
o
sin θ =
h
a
cos θ =
h
and we define the tangent as
sin θ
o
tan θ =
=
cos θ
a
The last is obtained by substituting from above. From
Pythagoras’ theorem
cos2 θ + sin2 θ = 1
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Introduction
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Transformations
Vectors
Matrices
Transformation matrices
Sums and differences of angles
It is possible to show that
sin(θ ± φ) = sin θ cos φ ± cos θ sin φ
cos(θ ± φ) = cos θ cos φ ∓ sin θ sin φ
tan θ ± tan φ
tan(θ ± φ) =
1 ∓ tan θ tan φ
The tan relationship is true providing θ + φ 6= (k + 12 )π, k ∈ Z.
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
From these sum-and-difference equations, one can show that
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
2 tan θ
tan 2θ =
1 − tan2 θ
The last of these requires that θ 6= (k + 12 )π, k ∈ Z.
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
It also follows that
2 cos2 θ = 1 + cos 2θ
2 sin2 θ = 1 − cos 2θ
2 sin θ cos φ = sin(θ − φ) + sin(θ + φ)
2 cos θ cos φ = cos(θ − φ) + cos(θ + φ)
2 sin θ sin φ = cos(θ − φ) − cos(θ + φ)
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D geometric transformations
There are three basic types of transformation:
translation
scaling
rotation
As we shall see, other types of transformation are obtained by
combinations of these primitive operations.
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D translation
y
Consider the
movement of the
point P = (x, y )0 to
P0 = (x 0 , y 0 )0 .
P
dy
We have the two
equations
P’
x 0 = x + dx
y 0 = y + dy
dx
x
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D scaling
y
If we have x 0 = xSx
and y 0 = ySy , we
have scaled the
point P to P0 .
Note that the scale
factor may be
different in x and y :
this would turn a
circle into an ellipse,
for example.
P’
P
x
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D rotation
y
P’
y’
r
P
y
r
θ
φ
x’
x
x
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
We have x = r cos φ and y = r sin φ and we can see that
x 0 = r cos(θ + φ) = r cos θ cos φ − r sin θ sin φ
y 0 = r sin(θ + φ) = r cos θ sin φ + r sin θ cos φ
Substituting for x and y , we get
x 0 = x cos θ − y sin θ
y 0 = x sin θ + y cos θ
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Properties of transformations
Points on a straight line remain on a straight line after
transformation
Consequently, we need transform only the end-points of
vectors
Parallel lines remain parallel (also apparent from our
example)
Intersecting lines remain intersecting — we can show this
by drawing a line across the diagonal of a square and then
rotating the square
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Vectors
Computer graphics
uses vector notation
extensively because
it is compact
The basic idea is
that a point with the
coordinates (x, y , z)
is written in the form
 
x
P = y 
z
i.e., as a 3 × 1 vector
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Where there isn’t room to write a column vector, people
write (say) P = (x, y , z)0 , the 0 denoting transpose
You can think of this as being the same as a line drawn
from the origin to (x, y , z)0 — in other words, a vector has a
magnitude (the length of the line) and a direction
Vectors (and matrices) are usually handwritten as
underlined but printed as bold
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Introduction
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Transformations
Vectors
Matrices
Transformation matrices
Unit vectors
The length of the vector P = (x, y , z)0 is simply
q
P = x 2 + y 2 + z 2.
(Pythagoras’s theorem in 3D).
A vector of unit length pointing in the same direction as P is
x 
P̂ =
P
y 
P
z
P
The ‘hat’ notation to identify unit vectors is widely used
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Introduction
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Transformations
Vectors
Matrices
Transformation matrices
Basis vectors
The transformations we have looked at have comparatively
simple forms only because the x- y - and z-axes are
orthogonal (at right angles) to each other
Any point in space can be represented by a combination of
the basis vectors
 
 
 
1
0
0
x̂ =  0  ŷ =  1  ẑ =  0 
0
0
1
It is possible to derive equivalents to all the above
equations for non-orthogonal axes
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
Matrices
Matrices are similar to, but more general than, vectors: the
dimensions may take any value, one does not have to be
unity
For example, the following matrix is of dimension 2 × 3:
a11 a12 a13
a21 a22 a23
As you’ll see, most matrices in 3D graphics are 4 × 4,
where they are used to represent transformations
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Introduction
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Transformations
Vectors
Matrices
Transformation matrices
Adding Vectors and Matrices
You can only add or subtract vectors and matrices if are
the same size, e.g. both 3 × 1. The addition or subtraction
is done element by element. For example
 
 


x1
x2
x1 + x2
P1 = y1  ;
P2 = y2  ;
P1 + P2 = y1 + y2 
z1
z2
z1 + z2
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Introduction
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Transformations
Vectors
Matrices
Transformation matrices
Multiplying Matrices
You can multiply matrices only if the second dimension of
the first matrix is the same as the first dimension of the
second one; so you can multiply a 1 × 3 and a 3 × 2 matrix
but not two 3 × 1 ones.
For example, if we have


Q11 Q12
P11 P12 P13
P=
Q = Q21 Q22 
P21 P22 P23
Q31 Q32
then their product is the 2 × 2 matrix
P11 Q11 + P12 Q21 + P13 Q31 P11 Q12 + P12 Q22 + P13 Q32
PQ =
P21 Q11 + P22 Q21 + P23 Q31 P21 Q12 + P22 Q22 + P23 Q32
It is important to remember that matrix multiplication is not
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
The scalar or ‘dot’ product
If we have two vectors, P and Q, of lengths P and Q
respectively, then we define
P.Q = PQ cos θ
where θ is the angle between P and Q
This is known as the scalar, dot or inner product and is a
number, not a vector or matrix
It is also possible to show that we can calculate P.Q as
Px Qx + Py Qy + Pz Qz
The physical interpretation of the scalar product is the
projection of P on Q or vice versa
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Introduction
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Transformations
Vectors
Matrices
Transformation matrices
The vector or ‘cross’ product
The vector, cross or outer product of P and Q is written as
P × Q or P ∧ Q
It calculates a vector which is at right angles to both P and
Q, something which is important when working out the
shading of surfaces
We calculate it using


Py Qz − Pz Qy
P × Q = Pz Qx − Px Qz 
Px Qy − Py Qx
assuming a right-angled coordinate system
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D Translation
We can write our translation formulae
x 0 = x + dx
y 0 = y + dy
in matrix notation as
x0
y0
=
x
y
+
dx
dy
or as
P0 = P + T
where T represents the translation
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D Scaling
We can write scaling as
0 x
Sx
=
y0
0
0
Sy
x
y
or as
P0 = SP
where S is the matrix that represents the scaling
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Introduction
Trig & triangles
Transformations
Vectors
Matrices
Transformation matrices
2D Rotation
We can write the equations
x 0 = x cos θ − y sin θ
y 0 = x sin θ + y cos θ
in matrix form as
0 x
cos θ − sin θ
x
=
y0
sin θ cos θ
y
or as
P0 = RP
where R is the matrix that represents the rotation
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