Cheat Sheet for Computer Graphics

Winter 2019/Summer 2020

This is a rudimentary cheat sheet for the lecture Computer Graphics, Winter Term 2019. It is
by no means complete, rather it is optimized to fit with the challenge set by the exam. That
is, it contains the formulas and matrices you are expected to recall during the exam.

1 Basic Math

1.1 Scalar Product

   
a1 b1
a ◦ b = a2  ◦ b2  = a1 b1 + a2 b2 + a3 b3 (1)
a3 b3

1.2 Cross Product

     
a1 b1 a2 b 2 − a3 b 2
a × b = a2  × b2  = a3 b1 − a1 b3  (2)
a3 b3 a1 b 2 − a2 b 1

2 Bresehenham Algorithm
Given two points A = (x0 , y0 ) and B = (x1 , y1 ), we want to rasterize a line from A to B. For
this, we first compute updates

∆DE = 2∆y and ∆DN E = 2(∆x − ∆y) (3)

and initialize

D := ∆x − 2∆y. (4)

Then keep incrementing x := x0 up to x1 . In each iteration, as long as x ≥ 0, we move to the

east (right) and update with ∆DE. If x < 0, we move north-east (increment y) and update
with ∆DN E.

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3 Homogeneous Coordinates
• Given a point (x, y, z) in 3D space, the homogeneous coordinate that represents that point
is (x, y, z, w = 1).
• To undo this conversion, compute
 
x
1 
y (5)
w
z

4 Affine Transformations

4.1 Shearing

Horizontal sheering keeps y fixed. On paper, it looks like the y-axis is moving. It maps the
axis vector (0, 1) to (sh , 1).
 
1 sh 0
0 1 0 (6)
0 0 1

Vertical sheering keeps x fixed. On paper, it looks like the x-axis is moving. It maps the axis
vector (1, 0) to (1, sv ).
 
1 0 0
sv 1 0 (7)
0 0 1

4.2 Rotation

Rotation by angle φ from origin (0, 0), anti-clockwise is implemented by

 
cos φ − sin φ 0
Rφ =  sin φ cos φ 0 (8)
0 0 1

In particular we get
     
0 −1 0 −1 0 0 0 1 0
R π2 = 1 0 0 , Rπ =  0 −1 0 , R 3π = −1 0 0 . (9)
2
0 0 1 0 0 1 0 0 1

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4.3 Reflection

To reflect on the x-axis, invert y with

 
1 0 0
0 −1 0 . (10)
0 0 0

To reflect on the y-axis, invert x with

 
−1 0 0
 0 1 0 . (11)
0 0 1

5 Multiple Transformations

5.1 Matrix Application

Given are transformations M1 and M2 . To first compute M1 and then M2 , multiply a given
point with the matrix product

M2 M1 , (12)

that is transformations are applied right to left.

5.2 Rotation and Transformation

Matrix M first translates by (tx , ty ) and then rotates by angle φ.

     
cos φ − sin φ 0 1 0 tx cos φ − sin φ tx cos φ − ty sin φ
M =  sin φ cos φ 0 · 0 1 ty  =  sin φ cos φ tx cos φ + ty sin φ (13)
0 0 1 0 0 1 0 0 1

Matrix N first rotates by angle φ and then translates by (tx , ty ).

     
1 0 tx cos φ − sin φ 0 cos φ − sin φ tx
N= 0  1 ty  ·  sin φ cos φ 0 =  sin φ cos φ ty  (14)
0 0 1 0 0 1 0 0 1

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6 Rotations in 3D Space
Anti-Clockwise rotation around an axis m by angle φ is achievable with matrix Rm as follows.
 
1 0 0 0
0 cos φ − sin φ 0
Rx = 
0 sin φ cos φ 0
 (15)
0 0 0 1
 
cos φ 0 sin φ 0
 0 1 0 0
Ry = 
− sin φ 0 cos φ 0
 (16)
0 0 0 1
 
cos φ − sin φ 0 0
 sin φ cos φ 0 0
Rz = 
 0
 (17)
0 1 0
0 0 0 1

7 Camera
A camera is often defined by eye vector e, gaze direction g and view-up vector t. However, in
practice it is useful to construct the (u, v, w) coordinate system for a given camera.
1 1
w=− g u= (t × w) v =w×u (18)
||g|| ||t × w||

8 Viewing Transformation
From camera coordinate system (u, v, w), we can construct viewing matrix
 
ux uy uz −uT e
 vx vy vz −v T e 
Mview =   (19)
wx wy wz −wT e
0 0 0 1

which moves the camera into the origin.

9 Phong Lighting Model

The Phong Lighting Model is based around three components, namely ambient light Lamb ,
diffuse reflection Ldif f and specular reflection Lspec .

9.1 Ambient Light

Ambient light is constant, parameterized by two arguments.

Lamb = kamb · Iin (20)

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9.2 Diffuse Reflection

Diffuse reflection imitates micro structure and random reflection. It is dependent on the angle φ
between surface normal n and light vector l. Both n and l have to be normalized.

Ldif f = kdif f · Iin · cos φ (21)

= kdif f · Iin · (n ◦ l) (22)

9.3 Specular Reflection

Specular reflection creates spotlight effects. It takes the eye position into account. To compute
it, we need the perfect reflection vector r and view vector v, both of which need to be normalized.
Angle ϕ is the angle between vectors v and r. Material constant m introduces a cutoff for the
highlight, higher values for m result in smaller highlights.

Lspec = kspec · Iin · cosm ϕ (23)

= kspec · Iin · (v ◦ r)m (24)

If perfect reflection r is not known, it can be computed in terms of

r = 2 · (n ◦ l) · n − l. (25)

10 Ray Tracing

10.1 Eye Ray

An eye ray r is a ray starting at eye position e in direction d.

r(t) = e + t · d (26)

10.2 Intersection Point

An intersection point x of an eye ray with some geometry is defined by distance tx the ray had
to travel from eye to geometry.

x = e + tx · d (27)

10.3 Intersection With a Plane

Given a plane defined by a normal n and some point c on the plane, we can compute intersection
point x by finding the right value for tx .

(c − e) ◦ n
tx = (28)
d◦n
We only care about real solutions for tx . We also only care for solutions tx > 0 as negative
values mean that we traveled in opposite of direction d.

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10.4 Intersection With a Sphere

Given a sphere with center c and radius r, find out tx for which an eye ray e + tx · d intersects
the sphere. The solution can be found by solving the quadratic equation in terms of
√
−b ± b2 − 4c
tx = (29)
2
with

b = 2 · d ◦ (e − c) and c = (e − c) ◦ (e − c) − r2 . (30)

If there is no real solution for tx , there is no intersection of the ray with the sphere.

11 Fin
Keine Macht für Niemand.