ReviewofBasicMathfor
ComputerGraphics
CENG315–ComputerGraphics
M.AbdullahBulbul
Sets
m∈S
n∈S
x∈T
y∈T
z∈T
m
n
setS
AxB:allpossiblepairs(a,b)wherea∈A,b∈B
A2=S,S
SxT={(m,x),(m,y),(m,z),
(n,x),(n,y),(n,z)}
S2={(m,m),(m,n),(n,m),(n,n)}
x
y
z
setT
Usefulsets
Trigonometry
•  Radians
•  Angle=lengthoftheunitcirclethatiscutbythetwodirecOons.
•  Angles
•  degrees=radians*180/π
•  Pythagoreantheorem
•  Righttriangle
• 
• 
• 
• 
• 
• 
sin(x)=?
cos(x)=?
tan(x)=?
cot(x)=?
csc(x)=1/sin(x)
sec(x)=1/cos(x)
•  Unitcircle
Usefulidentities
•  sin(-x)=-sin(x)
•  cos(-x)=cos(x)
•  tan(-x)=-tan(x)
•  sin(π/2-x)=cos(x)
•  cos(π/2-x)=sin(x)
•  tan(π/2-x)=cot(x)
Usefulidentities
•  PythagoreanidenOOes:
•  sin2(x)+cos2(x)=?
•  sec2(x)–tan2(x)=?
•  sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
•  sin(2a)=2sin(a)cos(a)
•  sin(a-b)=sin(a)cos(b)-sin(b)cos(a)
•  cos(a+b)=cos(a)cos(b)–sin(a)sin(b)
•  cos(2a)=cos2(a)–sin2(a)
•  cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
Vectors
•  Avector:alength&adirecOon
a
•  Usuallywri\enasa(bold)
•  AbsoluteposiOonisnotimportant
•  Magnitude=||a||
•  Usedtostoreoffsets,displacements,locaOons
•  TodefinealocaOonyouneedanorigin
Vectoraddition
a+b
b
a
commutaOve:a+b=b+a
a
b
b+a
Unaryminusandsubtraction
a
-a
•  ChecksubtracOononaparallelogram
2DCartesianSpace
•  a=
xa
ya
•  aT= xaya
4x
•  a= 3y
2DCartesianSpace
•  A2Dvectorcanbewri\enasacombinaOonofanytwononzerovectorsthatarenotparallel
•  Linearindependence
•  c=k1a+k2b
•  Exercise
Linearindependence
Linearindependence&basis
•  Anytwoparallelsthatarenotparallelformsa2Dbasis.
•  CangoanywherebyacombinaOonofthem
•  Unitvectorsarespecialcase
•  Orthonormalbasis
•  Simplifiesthings
VectorMultiplication
• Dot(scalar)product
• Crossproduct
Dot(scalar)product
b
•  a.b=b.a=?
x
a
Dot(scalar)product
b
x
•  a.b=b.a=?
•  =||a||||b||cos(x)
a
•  DistribuOve:a.(b+c)=a.b+a.c
•  x=cos-1(a.b/(||a||||b||))(mostcommonuse)
Dotproductincartesiancoordinates
xa
xb
•  a.b=.=?
ya
yb
Dotproductincartesiancoordinates
xa
xb
•  a.b=.=?
ya
(Fairlysimple)
=xaxb+yayb
yb
Commonusesofdotproduct
•  Findanglebetweentwovectors(lightandsurfaceexample)
•  ProjecOonofavectoronanotherone
•  Computedeasilyoncartesiancomponents
Projectionofavectoronanother
•  ba
b
x
ba
a
Projectionofavectoronanother
•  ba
b
x
ba
•  inferit(5minutes)
a
Crossproduct
•  Theresultofadotproductwasscalar
•  Theresultofacrossproductisavector
•  Perpendiculartobothvectors
•  Length:
•  Areaoftheparallelogram
•  ||a×b||=||a||||b||sin(ϕ)
•  DirecOon:
•  Righthandrule
Crossproduct
•  X×Y=+Z
•  Y×X=-Z
•  Y×Z=+X
•  Z×Y=-X
•  a×b=-b×a
•  a×a=?
•  a×(b+c)=a×b+a×c
•  Z×X=+Y
•  X×Z=-Y
•  Be\erontheboard
CrossproductonCartesianspace
xyz
aybz–azby
•  a×b==
axayaz
azbx–axbz
bxbybz
axby–aybx
Orthonormalbases
•  Managingcoordinatesystemsisimportant!
•  Orthonormalbasesisthekey.
In2D:uandvformanorthonormalbasisiftheyare
•  Eachofthemareunitlength
||u||=||v||=1
•  Orthogonaltoeachother
u.v=0
Orthonormalbases
In3D:u,vandwformanorthonormalbasisiftheyare
•  Allofunitlength
||u||=||v||=||w||=1
•  Anypairofthemareorthogonaltoeachother
u.v=0,v.w=0,u.w=0
•  Righthandedifw=u×v
•  Otherwiselenhanded
•  Thereareinfinitelymanyorthonormalbases
Orthonormalbases
In3D:u,vandwformanorthonormalbasisiftheyare
•  Allofunitlength
||u||=||v||=||w||=1
•  Anypairofthemareorthogonaltoeachother
u.v=0,v.w=0,u.w=0
•  Righthandedifw=u×v
•  Otherwiselenhanded
•  Thereareinfinitelymanyorthonormalbases
Canonicalorthonormalbasis
•  Cartesiancanonicalorthonormalbasis
•  x=(1,0,0)
•  y=(0,1,0)
•  z=(0,0,1)
Constructinganorthonormalbasis
givenasinglevector
•  Easierontheboard
Matrix
•  Whatisamatrix
•  IdenOtymatrix
•  Transpose
•  Determinant
•  MatrixvectormulOplicaOon
•  MatrixmatrixmulOplicaOon