UPI YPTK - Padang

Mathematical Foundation
for Graphics

Eko Syamsuddin Hasrito, PhD

204481 Foundation of Computer Graphics April 18, 2020 1

 Acknowledgement

 This lecture note has been summarized from

lecture note on Foundation of Computer
Graphics, Introduction to Computer Graphics
all over the world. I can’t remember where
those slide come from. However, I’d like to
thank all professors who create such a good
work on those lecture notes. Without those
lectures, this slide can’t be finished.

204481 Foundation of Computer Graphics April 18, 2020 2

 Objectives

 Introduce the elements of geometry

 Scalars
 Vectors
 Points
 Develop mathematical operations among them
in a coordinate-free manner
 Define basic primitives
 Line segments
 Polygons

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 Basic Elements

 Geometry is the study of the relationships

among objects in an n-dimensional space
 In computer graphics, we are interested in objects
that exist in three dimensions
 Want a minimum set of primitives from which we
can build more sophisticated objects
 We will need three basic elements

 Scalars
 Vectors

 Points

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 Coordinate-Free Geometry

 When we learned simple geometry, most of us

started with a Cartesian approach
 Points were at locations in space p=(x,y,z)
 We derived results by algebraic manipulations
involving these coordinates
 This approach was nonphysical
 Physically, points exist regardless of the location of
an arbitrary coordinate system
 Most geometric results are independent of the
coordinate system
 Euclidean geometry: two triangles are identical if two
corresponding sides and the angle between them are
identical
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 Geometry

 Geometry provides a mathematical foundation

for much of computer graphics:
 Geometric Spaces: Vector, Affine, Euclidean,
Cartesian, Projective
 Affine Geometry
 Affine Transformations
 Perspective
 Projective Transformations
 Matrix Representation of Transformations
 Viewing Transformations

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 Introduction (1/2)

 Points are associated with locations of space

 Vectors represent displacements between
points or directions
 Points, vectors, and operators that combine
them are the common tools for solving many
geometric problems that arise in Geometric
Modeling, Computer Graphics, Animation,
Visualization, and Computational Geometry.

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 Introduction (2/2)

 The three most fundamental operators on

vectors are the dot product, the cross
product, and the mixed product (sometimes
called the triple product).
 Although a point and a vector may be
represented by its three coordinates, they are
of different type and should not be mixed
 Avoid, whenever possible, using coordinates
when formulating geometric constructions.
Instead use vectors and points.

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 What will you learn here? (1/2)

 Properties of dot, cross, and mixed products

 How to write simple tests for
 4-points and 2-lines coplanarity

 Intersection of two coplanar edges

 Parallelism of two edges or of two lines

 Clockwise orientation of a triangle from a

viewpoint
 Positive orientation of a tetrahedron

 Edge/triangle and ray/triangle intersection

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 What will you learn here? (2/2)

 How to compute
 Center of mass of a triangle

 Volume of a tetrahedron

 “Shadow” (orthogonal projection) of a vector

on a plane
 Line/plane intersection

 Plane/plane intersection

 Plane/plane/plane intersection

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 Terminology

 Orthogonal to = Normal to = forms a 90o angle

with
 Norm of a vector = length of the vector
 Are coplanar = there is a plane containing them

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 Notation

 means “is always equal to” or “by definition”

== means the test “is equal to” used in Boolean expressions
:= is the assignment “is computed as” and is used in construction
algorithms

s (italic type) Boolean

v (normal type) scalar

a (bold type lower-case) point
A (bold upper-case) pointset (edge, curve, triangle, surface, solid)

U (bold underscored upper-case) vector

u (bold underscored lower-case) unit vector
T (bold upper-case) transformation

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 Scalars

 Need three basic elements in geometry

 Scalars, Vectors, Points

 Scalars can be defined as members of sets

which can be combined by two operations
(addition and multiplication) obeying some
fundamental axioms (associativity,
commutivity, inverses)
 Examples include the real and complex
number systems under the ordinary rules with
which we are familiar
 Scalars alone have no geometric properties

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 Vectors And Point

 We commonly use vectors to represent:

 Points in space (i.e., location)
 Displacements from point to point
 Direction (i.e., orientation)
 But we want points and directions to behave
differently
 Ex: To translate something means to move it without
changing its orientation
 Translation of a point = different point
 Translation of a direction = same direction

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 Vectors

 Physical definition: a vector is a quantity with

two attributes
 Direction
 Magnitude

 Examples include
 Force
 Velocity

 Directed line segments

 Most important example for graphics

v
 Can map to other types

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 Vector Operations
 Every vector has an inverse
 Same magnitude but points in opposite
direction
 Every vector can be multiplied by a scalar

 There is a zero vector

 Zero magnitude, undefined orientation

 The sum of any two vectors is a vector

 Use head-to-tail axiom

v w
v -v v
u
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 Vectors Lack Position

 These vectors are identical

 Same length and magnitude

 Vectors spaces insufficient for geometry

 Need points

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 Vectors (1/2)

 Vector space (analytic definition from Descartes in the

1600s)
 U+V=V+U (vector addition)

 U+(V+W)=(U+V)+W

 0 is the zero (or null) vector if V, 0+V=V

 –V is the inverse of V, so that the vector subtraction

V–V= 0
 s(U+V)=sU+sV (scalar multiplication)

 U/s is the scalar division (same as multiplication by

s–1)
 (a+b)V=aV+bV

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 Vectors (2/2)

 Vectors are used to represent displacement

between points
 Each vector V has a norm (length) denoted ||V||
 V / ||V|| is the unit vector (length 1) of V. We denote
it Vu
 If ||V||==0, then V is the null vector 0
 Unit vectors are used to represent
 Basis vectors of a coordinate system
 Directions of tangents or normals in definitions of
lines or planes

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 Vector Spaces

 A linear combination of vectors results in a new

vector:
v = 1v1 + 2v2 + … + nvn
 If the only set of scalars such that
1v1 + 2v2 + … + nvn = 0
is 1 = 2 = … = 3 = 0
then we say the vectors are linearly independent
 The dimension of a space is the greatest number of
linearly independent vectors possible in a vector set
 For a vector space of dimension n, any set of n linearly
independent vectors form a basis

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 Vector Spaces: An Example

 Our common notion of vectors in a 2D plane

is (you guessed it) a vector space:
 Vectors are “arrows” rooted at the origin
 Scalar multiplication “streches” the arrow,
changing its length (magnitude) but not its direction
 Addition uses the “trapezoid rule”:

u+v
y
v
u
x
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 Vector Spaces: Basis Vectors

 Given a basis for a vector space:

 Each vector in the space is a unique linear
combination of the basis vectors
 The coordinates of a vector are the scalars from this
linear combination
 Best-known example: Cartesian coordinates
 Draw example on the board

 Note that a given vector v will have different

coordinates for different bases

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 Points

 Location in space
 Operations allowed between points and vectors
 Point-point subtraction yields a vector
 Equivalent to point-vector addition

v=P-Q

P=v+Q

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 Affine Spaces

 Point + a vector space

 Operations
Vector-vector addition
 Scalar-vector multiplication

 Point-vector addition

 Scalar-scalar operations

 For any point define

1 • P = P

 0 • P = 0 (zero vector)

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 Affine Space (1/2)

Definition: Set of Vectors V and a Set of Points

P
 Vectors V form a vector space.

 Points can be combined with vectors to make

new points:

P +v =Q with P,Q  P and v  V

Dimension: The dimension of an affine space is

the same as that of V .

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 Affine Space (2/2)

Note:
 All other point operations are just variations
on the
P + v operation.
 No distinguished origin
 No notion of distances or angles.
 Most of what we do in graphics is affine.

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 Affine space

 Defined together with a vector space and the

point difference mapping
 V=b–a (vector V is the displacement from point
a to point b)
 Notation: ab stands for the vector b – a
 c– a = (c – b) + (b – a). Written ac = ab + bc
 Point b is uniquely defined, given point a and
vector (b – a)

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 Affine space

 If we pick a as origin, then there is a 1-to-1 mapping

between a point b and the vector b –a.
 We may extend the vector operations to points, but
usually only when the result is independent of the
choice of the origin!
 A + b is not allowed (the result depends on the
origin)
 (a + b)/2 is OK

 Points may be used to represent

 The vertices of a triangle or polyhedron

 The origin of a coordinate system

 Points in the definition of lines or plane

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 Lines

 Consider all points of the form

 P()=P0 +  d
 Set of all points that pass through P0 in the direction
of the vector d

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 Parametric Form

 This form is known as the parametric form of the

line
 More robust and general than other forms
 Extends to curves and surfaces

 Two-dimensional forms
 Explicit: y = mx +h
 Implicit: ax + by +c =0

 Parametric:

x() = x0 + (1-)x1

y() = y0 + (1-)y1

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 Rays and Line Segments

 If  >= 0, then P() is the ray leaving P0 in the

direction d
If we use two points to define v, then
P( ) = Q +  (R-Q)=Q+v
=R + (1-)Q
For 0<=<=1 we get all the
points on the line segment
joining R and Q

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 Convexity

 An object is convex iff for any two points in the

object all points on the line segment between
these points are also in the object

P
P

Q Q

not convex
convex
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 Affine Sums

 Consider the “sum”

P=1P1+2P2+…..+nPn
Can show by induction that this sum makes
sense iff
1+2+…..n=1
in which case we have the affine sum of the
points P1,P2,…..Pn
 If, in addition, i>=0, we have the convex hull of
P1,P2,…..Pn

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 Convex Hull

 Smallest convex object containing P1,P2,…..Pn

 Formed by “shrink wrapping” points

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 Curves and Surfaces

 Curves are one parameter entities of the form

P() where the function is nonlinear
 Surfaces are formed from two-parameter
functions P(, b)
 Linear functions give planes and polygons

P() P(, b)
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 Planes

 A plane be determined by a point and two

vectors or by three points

P(,b)=R+u+bv P(,b)=R+(Q-R)+b(P-Q)

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 Triangles

convex sum of S() and R

convex sum of P and Q

for 0<=,b<=1, we get all points in triangle

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 Normals

 Every plane has a vector n normal (perpendicular,

orthogonal) to it
 From point-two vector form P(,b)=R+u+bv, we know
we can use the cross product to find n = u  v and
the equivalent form
(P()-P)  n=0

u
P
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 Dot Product (1/2)

UV denotes the dot product (also called the

inner product)
UV = ||U||||V||cos(angle(U,V))
 UV is a scalar.
 UV==0  ( U==0 or V==0 or (U and V are
orthogonal))
v

cos(angle(u,v)) u
VU = UV > 0 here VU = UV < 0 here

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 Dot Product (2/2)

 UV is positive if the angle between U and V is

less than 90o
 Note that UV = VU, because: cos(a)=cos(–a).
 uv = cos(angle(u,v) # unit vectors: ||u||  1
 the dot product of two unit vectors is the cosine of
their angle
 Vu = length of the orthogonal projection of V
onto the direction of u
 ||U||= sqrt(UU) = length of U = norm of U

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 Dot Product

 The dot product or, more generally, inner

product of two vectors is a scalar:
v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)
 Useful for many purposes
 Computing the length of a vector: length(v) = sqrt(v • v)
 Normalizing a vector, making it unit-length
 Computing the angle between two vectors:
u • v = |u| |v| cos(θ)
 Checking two vectors for orthogonality
 Projecting one vector onto another v
θ
u
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 Computing the reflection vector

 Given the unit normal n to a mirror surface and

the unit direction l towards the light, compute
the direction r of the reflected light.

By symmetry, l+r is parallel to n and

has a norm that is twice the length of
the projection nl of l upon n. Hence:
r = 2(nl)n–l
n –l
l r
l r

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 Computing a “shadow” vector (projection)

 Given the unit up-vector u and the unit

direction d, compute the shadow T of d onto
the floor (orthogonal to u).

T and u are orthogonal.

u
d is the vector sum T+ (du)u . d
Hence:
T = d – (du)u T

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 Cross product (1/3)

UV denotes the cross product

UV is either 0 or a vector orthogonal to both U
and V
 When U==0 or V==0 or U//V (parallel) then
UV= 0
 Otherwise, UV is orthogonal to both U and V

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 Cross product (2/3)

The direction of UV is defined by the thumb of

the right hand
 Curling the fingers from U to V
 Or standing parallel to U and looking at V, UV goes
left

||UV||  ||U||||V||sin(angle(U,V))
 sin(angle(u,v)2  1–(uv)2 # unit vectors

 (uv== 0)  u//v (parallel)

 UV  –VU
 Useful identity: U(VW)  (UW)V – (UV)W

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 Cross Product (3/3)

 The cross product or vector product of two

vectors is a vector:
 y1 z 2  y 2 z1 
v1  v 2   ( x1 z 2  x 2 z1)
 x1 y 2  x 2 y1 
 The cross product of two vectors is orthogonal
to both
 Right-hand rule dictates direction of cross
product

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 When are two edges parallel in 3D?

 Edge(a,b) is parallel to Edge(c,d) if abcd == 0

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 Mixed product

U(VW) is called a mixed product

 U(VW) is a scalar

 U(VW) ==0 when one of the vectors is null or all 3

are coplanar
 U(VW) is the determinant | U V W |

 U(VW)  V(WU)  – U(WV) # cyclic

permutation

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 Testing whether a triangle is front-facing

 When does the triangle a, b, c appear

clockwise from d?
when da(dbdc) > 0

c
a

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 Volume of a tetrahedron

Volume of a tetrahedron with vertices a, b, c and d is

v  | da(dbdc) | / 6
a

dbdc da

c
dc

d db b

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 z, s, and v functions (1/2)
zero: z(a,b,c,d)  da(dbdc)==0 # tests co-planarity of 4 points
 Returns Boolean TRUE when a,b,c,d are coplanar
sign: s(a,b,c,d)  da(dbdc)>0 # test orientation or side
 Returns Boolean TRUE when a,b,c appear clockwise from d
 Used to test whether d in on the “good” side of plane through a,b,c
a
a
dbdc da
d
c
dc
b
c d b
db

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 z, s, and v functions (2/2)
value: v(a,b,c,d)  da(dbdc) # compute volume
 Returns scalar whose absolute value is 6 times the volume of
tetrahedron a,b,c,d
 v(a,b,c,d)  v(d,a,b,c)  –v(b,a,c,d)

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 Testing whether an edge is concave

 How to test whether the edge (c,b) shared by triangles

(a,b,c) and (c,b,d) is concave?

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 When do two edges intersect in 2D?

 Write a geometric expression that returns true when

two coplanar edges (a,b) and (c,d) intersect

0  (abac)•(abad) and 0  (cdca)•(cdcb)

d
d

a b
c b
a
c

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 When do two edges intersect in 3D?

Consider two edges: edge(a,b) and edge(c,d)

 When are they co-planar? When z(a,b,c,d)

 When do they intersect?

When they are co-planar and
0  (abac)•(abad) and 0  (cdca)•(cdcb)
d d
q
a b a b
p

c c

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 What is the common normal to 2 edges in 3D?

Consider two edges: edge(a,b) and edge(c,d)

 What is their common normal n? n=(abcd)u

n
a b

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 When is a point inside a tetrahedron?

When does point p lie inside tetrahedron a, b, c, d ?

 Assume z(a,b,c,d) is FALSE (not coplanar)
When
s(a,b,c,d), d
s(p,b,c,d),
s(a,p,c,d), p
s(a,b,p,d), and c
s(a,b,c,p)
(i.e. all return TRUE or all a b
are identical. return FALSE)

 When is p on the boundary of the tetrahedron?

 Do as an exercise for practice.

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 A faster point-in-tetrahedron test ?

 Suggested by Nguyen Truong

 Write ap = sab+tac+uad

 Solve for s, t, u (linear system of 3 equations)

 Requires 17 multiplications, 3 divisions, and 11 additions

 Check that s, t, and u are positive and that s+u+t<1

 A more expensive Variation:

d
Compute s, t, u, w
 w = v(a,b,c,d)

 s = v(a,p,c,d)
p
 u = v(a,b,p,d) c
 t = v(a,b,c,p)

Check that a b
 w, s, t, u have the same sign and that s+u+t<w

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 When is a 3D point inside a triangle

When does point p lie inside triangle with vertices a, b, c ?

When z(p,a,b,c) and (abap)•(bcbp)0 and (bcbp)•(cacp)0

a
p p

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 Parametric representation of a line

 Let Line(p,t) be the line through

point p with tangent t
 Its parametric form associates a
scalar s with a point q(s) on the p
line
t
• s defines the distance from p
to q(s) s
• q(s) = p + st
 Note that the direction of t q
gives an orientation to the line
(direction where s is positive)
 We can represent a line by p and t

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 Implicit representation of a plane

 Let Plane(r,n) be the plane through point r with normal n

 Its implicit form states that a point q lies on Plane(r,n) when
rqn=0
 Remember that rq = q – r

 Note that the direction of n defines an orientation of the plane

p
p not in plane
n n
q r
r

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 Line/plane intersection

Compute the point q of intersection between Line(p,t) and

Plane(r,n)

Replacing q by p+st in (q–r)n=0 yields p

(p–r+st)n=0
Solving for s yields t
rpn+stn=0 and s
s = – rpn / tn n
s = prn / tn q r
Hence q = p + (prn)t / (tn)

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 When are two lines coplanar in 3D?

 L(p,t) and L(q,u) are coplanar

When tu == 0 OR pq•(tu) == 0

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 What is the intersection of two planes

 Consider two planes Plane(p,n) and Plane(q,m)

 Assume that n and m are not parallel (i.e. nm ≠ 0)
 Their intersection is a line Line(r,t). m
 How can one compute r and t ?
q
t := (nm)u
let u := nt
r := p+(pq•m)u/(u•m) t n

r u p
If correct, then provide the derivation.
Otherwise, provided the correct answer.

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 What is the intersection of three planes

 Consider planes Plane(p,m), Plane(q,n), and

Plane(r,m).
 How do you compute their intersection w ?
r
Write w=p+an+bm+ct then
solve the linear system t
{pw•n=0, qw•m=0, rw•t=0}
for a, b, and c q m w
n
or compute the line of
intersection between two of p
these planes and then intersect
it with the third one.

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 Implementation

 Points and vectors are each represented by their 3

coordinates
 p=(px,py,pz) and v=(vx,vy,vz)
 U•V = UxVx + UyVy + UzVz
 UV = (UyVz – UzVy) – (UxVz – UzVx) + (UxVy – UyVx)

 Implement:
 Points and vectors
 Dot, cross, and mixed products
 z, s, and v functions from mixed products
 Edge/triangle intersection
 Lines and Planes
 Line/Plane, Plane/Plane, and Plane/Plane/Plane intersections
 Return exception for singular cases (parallelism…)

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 Linear Transformations

 A linear transformation:
 Maps one vector to another
 Preserves linear combinations
 Thus behavior of linear transformation is
completely determined by what it does to a
basis
 Turns out any linear transform can be
represented by a matrix

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 Matrices

 By convention, matrix element Mrc is located

at row r and column c:
 M11 M12  M1n 
 M21 M22  M2n 
M
     
 
Mm1 Mm2  Mmn 

 By (OpenGL) convention,  v1 
vectors are columns: 
v   v 2 
 v 3 
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 Matrices

 Matrix-vector multiplication applies a linear

transformation to a vector:

 M11 M12 M13   vx 

M  v  M 21 M 22 M 23   vy 
M 31 M 32 M 33   vz 
 Recall how to do matrix multiplication

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 Matrix Transformations

 A sequence or composition of linear

transformations corresponds to the product of
the corresponding matrices
 Note: the matrices to the right affect vector first
 Note: order of matrices matters!
 The identity matrix I has no effect in
multiplication
 Some (not all) matrices have an inverse:

M 1 Mv   v

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 Source :

Pradondet Nilagupta
Dept. of Computer Engineering
Kasetsart University

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 Pertanyaan untuk Mahasiswa (3)

Masing-masing mahasiswa harus menjawab / memberi komentar terhadap minimal 1

pertanyaan di bawah ini di kolom komentar :
1. Jelaskan mengenai Geometric Spaces pada Komputer Grafik ?
2. Jelaskan mengenai konsep “mixed Product” pada Komputer Grafik ?
3. Jelaskan dalam Bahasa Indonesia penjelasan pada Slide No.14 ?
4. Jelaskan mengenai konsep “convexity” pada Komputer Grafik ?
5. Jelaskan dalam Bahasa Indonesia penjelasan dan gambar pada Slide No.21?
6. Jelaskan mengenai perbedaan konsep “perpotongan 2 garis pada 2
Dimensi dan ruang 3 Dimensi” pada Komputer Grafik
7. Jelaskan dalam Bahasa Indonesia penjelasan dan gambar pada Slide No.50?
8. Jelaskan dalam Bahasa Indonesia penjelasan pada Slide No.70?

Kesemua 8 pertanyaan ini harus dibahas oleh semua anggota kelas, maka 7 mahasiswa terakhir
yang akan jawaban pertanyaan harus menjawab topik-topik yang belum pernah terbahas ??
Setelah itu yang lain boleh beri komentar / koreksian atas jawaban temannya ? Nilai keaktifan !!

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