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CS 543: Computer Graphics Lecture 3 (Part II) : Points, Scalars and Vectors

Points define a location in space using coordinates from a reference frame, while vectors define direction and magnitude without specifying position. Common vector operations include addition, subtraction, and scaling by scalars. The dot product between two vectors yields a scalar value that can indicate the cosine of the angle between the vectors. The cross product produces a vector perpendicular to both input vectors and is useful for computing properties like surface normals in 3D graphics.

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51 views23 pages

CS 543: Computer Graphics Lecture 3 (Part II) : Points, Scalars and Vectors

Points define a location in space using coordinates from a reference frame, while vectors define direction and magnitude without specifying position. Common vector operations include addition, subtraction, and scaling by scalars. The dot product between two vectors yields a scalar value that can indicate the cosine of the angle between the vectors. The cross product produces a vector perpendicular to both input vectors and is useful for computing properties like surface normals in 3D graphics.

Uploaded by

Chi Hana
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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CS 543: Computer Graphics

Lecture 3 (Part II): Points, Scalars and Vectors

Emmanuel Agu
Points, Scalars and Vectors

n Points, vectors defined relative to a coordinate system


Vectors

n Magnitude
n Direction
n NO position
n Can be added, scaled, rotated Length
n CG vectors: 2, 3 or 4
dimensions

Angle
Points

n Location in coordinate system


n Cannot add or scale
n Subtract 2 points = vector
Point
Vector-Point Relationship

n Diff. b/w 2 points = vector


v=Q–P

n Sum of point and vector =


P
point
v
v+P=Q
Q
Vector Operations

n Define vectors
Then vector addition:
a = (a1, a2 , a3 )
b = (b1,b2 , b3 ) a + b = (a1 + b1, a2 + b2 , a3 + b3 )

n and scalar, s

a
a+b

b
Vector Operations

n Scaling vector by a scalar


Note vector subtraction:

as = (a1s, a2 s, a3 s ) a−b
= (a1 + (−b1 ), a2 + (−b2 ), a3 + (−b3 ))

a a-b
a

2.5a b
Vector Operations: Examples

n Scaling vector by a scalar •Vector addition:

as = (a1s, a2 s, a3 s ) a + b = (a1 + b1, a2 + b2 , a3 + b3 )

n For example, if a=(2,5,6) and


b=(-2,7,1) and s=6, then

a + b = (a1 + b1, a2 + b2 , a3 + b3 ) = (0,12,7)

as = (a1s, a2 s, a3 s ) = (12,30,36)
Affine Combination

n Summation of all components = 1

a1 + a2 + .........an = 1

n Convex affine = affine + no negative component

a1 , a2 ,.........an = non − negative


Magnitude of a Vector

n Magnitude of a

| a |= a1 + a2 .......... + an
2 2 2

n Normalizing a vector (unit vector)


a vector
â = =
a magnitude
n Note magnitude of normalized vector = 1. i.e

a1 + a2 .......... + an = 1
2 2 2
Dot Product (Scalar product)

n Dot product,

d = a ⋅ b = a1 ⋅ b1 + a2 ⋅ b2 ........ + a3 ⋅ b3

n For example, if a=(2,3,1) and b=(0,4,-1)


then
a ⋅b = 2 ⋅ 0 + 3 ⋅ 4 + 1 ⋅ −1

= 0 + 12 − 1 = 11
Dot Product

n Try your hands at these:


n ( 2, 2, 2, 2)•( 4, 1, 2, 1.1)
n ( 2, 3, 1)•( 0, 4, -1)
Dot Product

n Try your hands at these:


n ( 2, 2, 2, 2)•( 4, 1, 2, 1.1) = 8 + 2 + 4 + 2.2 = 16.2
n ( 2, 3, 1)•( 0, 4, -1) = 0 + 12 –1 = 11
Properties of Dot Products

n Symmetry (or commutative):

a⋅b = b ⋅a
n Linearity:
(a + c) ⋅ b = a ⋅ b + c ⋅ b
n Homogeneity:

( sa) ⋅ b = s (a ⋅ b)
n And
b2 = b ⋅ b
Angle Between Two Vectors

y
c b = ( b cos φb , b sin φb )

c = ( c cos φc , c sin φc )
θ b
φc b ⋅ c = b c cosθ
φb
x
b b
b
Sign of b.c:
c c c
b.c > 0
b.c = 0 b.c < 0
Angle Between Two Vectors

n Find the angle b/w the vectors b = (3,4) and c = (5,2)


Angle Between Two Vectors

n Find the angle b/w the vectors b = (3,4) and c = (5,2)


n |b|= 5, |c|= 5.385

3 4
b̂ =  ,  bˆ • cˆ = 0.85422 = cosθ
5 5

θ = 31.326o
Standard Unit Vectors

Define y

i = (1,0,0)
k i

j = (0,1,0)

k = (0,0,1)
0 j

z x
So that any vector,

v = (a, b, c ) = ai + bj + ck
Cross Product (Vector product)

If
a = (a x , a y , a z ) b = (bx , by , bz )
Then
a × b = (a y bz − a z by )i − (a x bz − a z bx ) j + (a x by − a y bx )k
Remember using determinant
i j k
ax ay az
bx by bz

Note: a x b is perpendicular to a and b


Cross Product

Note: a x b is perpendicular to both a and b

axb

0
a b
Cross Product

Calculate a x b if a = (3,0,2) and b = (4,1,8)


Cross Product

Calculate a x b if a = (3,0,2) and b = (4,1,8)

a x b = -2i – 16j + 3k
References

n Hill, chapter 4.2 - 4.4

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