csc 123 computational art points and vectors
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CSC 123 – Computational Art Points and Vectors. Zo ë Wood. Representing Points and Vectors. A 2D point: Represents a location with respect to some coordinate system A 2D vector: Represents a displacement from a position. Vectors. Objects with length and direction .

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representing points and vectors
Representing Points and Vectors
  • A 2D point:
    • Represents a location with respect to some coordinate system
  • A 2D vector:
    • Represents a displacement from a position
vectors
Vectors
  • Objects with length and direction.
  • Represent motion between points in space.
  • We will think of a vector as a displacement from one point to another.
  • We use bold typeface to indicate vectors.

v

vectors as displacements

v

v

v

v

v

Vectors as Displacements
  • As a displacement, a vector has length and direction, but no inherent location.
  • It doesn’t matter if you step one foot forward in Boston or in New York.

v

vector notation
Vector Notation
  • We represent vectors as a list of components.
  • By convention, we write vectors as a column matrix:
  • We can also write them as row matrix, indicated by vT:

vT = [x0, x1] or wT = [x0, x1 , x2]

  • All components of the zero vector0 are 0.

or

vector operations
Vector Operations
  • Addition and multiplication by a scalar:
    • a + b
    • s*a
  • Examples:
vector operations1
Vector Operations
  • Addition and multiplication by a scalar:
    • a + b
    • s*a
  • Examples:
vector scaling geometric interpretation

v

Vector Scaling – geometric interpretation
  • Multiplying a vector by a scalar multiplies the vector’s length without changing its direction:

3.5 *v

2 *v

vector scalar multiplication
Vector scalar Multiplication
  • Given wT = [x0, x1 , x2] and scalars s and t

s *wT = [s *x0, s *x1 , s * x2]

  • Properties of Vector multiplication
    • Associative: (st)V = s(tV)
    • Multiplicative Identity: 1V = V
    • Scalar Distribution: (s+t)V = sV+tV
    • Vector Distribution: s(V+W) = sV+sW
vector addition geometric interpretation
Vector addition – geometric interpretation

Paralleologram rule

  • To visualize what vector addition is doing, here is a 2D example:

V+W

V

W

vector addition
Vector Addition
  • Head to tail: At the end of a, place the start of b.

a + b = b + a

  • Components: a + b = [a0 + b0, a1 + b1]T

a+b

b

a

unary minus

v

-v

Unary Minus
  • Algebraically, -v = (-1) v.
  • The geometric interpretation of -v is a vector in the opposite direction:
vector subtraction

-a+b

Vector Subtraction
  • b - a = -a + b
  • Components: b - a = [b0 - a0, b1 - a1]T

b

a

b-a

linear combinations
Linear Combinations
  • A linear combination of m vectors

is a vector of the form:

where a1, a2, …, am are scalars.

  • Example:
linear combinations1
Linear Combinations
  • A linear combination of m vectors

is a vector of the form:

where a1, a2, …, am are scalars.

  • Example:
vector addition1
Vector Addition
  • Given vT = [x0, x1 , x2] and wT = [y0, y1 , y2]
    • V+W = [x0 +y0, x1 + y1 , x2 + y2]
  • Properties of Vector addition
    • Commutative: V+W=W+V
    • Associative (U+V)+W = U+(V+W)
    • Additive Identity: V+0 = V
    • Additive Inverse: V+W = 0, W=-V
points and vectors
Points and Vectors
  • We normally say v = [3 2 7]T.
  • Similarly, for a point we say P = [3 2 7]T.
  • But points and vectors are very different:
    • Points: have a location, but no direction.
    • Vectors: have direction, but no location.
  • Point: A fixed location in a geometric world.
  • Vector: Displacement between points in the world.
vectors to specify locations
Vectors to Specify Locations
  • Locations can be represented as displacements from another location.
  • There is some understood origin location O from which all other locations are stored as offsets.
  • Note that locations are not vectors!
  • Locations are described using points.

O

v

P

operations on vectors and points
Operations on Vectors and Points
  • Linear operations on vectors make sense, but the same operations do not make sense for points:
  • Addition:
    • Vectors: concatenation of displacements.
    • Points: ?? (SLO + LA = ?)
  • Multiplication by a scalar:
    • Vectors: increases displacement by a factor.
    • Points: ?? (What is 7 times the North Pole?)
  • Zero element:
    • Vectors: The zero vector represents no displacement.
    • Points: ?? (Is there a zero point?)
operations on points
Operations on Points
  • However, some operations with points make sense.
  • We can subtract two points. The result is the motion from one point to the other:

P - Q = v

  • We can also start with a point and move to another point:

Q + v = P

other operations
Other operations
  • There are other operations on 3D vectors
    • dot product
    • cross product
  • We won’t talk about these
vector basis
Vector Basis
  • Vectors which are linearly independent form a basis.
  • The vectors are called basis vectors.
  • Any vector can be expressed as linear combination of the basis vectors:
  • Note that the coefficients ac and bc are unique.
orthogonal basis
Orthogonal Basis
  • If the two vectors are at right angles to each other, we call the it an orthogonal basis.
  • If they are also vectors with length one, we call it an orthonormal basis.
  • In 2D, we can use two such special vectors x and y to define the Cartesian basis vectors
  • x = [1 0] y = [0 1]

y

x

cartisian
Cartisian
  • Descarté
  • One reason why cartesian coordinates are nice

By Pythagorean theorem:

length of c is:

y

c

0.5

2

x

normalizing a vector
Normalizing a vector
  • In order to make a vector be a ‘normal’ vector it must be of unit length.
  • We can normalize a vector by dividing each component by its length
how can we use these
How can we use these?
  • When we want to control movement, we can use vectors
  • For example if we want to have a faces’ eyeballs follow the mouse…
problem
Problem
  • Move eyeballs to the edge of a circle in a given direction
    • Compute the vector, which is a displacement fromthe current position towhere the mouse wasclicked
    • Scale the vector to only be aslong as the eye’s radius
    • Displace the eyeball to rimof the eye
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