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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### CSC 123 – Computational ArtPoints and Vectors

Zoë 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.
• 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

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
• 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
• Addition and multiplication by a scalar:
• a + b
• s*a
• Examples:
Vector Operations
• Addition and multiplication by a scalar:
• a + b
• s*a
• Examples:

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
• 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

Paralleologram rule

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

V+W

V

W

• 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

v

-v

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

-a+b

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

b

a

b-a

Linear Combinations
• A linear combination of m vectors

is a vector of the form:

where a1, a2, …, am are scalars.

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

is a vector of the form:

where a1, a2, …, am are scalars.

• Example:
• Given vT = [x0, x1 , x2] and wT = [y0, y1 , y2]
• V+W = [x0 +y0, x1 + y1 , x2 + y2]
• 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
• 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
• 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
• Linear operations on vectors make sense, but the same operations do not make sense for points:
• 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
• 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
• There are other operations on 3D vectors
• dot product
• cross product
• We won’t talk about these
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
• 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
• Descarté
• One reason why cartesian coordinates are nice

By Pythagorean theorem:

length of c is:

y

c

0.5

2

x

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?
• When we want to control movement, we can use vectors
• For example if we want to have a faces’ eyeballs follow the mouse…
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