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 Art Points and Vectors

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Csc 123 computational art points and vectors

CSC 123 – Computational ArtPoints and Vectors

Zoë Wood


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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