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CSC 123 – Computational Art Points and Vectors

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CSC 123 – Computational ArtPoints and Vectors

Zoë Wood

- A 2D point:
- Represents a location with respect to some coordinate system

- A 2D vector:
- Represents a displacement from a position

- 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

- 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

- 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

- Addition and multiplication by a scalar:
- a + b
- s*a

- Examples:

- Addition and multiplication by a scalar:
- a + b
- s*a

- Examples:

v

- Multiplying a vector by a scalar multiplies the vector’s length without changing its direction:

3.5 *v

2 *v

- 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

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

-a+b

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

b

a

b-a

- A linear combination of m vectors
is a vector of the form:

where a1, a2, …, am are scalars.

- Example:

- 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]

- 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

- 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.

- 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

- 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?)

- 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

- There are other operations on 3D vectors
- dot product
- cross product

- We won’t talk about these

- 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.

- 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

- Descarté
- One reason why cartesian coordinates are nice

By Pythagorean theorem:

length of c is:

y

c

0.5

2

x

- 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

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

- 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