Seven measuring the world (geo/metry). Measuring space This course is fundamentally about spaces of various kinds Physical space Image space Auditory

seven

measuring the world

(geo/metry)

Measuring space

This course is fundamentally about spaces of various kinds

Physical space Image space Auditory space Cyber space

One of our fundamental questions is how we measure objects in space

Their position Their size Their orientation Their brightness The color …

Basic questions

Measuring size How big is that bail of hay?

Measuring position Where does my land end and your land begin

Measuring angle Which way is home? What time is it?

Measuring length

Choose some reference length to act as a unit of measure

Measuring length

Choose some reference length to act as a unit of measure

Duplicate it to determine the length of another object

Measuring position (1D)

We can measure the position of something


We can measure the position of something By choosing a reference point


We can measure the position of something By choosing a reference point And measuring the length of the space in between

Remember that the reference point is arbitrary

Measuring position in 2D

2D is more complicated


2D is more complicated We need not only

A reference point



A reference point And a unit of measure




But two directions




But two directions along which to measure

position

Cartesian coordinates

Descartes developed the method of specifying position in terms of A coordinate system

Reference point (origin) Directions (axes)

Distances along the axes (coordinates)

[point 4 4]

[point 0 0]

Coordinate systems

You can use any coordinate system that’s convenient

By choosing a different origin

[point 3 2.5]

[point 0 0]

Coordinate systems



Different axes

[point 3 2.5]

[point 0 0]

Coordinate systems



Different axes Or a different scale

[point 1 0.833]

[point 0 0]

3D

3D is the same except: We choose 3 axes And represent position

with 3 coordinates

(And it’s harder to draw convincingly)

Vectors

Vectors measure the displacement (shifts) between to points

They can also be represented as coordinate pairs

So we’ll mostly ignore the difference between points and vectors

Indeed, they’re the same thing in most computer graphics packages (including Meta)

[vector 3 2]

Combining vectors If you shift a point

First one way And then another

Then the resulting overall shift is The total shift along the X axis Plus the total shift along the Y axis

So it makes sense to talk about combining vectors

Since the total shift is The sum of the X coordinates and the sum of the Y coordinates, We’ll call this adding the vectors

It also corresponds to just adding their X and Y components

[vector 3 2]

[vector -2 1]

[vector 1 3]= [+ [vector -2 1] [vector 3 2]]

Scaling vectors

You can also talk about doubling, halving or otherwise multiplying a vector by some scale factor

Again, the result is just what you get from multiplying the individual components

[vector 1 3]

[vector .5 1.5]= [vector 1 3] / 2

[vector 2 6]= 2×[vector 1 3]

What you needto know about vector arithmetic

Single numbers are called scalars

Coordinate pairs are called vectors or points

We won’t worry about the distinction between the two

Addition and multiplication have natural geometric interpretations

Addition means shifting (translating)

Multiplication by a scalar means stretching and shrinking the vector

Arithmetic rules: Shifting a vector

(x1, y1) + (x2,y2) means (x1+x2, y1+y2)

Growing/shrinking a vector k × (x,y) a.k.a. k(x,y) means (kx, ky)

Can’t multiply or divide two vectors

What would it mean?

Another picture

a 1.5a (50% longer)

2a (twice as long)

b+a b+1.5a

b+2a

b

origin-0.5a

?

Angle

How do we measure the angle between two lines?

Angle


Draw a circle around their intersection

Give it a radius of 1

Angle



Give it a radius of 1 Say that the angle between

the lines Is the distance between them

along the circle

Angle



Give it a radius of 1 Say that the angle between

the lines Is the distance between them

along the circle

This distance-based unit of angle is called the radian

360 degrees = 2π radians 180 degrees = π radians 90 degrees = π/2 radians

Circles

A circle is the set of points that are a given distance of a given point

The point is the center The distance is the radius

So we can use the Pythagorean theorem to work out which points those are

Remember the distance squared between two points

Is the sum of the squares of the differences of their coordinates

circle = all points for which x2+y2=r2

r is the radius

(x,y)

(0,0)

Sine and cosine

The sine and cosine functions are unbelievably useful

Given an angle, they give you the coordinates of a point on a “unit circle”

A circle with radius 1 About the origin (0,0)

Angles are measured in Degrees, or Radians: distance about the

unit circle

circle = all points: [point [cos θ] [sin θ]]

for every 0≤θ≤2π

(cos θ, sin θ)

θsin θ

cos θ

1

(0,0)

Sine and cosine(cos θ2, sin θ2)

cos θ2

sin θ2 θ2

The sine and cosine functions are unbelievably useful

Given an angle, they give you the coordinates of a point on a “unit circle”

A circle with radius 1 About the origin (0,0)

Angles are measured in Degrees, or Radians: distance about the

unit circle

circle = all points: [point [cos θ] [sin θ]]

for every 0≤θ≤2π

Who cares?

This gives us a way to make vectors pointing in any direction:

[vector [cos θ] [sin θ]]Gives us a vector

Pointing in direction θ Of length 1

It will also help explain how waves work later

Documents

Seven measuring the world (geo/metry). Measuring space This course is fundamentally about spaces of various kinds Physical space Image space Auditory