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

Measuring position (1D) We can measure the position of something By choosing a reference point

Measuring position (1D) 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

Measuring position in 2D 2D is more complicated We need not only A reference point

Measuring position in 2D 2D is more complicated We need not only A reference point And a unit of measure

Measuring position in 2D 2D is more complicated We need not only A reference point And a unit of measure But two directions

Measuring position in 2D 2D is more complicated We need not only A reference point And a unit of measure 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 You can use any coordinate system that’s convenient By choosing a different origin Different axes [point 3 2.5] [point 0 0]

Coordinate systems You can use any coordinate system that’s convenient By choosing a different origin Different axes Or a different scale [point ] [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 need to 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 (x 1, y 1 ) + (x 2,y 2 ) means (x 1 +x 2, y 1 +y 2 ) 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 How do we measure the angle between two lines? Draw a circle around their intersection Give it a radius of 1

Angle How do we measure the angle between two lines? Draw a circle around their intersection Give it a radius of 1 Say that the angle between the lines Is the distance between them along the circle

Angle How do we measure the angle between two lines? Draw a circle around their intersection 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 x 2 +y 2 =r 2 r is the radius (x,y)(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 ≤2 π (cos θ, sin θ) θ sin θ cos θ 1 (0,0)

Sine and cosine (cos θ 2, sin θ 2 ) cos θ 2 sin θ 2 θ2θ2 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