Seeing numbers Perception R.W. Oldford University of Waterloo - - PowerPoint PPT Presentation

Seeing numbers

Perception R.W. Oldford

University of Waterloo

What is in a number?

Properties Some possible properties a number might have are

◮ a visual representation

◮ a picture encoding a value which can be decoded ◮ easily, efficiently, accurately (both encoding and decoding)

◮ unique identity (can tell when we see the same number again; distinguish between numbers) ◮ comparable, having quantity, size, or extent

◮ numbers can be (at least partially?) ordered by size, i.e. ranked ◮ has magnitude: quantity, size, extent ◮ cardinality, how many?

(identify and count; e.g. how many are red?)

◮ difference in values have meaning

(interval-scale; e.g. temperature)

◮ ratio of values is in turn a meaningful number

(ratio scale; e.g. height)

Identity

Discrimination

Both QR codes and UPC-A bar codes satisfy identity – can visually discriminate one number from another.

Exercise: what are the four numbers?

Identity

Discrimination This is all we need, if the data we are looking at are categorical.

For example: ◮ coin toss: heads, tails ◮ state of health: well, ill, dead ◮ mammal species: baboon, rat, dog, human, humpback whale, gorilla, . . . ◮ sex: male, female ◮ colour: red, orange, yellow, green, blue, violet ◮ person: Justin, Ameer, Ana, Chris, Andrew, Xiaoqing, Karan, Jacob, Kayla, Kofi, Matt, . . . Any set of distinguishable visual representations will do, but some will be better than

• thers.

Identity

Visual representations - Letters, simple stick shapes

Simple line combinations as used in block letters, etc. can be useful, provided the stick shapes are easily distinguished. For example, what distinctions appear here?

Identity

Visual representations - Letters, simple stick shapes

Simple line combinations as used in block letters, etc. can be useful, provided the stick shapes are easily distinguished. For example, what distinctions appear here? Angle (or orientation), spatial location, letter value (T or L). How many Ls are there? 1? 2? 3? 4? 5? 6? 7? It is kind of hard to distinguish the Ls from the Ts here; this might have been easier if

• rientation was not changing at the same time.

Identity

Visual representations - Shapes

Simple geometric shapes can also be distinguished. For example, how many different shapes are here? What distinctions are being made?

Identity

Visual representations - Shapes

Simple geometric shapes can also be distinguished. For example, how many different shapes are here? What distinctions are being made? How many different shapes? 5? 6? 7? 8? 9? 10? 11? 12? 13? 14? What distinctions are being made? Shape, location, orientation, size, reflection.

Questions

◮

Might regular shapes be more easily recognized?

◮

More easily distinguished?

◮

Is shape invariant to orientation? size? reflection?

Identity

Visual representations - visual location

An often overlooked means of discriminating categories is simply spatial location ◮ points in the same category are grouped spatially (close to each other) ◮ categories are separated spatially (more space between than within categories) ◮ add a boundary (here a rectangle) to enclose each group Works well with few categories.

Identity

Visual representations - Names

Simple labels provide a large range of different values that are relatively easily distinguished.

Identity

Visual representations - Names

Simple labels provide a large range of different values that are relatively easily distinguished. How many categories are here? How many labels in each category?

Identity

Visual representations - Names

Simple labels provide a large range of different values that are relatively easily distinguished. How many categories are here? How many labels in each category?

red? green? blue?

Note that in some cases, such as colour name, the colour of the label could induce confusion rather than further separation.

Identity

Visual representation - thinking outside of the box We are very good at distinguishing faces:

We can even recognize familiar faces and distinguish these from unfamiliar faces. We might use a different face for every category.

Identity

Visual representation - faces

It is important to our survival that we have evolved the ability to recognize faces (e.g. predators, family, friend, and foe) and even to distinguish facial expressions (e.g. to understand social situations). Facial recognition occurs very quickly within a particular region of the brain. Moreover, there are neurons in our brain that fire much more frequently whenever we see a face. Here the firing rate of the neuron is much higher when presented with something that looks like a face.

Identity

Visual representations - faces

Can easily recognize familiar faces. For example, who is this?

Identity

Visual representations - faces

Can even recognize faces, as faces, when upside down!

Identity

Visual representations - faces

Can recognize faces when the orientation changes!

Identity

Visual representations - faces

Can recognize faces when the orientation changes!

Identity

Visual representations - faces

Can recognize faces when the orientation changes!

Identity

Visual representations - faces

Can recognize faces when the orientation changes!

Identity

Visual representations - faces

Can recognize faces when the orientation changes! · · · um, wait a second?

Identity

Visual representations - faces

· · · huh?

Identity

Visual representations - faces

· · · huh?

Identity

Visual representations - faces

· · · huh?

Identity

Visual representations - faces

· · · whaaa . . . ?

Identity

Visual representations - faces

· · · whaaa . . . ?

Identity

Visual representations - faces

· · · whaaa . . . ?

Identity

Visual representations - faces

· · · OK, that’s messed up! What happened?

Identity

Visual representations - faces

These are identical images! We focus on different features, with different priorities, as determined largely by our evolutionary path and cultural training. What we “see” is a mental construction . . .

Identity

Visual representation - faces

The importance of faces along our evolutionary path unfortunately means we see them everywhere, even where they are not – we construct what we see!

Face on Mars Grilled cheese sandwich sold on EBay

Identity

Visual representations - colour

The human visual system also distinguishes COLOUR . The human retina contains about 6 million colour photoreceptors called “cones”, and about 120 million photoreceptors called “rods”

Human eye Human retina and photo receptors Distribution of receptors So let’s try it.

Identity

Visual representations - categories from hues

The safest colour use for identifying categories is having each category be coloured with a different hue: Even here there can be some challenges distinguishing hues that are close to one

• another. These work best when there are few categories so that hues can be chosen

that are widely separated.

Identity

Visual representations - colour

And then there is the problem of colour blindness (≈ 1 in 12 men, 1 in 200 women).

A number A number A path

Identity

Visual representations - colour

Rods: ◮ about 120 million rods in the human retina ◮ rods are distributed more evenly throughout our retina ◮ extremely light sensitive (a single rod can detect a single photon) ◮ do not detect colour (achromatic); instead register differences in light and dark ◮ provide most of our night vision (hence black and white at night) Cones: ◮ about 6 million cones in the human retina ◮ concentrate near the fovea (focal point) in our eye ◮ 3 different types of cone distinguishing three different sets of electromagnetic wavelengths (roughly corresponding to red, green, and blue) ◮ together they allow us to distinguish colours (much like an RGB display is able to produce the same)

Identity

Visual representations - colour

Cones: 3 different cone types react more to different wavelengths of light

(short, medium, and long roughly corresponding to blue, green, and red) Wavelength sensitivity of three cone types (source Lotto et al (2011) Optics & Laser Tech.)

Differences in the signal strength from the cones being stimulated help us perceive different colours.

(Note that red sensitivity was the last to evolve.)

Identity

Visual representations - colour

Deficiency in each colour (either cone, or pathway from cone):

• 1. Protanopia – deficiency in perceiving ‘red’ light
• 2. Deuteranopia – deficiency in perceiving ‘green’ light
• 3. Tritanopia – deficiency in perceiving ‘blue’ light.

Following images are from www.colourblindawareness.org

Normal Vision Protanopia – red deficiency Some care needs to be taken so that any visualization produced will also be accessible to the colour blind www.color-blindness.com/coblis-color-blindness-simulator.

Identity

Visual representations - colour

Deficiency in each colour (either cone, or pathway from cone):

• 1. Protanopia – deficiency in perceiving ‘red’ light
• 2. Deuteranopia – deficiency in perceiving ‘green’ light
• 3. Tritanopia – deficiency in perceiving ‘blue’ light.

Following images are from www.colourblindawareness.org

Normal Vision Deuteranopia – green deficiency Some care needs to be taken so that any visualization produced will also be accessible to the colour blind www.color-blindness.com/coblis-color-blindness-simulator.

Identity

Visual representations - colour

Deficiency in each colour (either cone, or pathway from cone):

• 1. Protanopia – deficiency in perceiving ‘red’ light
• 2. Deuteranopia – deficiency in perceiving ‘green’ light
• 3. Tritanopia – deficiency in perceiving ‘blue’ light.

Following images are from www.colourblindawareness.org

Normal Vision Tritanopia – blue deficiency Some care needs to be taken so that any visualization produced will also be accessible to the colour blind www.color-blindness.com/coblis-color-blindness-simulator. Try it!

Identity

Visual representations - colour

Simulating colour blindness for a statistical graphic.

Normal Vision Protanopia – red deficiency Deuteranopia – green deficiency Tritanopia – blue deficiency The images were created by uploading the normal one to www.colourblindawareness.org

Identity

Visual representations - colour gradients

Modern systems make it easy to introduce colour in all kinds of weird and wacky ways.

(Just try MS Excel if you want to see the truly bizarre world of business graphics!)

For example, colour gradients are very popular amongst the naive. Here two different identities (categories) are distinguished simply by the direction of the gradient:

Notice anything unusual? A three-dimensional effect maybe?

Identity

Visual representations - colour gradients

We implicitly assume light is from above.

Identity

Visual representations - colour gradients

We implicitly assume light is from above. Same picture upside down.

Identity

Visual representations - we assume light is from above.

The pictures are inversions of one another, producing a very different effect as to what is raised and what is depressed.

Identity

Visual representations - saturation changes

Another simple way of distinguishing (even ordering) categories through colour is to hold the hue of the colour fixed and change the saturation of the colour. Note: Even here care must be taken. It is easy to introduce unintentional effects caused by our human visual system. For example:

Notice anything unusual? This scalloping is called the Mach effect. It’s not really there! A gradient is constructed by our visual system to highlight edges.

Identity

Visual representations - saturation changes

Distinguishing categories through colour can be severely affected by the surrounding colour. For example: The centre dot is exactly the same colour in all blocks! This also holds with colours as well: here the red centre rectangle is exactly the same colour in both larger rectangles.

Identity

Visual representations - saturation changes

Distinguishing categories through colour can be severely affected by the surrounding colour. For example: The centre dot is exactly the same colour in all blocks! This also holds with colours as well: here the blue centre rectangle is exactly the same colour in both larger rectangles.

This is an example

• f an adjustment by our visual system called

colour constancy where we lighten the colour

• f objects in the dark !

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Perhaps the most dramatic example of how we construct what we see, particularly with respect to colours, is the following picture which will be revealed one step at a time. Starting with two optically identical grey regions which never change optically but do change visually!

Identity

Visual representations - colour is constructed by our visual system

Here we add a rectangle of exactly the same grey colour (optically) as the two checker board squares A and B. Notice how the grey rectangle has a gradient introduced entirely by our visual system!

Identity

Visual representations - colour is constructed by our visual system

The pictures at left and right are identical except that the rightmost picture has two identically coloured thin grey rectangles added. These run through and connect three identically grey checkerboard squares. The connections make them appear more alike but not perfectly alike.

Identity

Visual representations - colour is constructed by our visual system

To illustrate the nature of colour constancy, we add two optically identical orange spots to the checkerboard - one in the light, one in the dark. Our brain will lighten up the one in shadow! They will appear visually identical only if we remove the picture and its shadow and light context!

Identity

Visual representations - colour is constructed by our visual system

To illustrate the nature of colour constancy, we add two optically identical orange spots to the checkerboard - one in the light, one in the dark. Our brain will lighten up the one in shadow! They will appear visually identical only if we remove the picture and its shadow and light context!

Identity

Visual representations - colour is constructed by our visual system

To illustrate the nature of colour constancy, we add two optically identical orange spots to the checkerboard - one in the light, one in the dark. Our brain will lighten up the one in shadow! They will appear visually identical only if we remove the picture and its shadow and light context!

Identity

Visual representations - colour is constructed by our visual system

To illustrate the nature of colour constancy, we add two optically identical orange spots to the checkerboard - one in the light, one in the dark. Our brain will lighten up the one in shadow! They will appear visually identical only if we remove the picture and its shadow and light context! Here they are close up!

Magnitude

The size of numbers

We are finally getting down to more what we mean by numbers – their magnitude. For much data, we need to be considering measurements, which will have arbitrary value of varying precision. In either case, when a magnitude exists for each datum, interest lies in ◮ discriminating between the numbers ◮ ranking the numbers ◮ judging the relative size of the numbers

◮ for some purposes it might be enough to just distinguish “smaller than”,“much smaller than” , “about the same”, “larger than”, and “much larger than” ◮ for other applications, we will want to be able to tell the size of the difference ◮ for still others, we want to be able to see the ratio of the sizes ◮ for some we will want to be able to determine the actual magnitude of the values from the visualization

Magnitude

Visual representations

For what we have called “natural numbers”, namely integers which are not too large as to beyond comprehension, we have considered a number of ways to give them pictorial form. Our objective here is to think of some visual representations that will encode an arbitrary numerical value that is easily and reliably decoded. The visual representations used for categorical values that might also be used here include ◮ colour (hue, saturation, etc.) ◮ shape (perhaps size of shape) ◮ visual location Given the discussion on colour for categories, this seems least promising for numerical values.

Magnitude

Visual representation - order by colour

Minimally we must be able to order the numerical values. This requires choosing colour which also displays order.

If there are only a few values to distinguish, then we might use a sequential palette of colours all sharing the same hue but different amounts of lightness or saturation. (See colourPalettes.R on course website.) A 5 value sequence A 7 value sequence Clearly as the number of different values increases it becomes more difficult to distinguish between the colours. If there are two many, ordering will be subject to considerable error.

Magnitude

Visual representation - order by colour

A divergent palette, where two different sequential hues are used, can be of especial value in ordering values where there is a natural origin or zero for the values This can effectively double the number of different values that can be ordered.

Magnitude

Visual representation - continuous scale by colour

Changing saturation or lightness Changing saturation (or lightness) and hue Oppositional hues Not so bad if values (e.g. on a continuous function) change along scale. Bad if they hop all over.

Magnitude

Visual representation - continuous scale by colour

A serious problem is to calibrate how our percep- tion of the value changes with the actual saturation (say). Is the change in our perception linear with the change in the saturation? Psychophysical studies show that when dealing with the senses (including sight) that the perceived inten- sity of the stimulus is not always a linear function of the magnitude of the physical stimulus (e.g. weight, sound, pressure, brightness, . . . , as perceived by the corresponding sense).

Source: Josef Albers (1963, p. 56) Interaction of Color

Magnitude

Visual representation - by area

Another possibility to carry over from the categorical case, is to look at shapes, and order them by their area This doesn’t look so promising either . . . unless the shapes are the same.

Magnitude

Visual representation - by length

Encode values as lengths

Magnitude

Visual representation - by position

Encode values as positions on some scale

Magnitude

Visual representation - by slope

Encode values as slopes

Magnitude

Visual representation - by angle

Encode values as angles

Magnitude

Visual representation - by volume

Encode values as volumes

Perceptual “laws”

Perceived magnitude

◮ Psychophysical researchers have tried to relate

◮ a physical stimulus of measured magnitude ◮ to intensity of the stimulus as perceived by the person receiving it ◮ E.g. weight, sound, sight, . . . as perceived by the corresponding senses.

◮ they have discovered “laws” empirically by experimentation

◮ consistency in the results, for different senses ◮ variation across individuals

◮ Of interest here are those which relate to the “visual magnitude”

perceived by viewers of the various visual representations of numbers.

Perceptual laws

Weber’s law Let

◮ x denote the physical magnitude of a stimulus ◮ and wp(x) > 0 be such that

◮ x + wp(x) is perceived to be larger than x with probability p

Then Weber’s law states that wp(x) = kpx where kp does not depend on x, but only on p (presumably increasing with p) . . . and,

• f course, the type stimulus.

(see also Weber-Fechner law)

Perceptual laws

Weber’s law

wp(x) = kpx

where kp does not depend on x but only on p Suggests that what really matters is proportional change, in that

x + wp(x) x = 1 + wp(x) x = 1 + kp

It seems to make sense for visually determining length

◮ E.g. says that it is easier to detect that a 2.5 cm line is longer than a 2.0 cm line than it is to detect that a 20.5 cm line is longer than a 20.0 cm. ◮ namely, the first is 25% longer; the second only 2.5% longer.

Perceptual laws

Weber’s law Example: comparing the length of rectangles. Which rectangle is longer A or B?

What does Weber’s law suggest might be the difficulty?

They are near each other in length so their proportional difference is small . . . hard.

Perceptual laws

Weber’s law Now which rectangle is longer A or B? Seems easier. Why? What does Weber’s law suggest?

By framing the rectangles the two shorter pieces (the remainders in the frames) have a greater proportional difference. Using framed rectangles should be (and is) much easier according to Weber’s law

Perceptual laws

Stevens’ law A person’s perceived magnitude of a stimulus of magnitude x is

p(x) = cxβ.

(sometimes called Stevens’ power law.)

Seems to work well on a wide variety of perceived scales, including:

◮ Length: 0.9 ≤ β ≤ 1.1 ◮ Area: 0.6 ≤ β ≤ 0.9 ◮ Volume: 0.5 ≤ β ≤ 0.8

All values empirically determined. There is person to person variation.

Perceptual laws

Stevens’ law

Consider visually comparing two areas of size a and b. And suppose that β = 0.7 (which has been observed empirically) Then, according to Stevens’ law, the ratio of the areas (i.e. a/b) will be perceived to be

a b 0.7

instead of (a/b). Example: Suppose objects of area 2, 4, and 8 were being compared visually, then the perceived ratio of the first to the second would be 2 4 0.7 ≈ 0.62 instead of 0.5, and the perceived ratio of the third to the second will be 8 4 0.7 ≈ 1.62 instead of 2. Small areas are perceived to be larger than they are; large areas smaller.

It’s even worse for volumes.

Perceptual laws

Stevens’ law This suggests an ordering for visual representations of magnitudes. If we are looking at visually encoding a magnitude x, then

◮ using length has smaller bias (zero if β = 1) than ◮ using area (β = 0.7 say), ◮ which in turn has smaller bias than using volume (β = 0.5 say)

If the variability in perception is the same for each, then we should expect a decreasing accuracy in encoding from length to area to volume.

Perceptual laws

Graphical Perception In a series of experiments, Cleveland and McGill (1984, 1985, 1987) investigated the quality of a variety of different encodings for magnitude. One such series (reported in Cleveland and McGill, 1985) considered

◮ angle ◮ area (circular and “blob”) ◮ position along a common scale ◮ position along identical but nonaligned scales ◮ length ◮ slope

Perceptual laws

Graphical Perception

positions, lengths, and slopes (local rates of change)

Perceptual laws

Graphical Perception

positions, lengths, slopes,angles, circle area, blob area

Perceptual laws

Graphical Perception Ordering of the elementary tasks (according to Cleveland, p. 254: Theory of visual perception, experiments in graphical perception, and “informal experimentation” suggest the following ordering from most accurate to least accurate:

• 1. Position along a common aligned scale
• 2. Position on identical but nonaligned scales
• 3. Lengths (N.B. line segments were all oriented horizontally or vertically, though nonaligned)
• 4. Angle

Slope (not close to 0, π/2, or π radians)

• 5. Area
• 6. Volume
• 7. Colour density, colour saturation
• 8. Colour hue

Note: This ordering is tentative/suggestive, as opposed to definitive. More research required.

Perceptual laws

Graphical Perception Actually the recommendation for slopes (or, better, angle with the horizontal) can be justified mathematically.

Let r be the ratio of the slope of BC to that of AB. The question is, how large is r in each of the 3 figures?

Perceptual laws

Graphical Perception Empirically comparing slopes:

r ≈ 1 in the rectangles – all three ratios are the identical. But visual decoding is poor when the angle of the slopes is near 90 (left), or 0 (bottom). Empirical studies suggest slopes (or tilts, or slants) are compared more via the angle they form with the horizontal.

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

• For small ∆θ this is approximately

1+∆θ tan′(θ) tan(θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

• For small ∆θ this is approximately

1+∆θ tan′(θ) tan(θ)

• = 1+∆θ
• 1

cos2(θ)

• sin(θ)

cos(θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

• For small ∆θ this is approximately

1+∆θ tan′(θ) tan(θ)

• = 1+∆θ
• 1

cos2(θ)

• sin(θ)

cos(θ)

= 1+∆θ

• 1

cos(θ) sin(θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

• For small ∆θ this is approximately

1+∆θ tan′(θ) tan(θ)

• = 1+∆θ
• 1

cos2(θ)

• sin(θ)

cos(θ)

= 1+∆θ

• 1

cos(θ) sin(θ)

• = 1+∆θ
• 2

sin(2θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

• For small ∆θ this is approximately

1+∆θ tan′(θ) tan(θ)

• = 1+∆θ
• 1

cos2(θ)

• sin(θ)

cos(θ)

= 1+∆θ

• 1

cos(θ) sin(θ)

• = 1+∆θ
• 2

sin(2θ)

• r for small ∆θ

|r − 1| ≈ |∆θ|

• 2

sin(2θ)

Perceptual laws

Graphical Perception Mathematically comparing slopes:

Let θ be the angle of a line segment, and ∆θ be a small difference in the angle (e.g. just small enough to detect a difference in the slope of the line). The two slopes are tan(θ + ∆θ) and tan(θ) and their ratio r: r = tan(θ + ∆θ) tan(θ) = 1+ tan(θ + ∆θ) − tan(θ) tan(θ) = 1+∆θ

• 1

tan(θ) tan(θ + ∆θ) − tan(θ) ∆θ

• For small ∆θ this is approximately

1+∆θ tan′(θ) tan(θ)

• = 1+∆θ
• 1

cos2(θ)

• sin(θ)

cos(θ)

= 1+∆θ

• 1

cos(θ) sin(θ)

• = 1+∆θ
• 2

sin(2θ)

• r for small ∆θ

|r − 1| ≈ |∆θ|

• 2

sin(2θ)

• Whenever θ approaches 0 or π/2 radians this difference approaches ∞ whatever the value of ∆θ.

Note however that the multiplier of |∆θ| is smallest (and has least effect) when θ = π/4 radians (45 degrees). When θ is near this value, different values of ∆θ should be easily detectable.