How vanishing points work (Part 3)

May 31, 2013

In the second part of this series, we discovered that any set of parallel lines has a single vanishing point.

Let’s see what happens when there are two non-parallel lines in our field of view. Imagine standing at the corner of a field where two boundary fences meet:

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It’s already pretty clear that these two fences, which sit at right angles to each other, are going to have different vanishing points, so I’ve marked them in as red and green “x”s. As always, we can locate the vanishing points using the rule laid out in the first post. We simply have to draw lines through the center of the eye that lie parallel to each fence. Here’s the bird’s eye view for the first fence:

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I’ve left out the individual light rays from each fence post, to reduce clutter. But I have followed the required rule: I’ve drawn a ray from the center of the eye that lies parallel to the first fence. The vanishing point is marked with a red “x”.

If I do the same for the second fence, I get:

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Bringing the two diagrams together, we have:

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And indeed, there are two vanishing points, one for each fence. Note that the red vanishing point lies to the left of the center of gaze, while the green vanishing point lies to the right of the center of gaze, just as we observe in the first figure in this post.

We can now add a new rule:

Each distinct set of parallel lines has its own vanishing point.

In my experience with drawing things like buildings, I’ve used exactly two vanishing points. This makes sense because building’s floor plans are traditionally made up of a bunch of rectangles stuck together, and a bunch of rectangles are composed of only two sets of parallel lines. Each set has its own vanishing point.

Consider the following pair of drawings as examples. The drawing on the left is an outline of a building’s floor plan, and the sketch on the right is a 3-D rendering of that plan using two vanishing points:

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A real 3-D building also has vertical lines, not just horizontal ones. So technically this third set of parallel lines (all the uprights in the right-hand sketch above) should have their own vanishing point, making the sketch look something like this:

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The reason we don’t see such extreme perspective in real-life is that the vanishing points are usually much farther apart on our field of view. In the above sketch, they’re pretty close together, yielding a sort of fish-eye lens look.

Things get even more complicated when we have rectangles that are at different angles to each other. Each rectangle now needs its own pair of vanishing points (ignoring the uprights for now). So, the following two simple rectangular buildings require a total of four vanishing points:

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I’ve labeled each vanishing point according to the block it belongs to. If we wanted to, we could add the extra vanishing point needed for the uprights, but I think the point has been made – it can get complicated!

And that wraps up this series. I hope to have more science-related posts published soon, but summer time is kicking in and the outdoors are beckoning!


How vanishing points work (Part 2)

May 30, 2013

In the first part of this series, we came up with a rule for finding the location on the retina that corresponds to the vanishing point of a straight line in our field of view.

What if we have more than one straight line – will we need more than one vanishing point?

It depends!

Let’s consider two special cases. First, we’ll look at two parallel lines in our field of view. Imagine being back on our country road, and this time there is a fence on each side of the road:

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The eye’s view of two parallel fences, as one might see on either side of a road. Should they both have the same vanishing point?

If we consider a single horizontal wire from each fence, we will have two lines that, in the real world, are parallel. The question is, do they have the same vanishing point? Well, let’s follow our rule on how to get the vanishing point from a line. We’ll apply it first to fence 1, then to fence 2.

First, fence 1:

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In the above figure, I’ve drawn a line through the center of the eye that is parallel to fence 1. It hits the edge of the eye at the red “x”. But this is exactly where the vanishing point was for the fence we looked at in the first post. In fact, if we put the light ray drawings for the two fences together, we get:

vanish08

The two fences have the same vanishing point. This gives us a second rule for vanishing points:

Any set of parallel lines has a single vanishing point.

(You might have noticed that I implicitly assumed this rule to be true when I showed the very first picture of the fence along the country road. The three horizontal wires in the fence, which are all parallel to one another, were assumed to have the same vanishing point. I didn’t have to make this assumption. I could have drawn the fence with only a single wire, and the result would have been the same. It just wouldn’t have looked much like a real fence!)

Here’s where I’d like to mention the direction that the eye is looking in. Notice, in the above figure, that the person is looking a little to the right of center (thick grey arrow). This means that the vanishing point of the two fences lies a little to the left of the center of her gaze. Looking back to the first image in this post, you’ll see that this is indeed the case. The vanishing point of the two fences is to the left of the figure’s center.

If the person were to move her eyes so that she was gazing in a different direction, the only thing we would have to change in the light ray diagram is the position of the thick gray arrow. For example, let’s say she looks sharply to the left. The light ray diagram now looks like this:

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The only thing that’s changed is the direction of the thick grey arrow. But this now means that the vanishing point lies to the right of the center of gaze. To the viewer, then, the scene now looks something like this:

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Now the vanishing point is to the right of the viewer’s field of view. It’s quite remarkable how much the scene changes due to the simple act of looking in a different direction. But I guess that’s really the point of looking in a different direction in the first place – to change the scene!

In the next post I’ll look at what happens when we have two perpendicular lines in our field of view.


How vanishing points work (Part 1)

May 29, 2013

This is another sciency post, following on from my discussion of parallel sun rays.

If you’ve ever attempted realistic perspective drawings, you’ve undoubtedly come across the issue of vanishing points – the point toward which a set of parallel lines appears to converge. How many vanishing points should there be? And why do they even exist in the first place? Read on for some answers!

The first thing I’d like to demonstrate is that what we see as an apparent convergence of parallel lines is an artifact of the (almost) spherical shape of our eyeballs.

To start, let’s imagine we’re standing by the side of a country road looking at a fence receding into the distance, represented by this idealized sketch:

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The eye’s view of a fence receding into the distance. The red “x” on the left indicates the vanishing point.

I’ve labeled the fence posts so that we can see how they map onto the eye’s retina. I’ve also indicated the vanishing point with a red “x”. This is the point at which the three horizontal wires in the fence appear to converge.

Let’s consider a bird’s eye view of both the fence and the eyeball of the person looking at it. I’m going to draw the eyeball REALLY big so that we can see what the light rays are doing inside it. I’m also going to leave out the details like the lens, the cornea, and all the other bits and pieces of the optical system. All I’m going to show is a simple sphere that focuses light through a point at the sphere’s center, like this:

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View from above. Light rays from the fence posts enter a simplified (and over-sized) eye and strike the retina at the back. The eye is looking in the direction indicated by the thick grey arrow.

The first thing to note in this picture is the set of fence posts from the first figure, as seen from above. A light ray (black line) leads from each fence post to a large circle representing the eye. Each light ray passes through the center of the circle and strikes the back of the eye – the retina, where I’ve repeated its original label.

The other thing I’ve included in this figure is the direction the eye is looking in. This is indicated by the thick grey arrow. For the most part, this direction won’t be important, but I’m including it for completeness (and it will come in handy a little later).

Notice that when moving from fence post A to G, the position of the post’s images on the retina get closer and closer together. This is why the fence posts appear to get closer and closer together in the perspective view (first figure), even though they are equally spaced in the real world.

There’s something annoying about the above figure, though: the image on the retina is backwards, which makes it a little harder to interpret. We don’t have this difficulty in real vision because our brain makes the necessary transformation to a normal, unreversed image. What I’m going to do, then, is make things simpler by looking at where the light rays strike the front of the eyeball, since this will give us an image that’s the right way around, and it will make the figures a little simpler, without affecting the basic physics.

So, here is the simplified version of the above figure:

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As before, the positions of the fence posts on the eye get closer and closer together as the fence posts get farther and farther away.

What happens if we add more fence posts even farther away? Will the images of these fence posts converge to a single point on the eyeball? Yes, indeed, they will. Consider the following picture, where I’ve added just one extra fence post that’s a lot farther away than the original set:

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Notice how the light ray from fence post H is almost parallel to the fence itself. And indeed, if we were to place yet another fence post an infinite distance away, the light ray from that post would, in fact, be parallel to the fence. Which means that the vanishing point for the fence is located directly up from the center of the eye, as indicated by a red “x” in the following picture:

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This gives us a nifty rule for finding the vanishing point for any straight line in our field of view:

To find the vanishing point of any straight line, draw a second line that:

1. starts from the center of the eye and

2. is parallel to the first line. 

The vanishing point is where this new line crosses the retina.

Note how the vanishing point is drawn on the eye. This is because the vanishing point is an artifact of the eye’s optics. There is no vanishing point in the real world.

In the second part of this series, I’ll look at what happens when we have more than one line in our field of view. How many vanishing points will we need?


Got logically consistent ethics?

May 24, 2013

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There is a contingent of ethical thinkers who believe that our moral instincts are the best guide to deciding what’s moral. Our moral instincts, these folks argue, constitute a reliable epistemology for ethics in a similar way that the scientific method constitutes a reliable epistemology for physics (and all that silly, unimportant stuff that relies on physics, like biology and chemistry*).

But the evidence seems to indicate that our moral instincts arrived at their current state through evolution, just like our other instincts. And since evolution is imperfect, inefficient, and undirected, we are not justified in concluding that our moral instincts are perfectly reliable.

In fact, there’s a way of testing this. It arises from the observation that some of our moral instincts are broader than others. For instance, our penchant for fairness is broader than our aversion to lying. Lying is quite a specific area of moral interest, while fairness features in almost every moral topic.

The test of our moral instincts is therefore this: Are our more specific instincts consistent with the broader instincts they fall under? For instance, does our aversion to lying ever interfere with our desire to be fair? (Put differently, are there instances in which telling a lie would be the most fair thing to do?).

If a specific moral instinct prompts us to go against a broader moral instinct, then our instincts are not logically consistent with one another, and we need to rethink our reliance on those instincts. And the only way to determine if one instinct conflicts in a logical sense with another, is to take the broadest (and therefore most fundamental) of all the instincts, extend its logical structure down to the more detailed levels, and see what these more specific instincts ought to be telling us under conditions of logical coherence.

This, in essence, was the purpose behind my essay (now short book) on morality: To take a simple and fundamental idea like the desire for happiness, and see where it logically leads.

So, while moral instincts can certainly be useful as a quick guide to moral thinking, they should constantly be checked to see if they support the basic principles of the moral system being used.

—–

* Only kidding.


Gods and sproogles

May 13, 2013

I recently got involved in a dialogue that started with a fairly frequent complaint among theists: Why does science so quickly and casually rule out the possibility of supernatural explanations in favor of natural ones?

My response began with the following analogy.

A friend of mine ate a piece of fruit the other day. She said it was a sproogle.

I asked her what a sproogle was, and she said it wasn’t an apple, a banana, a pear, or any other fruit I had likely come across.

When I asked her what it looked or tasted like, she said it didn’t look or taste like an apple, banana, pear, or any other fruit I had likely come across.

You can see where this is going. Try as I may, I could not get her to describe a single unique, positive attribute of this strange fruit. Her descriptions were purely in terms of characteristics the sproogle did not have.

I wondered how I could prove that what she’d eaten was, in fact, a sproogle rather than a better known fruit, perhaps one eaten by folks in another part of the world.

It seemed my only option was to eliminate all the known fruits from the list of candidates, so that only the sproogle remained. I would have to collect a specimen of every fruit and ask my friend if it was like the one she had eaten.

Science is very much in the same boat when it comes to investigating supernatural claims: There is no positive definition of the supernatural. The very word itself means “of, pertaining to, or being above or beyond what is natural”. It is defined by its contrast with something else, not in terms of its own unique properties.

So if science wants to determine if a supernatural explanation is correct, it must first eliminate all possible natural explanations, in the same way that I must first eliminate all known fruits to determine if my friend ate a sproogle.

Science therefore has no choice but to consider as many natural explanations as possible before settling on a supernatural conclusion. It’s the only option.

And it’s a laborious option. Given that we have gaps in our understanding of the natural world, there could be feasible natural explanations we aren’t even aware of yet. An exhaustive elimination of all natural explanations is therefore practically impossible, and a supernatural explanation will always remain elusive.

Of course, theists could make the entire enterprise a whole lot easier if they discovered some unique identifiers of the supernatural – properties that were not shared by natural phenomena. Imagine if my friend told me that the sproogle she ate was luminous purple with a smattering of tiny orange stars, and two long, thin grey leaves emanating from each end. On the basis of this information, it would be reasonable to conclude, with little further investigation, that she had eaten a strange new fruit that could not be confused with known varieties.

But no attributes unique to the supernatural have been proffered. Not even the “omni” properties often attributed to God (omniscience, omnipotence, etc.) are necessarily supernatural. It is entirely feasible to imagine a purely natural being with these properties – they are simply natural properties writ large. Even properties like those of pantheistic conceptions of God (the all pervasive presence) cannot escape from a natural interpretation. After all, gravitational fields also pervade all space, and they are perfectly natural.

In the end, I don’t think this problem is solvable. We are stuck inside nature. We have no possible way of observing or understanding anything else. We’re built by nature, for nature. Therefore, any property we think of assigning to supernatural objects will always be borrowed from nature. We can’t help it.

What this means for the theist, like it or not, is that the supernatural inevitably boils down to a vanishing act. “Supernatural” refers to objects that are hidden from investigation. They are indistinguishable from imagined objects but, at the mere insistence of their supporters, remain card-carrying members of “things that exist”.

Things that exist, but cannot be seen, heard, or detected in any way. It’s a pretty good gig, if you can get it.


Using intuition to assess moral systems

May 10, 2013

If you want to poke holes in someone’s ethical system, you can come up with a bizarre, highly implausible scenario and show that the ethical system recommends an apparently perverse or reprehensible response.

I’ve seen this quite a lot in philosophical and religious discussions. Indeed, I’m pretty sure I’ve used it myself. (This despite the fact that, at the end of the day, I don’t think we should rely too heavily on our moral intuitions, since there is no guarantee they are optimal on any particular metric).

But, assuming that moral intuition is useful to some extent, I’d like to propose an alternative to the one-off implausible scenario test.

For starters, implausibility should be explicitly taken into account. It seems to me that a moral system has much bigger problems if it produces unreasonable solutions to common, everyday problems than to rare, highly contrived ones.

So let’s imagine applying more than one moral scenario to the ethical system under question. Let’s put a list of scenarios together, some of which are common (should I cut in line at the deli if I’m in a hurry?) and some of which are rare (if I’m forced to choose between killing one person or five, what should I do?).

For each scenario in the list, the ethical system will produce a prescribed course of action that either conforms well to our moral intuition (it “sounds right”) or does not (“sounds wrong”).

We therefore have two scales. We have the frequency of the moral scenario (how commonly it occurs) and the degree of conformity its ethical solution has to our moral intuition (how “right” it sounds).

How do we combine these scales to get a reasonable idea of the ethical system’s performance?

I’ll present one possible solution here, with the use of an example. Let’s say that our list of moral scenarios contains ten items ordered by frequency: The first scenario in the list is the most common, while the last item is the most rare. We then label them A through J. Here they are:

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(Exactly how frequent each scenario is, is not important, so I’ve not included labels on the vertical axis.)

Now let’s apply these scenarios to our (hypothetical) ethical system and see how intuitive the proposed solutions are:

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It turns out that, in this particular case, the intuitiveness of each moral scenario decreases from left to right, just like the frequency data: The most commonly occurring moral scenario (Scenario A) also produces the most intuitive moral prescription, while the rarest scenario (J) produces the least intuitive moral prescription.

This sounds like the sort of situation we could live with because, at the very least, an ethical system should get the most commonly occurring moral problems “right”. And if it gets some problems “wrong”, it’s better that these be very rare.

We can get a more specific measure of this combination of frequency and intuitiveness by multiplying the above two graphs together. Let’s multiply scenario A’s frequency by its intuitiveness. This will give me a “reasonableness” score for scenario A. Then I’ll do the same for scenarios B, C, D, etc. Here’s what I get:

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Moral scenario A is the most “reasonable” because not only does it produce an intuitive result when applied to our ethical system, but it’s also quite a common scenario, which means that a lot of intuitive moral decision-making will be going on.

Scenario J, on the other hand, gets a “reasonableness” score of almost zero because it’s associated with a counter-intuitive moral solution. It’s impact is also reduced thanks to its rarity.

One way to make a single numerical assessment of these findings is to add up the reasonableness scores of all ten scenarios. With the numbers I’ve used to generate the above graphs, I get a total of 220 points.

Now let’s see what would happen if the trend of intuitiveness was reversed:

intuitiveness-bad

Now our ethical system is giving us a highly counter-intuitive solution to Scenario A. This is unfortunate, because Scenario A is still the most commonly occurring one. This means that a whole lot of counter-intuitive moral decision-making will be going on.

At the other end of our list, Scenario J produces a nice intuitive result. But this is not much consolation because Scenario J is so rare.

Once again, let’s multiple the frequency of each scenario by its intuitiveness. Here’s what we get:

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Overall, the reasonableness scores are lower than before. Scenario A now has a low score because it has a counter-intuitive moral solution. Scenario J (like before) has a low score because it’s rare. When I add up the reasonableness scores, I get a mere 32 points.

Ideally, we want an ethical system that will produce an intuitive solution for all scenarios, regardless of how common they are. If this were the case – if scenarios A through J all had the maximum intuitiveness value seen on the blue graphs above – then the resulting reasonableness scores would add up to a whopping 1100 points.

It’s fun playing with these numbers, but the central message is that there are more sophisticated ways of assessing a moral system than simply throwing one highly implausible scenario at it, and judging it on the basis of that scenario alone.

Besides, no ethical system I’ve come across can escape at least one or two counter-intuitive results. Unless we’re prepared to throw all ethical systems out, we need to find a better way of assessing them.


When style becomes substance

May 7, 2013

Surely God MUST exist in a place like this?

Science is very hands-on. It’s all about experimenting, observing, manipulating. You can see science happen.

The same applies to sports. And the arts. And almost every other human endeavor. The core ideas and beliefs refer to real, physical things.

Religion is a little different. It’s core ideas and beliefs refer to an entity that cannot be seen. Its narratives involve events that occurred hundreds or thousands of years ago, and whose participants are therefore long gone.

This presents a bit of a problem. How do you make a system of beliefs seem real and relevant if its main players can’t be detected and its founding events are long over?

The solution, I think, is to construct rich physical manifestations of those beliefs. Build elaborate and beautiful cathedrals, temples, mosques, synagogues, and megachurches. Put your leaders in elaborate costumes. Compose music, create art. Build a Wailing Wall or a Kaaba. Devise traditions like the Eucharist.

It’s good to have your beliefs confirmed by the senses, to be shown how concrete they are.

So perhaps religious paraphernalia are more than expressions of religious worship. Perhaps they are also vitally necessary components of a stable religious faith. Without them, the core beliefs become naked ideas – less grand, less compelling, and less relevant without their physical expression.

I would even venture to say that the physical manifestations of religion can serve to misdirect. What the cathedral visitor interprets as a sense of God’s presence may simply be his feeling of awe at the magnificent architecture. The sense of reverence he feels during the Eucharist may be due, in large part, to the ornate design of the altar area and the elaborate robes and hats worn by the priests. The religious fervor felt during the singing of a favorite hymn may simply be due to the believer’s innate enjoyment of music. And so on.

Personally speaking, the bottom line of all this is very positive: Even if the beliefs underlying religion may not be accurate, they have inspired people to produce all manner of wonderful art and architecture. And we can all enjoy it.