Why Mirrored Images Feel Like Bad Reflections

Many artists and observant folks are already well aware that creating an image with a reflection is an art and that it can be very jarring when one is poorly crafted.  This is a brief post where I explore specifically what optically is wrong when a reflection is created by mirroring an image.

First we have to understand how real reflections look.  Let’s say we have a situation where an observer sees an object that is being reflected by the ground (it’s made of glass, there’s a layer of water, whatever) in Fig A below.

Fig A:

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In this case, the observer sees the physical object at one angle and the reflection at a slightly different angle.  In Fig B, I’ve updated Fig A to show a 90 degree viewing angle to the physical object and an approximately 45 degree viewing angle to the image.  Depending on the shape of the object, this may allow the observer to see more of the “underneath” surfaces in the reflection.

Fig B:

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Now, if I am in the process of creating a fake reflection where I’ve simply taken an image of an object and flipped the pixels, the flipped image because it is 2D is at the same viewing angle to the observer as the unflipped image.  Since in a real optical setting, the image should appear at a steeper angle, now it appears to be leaned in towards the observer (Fig C).

Fig C:

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And an even more fascinating realization is that for such a case to even be physically possible, the reflective surface would have to be bifurcating the angle between the image and its reflection and be perfectly in line with the observer’s eye (Fig D).  The fact that this is a physically impossible situation means that seeing such a reflection should feel completely alien and unsettling, especially in a doctored or Photoshopped or drawn image.

Fig D:

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Now of course this is an extreme simplification of the situation.  Clever artists know that distance matters as well.  Shrinking the reflection to accommodate for it’s perceived increased distance helps the reflection feel like it is in the right place.  In addition, if the reflective surface has texture and is not completely clear, the observer’s mind will fill in the gaps and think that the image should make sense.  Finally, as the distance between the observer and the object increases, the delta between the viewing angles of the object and the reflection becomes a negligible difference anyway.

Nonetheless, I found it to be an interesting thought exercise to explore the physical reality (or non-reality) that may sometimes underlie a gut feeling of a mirrored image being “wrong.”

Clock Hand Motion

Saw an analog clock today at a school, and it surprised me.  The minute hand moved every 15 seconds, and moved a quarter of the way to the next minute each of those times.

Now, the analog clock I have at home is continuously geared.  The minute hand is always moving, just at 1/60th the speed of the second hand [illustrated below]

I couldn’t help but to wonder exactly how the internal gearing of that clock in the school worked.  While we live in an age where a simple internet search and digging through some manuals and videos could explain this very quickly, it seemed remiss to pass on the chance to see what my own knowledge can afford me in an explanation.

My instinct in reverse-engineering something like this would be to think of the states that each hand can be.  The second hand has 60 states, so give it a gear with 60 teeth.  The minute hand moves every quarter minute and therefore must have 240 states, so give it a gear with 240 teeth.  Now how to convert between the two gears?  The seconds gear changes the state of the minutes gear only every 15 times it moves, so perhaps another gear on the same axle but with four teeth, corresponding to every 15 on the seconds gear [illustrated below].  The discontinuities between the four teeth would create the lack of continuous motion in between.

However, there was still one last puzzle.  The minute hand and second hand moved exactly in sync during that one tick on the clock I saw in the school, which implied to me that the teeth must be the same radial size.  Moreover, now that I knew the 1:15 relationship, I could imagine an easier gear pairing to make: Rather than a gear with 60 tiny teeth and another with only 4 teeth the same tiny size, how about a gear with 15 well-defined teeth and another gear with a single 1/15 tooth [illustrated below].  These can then be linked up to the appropriate 60-tooth and 240-tooth gears.  Moreover, in this arrangement, instead of power traveling from the seconds hand then to the minutes hand and having to overcome torque of each in the chain, the power that drives the clock comes from a single axle that transfers power to both the seconds hand and the minutes hand at the right ratios.

String Held Up in Tension

Saw this image online today, and was pretty fascinated by it.

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I decided to do a quick free body diagram to get a better sense of how exactly tension was distributed across the string.

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I color-coded the left-side against the right-side so that you can see the forces balanced by being from the same “spring.” As with any freebody diagram, taking any specific section of the string and summing up the forces coming into it from either direction gives us a sense of the net forces on it.

We see here that sections 1,3,5,7,9 are equivalent and 2,4,6,8 are equivalent.  The odd-numbered sections are in tension with force F, while the shorter even-numbered sections are in tension with force 2F.

This of course is a simplified view of the system because it does not take into account any moments of rotation generated by the masses hanging on the string.

Geometry 6/7/2016

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Stared at some ceiling tiles today, which was a bad call for my productivity because I immediately wondered how the contractors cut and installed the cross pieces, and whether or not their was a standardizable piece that allowed optimal tiling.

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So I started with some exploratory thought experiments.  Pieces that have one intersection and one edge can perfectly cover one square (in pink below) so let’s go from there.  Unfortunately as we add to it (green pieces), we create spaces that standard pieces don’t fit into.  A failure.

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What about a longer pieces that spans an intersection and two edges?  This creates a very familiar weave-like pattern that definitely tessellates (below).

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However, we aren’t close to a generalize-able rule.  Indeed, one could imagine many other strange angled or partial length shapes that I have not tried yet, and I wondered if there were at least some property that influenced their tessellate-ability.

I began to think that it has to do with the number of intersections and edges that a piece crossed since the relationship between those helps to define how much space the cross pieces need to cover.

An inductive analysis can be used: a single tile has 4 corners and 4 edges, then four tiles have 9 corners and 12 edges, and then nine tiles have 16 corners and 24 edges (image below).  This is solvable by looking for the pattern (n^2 corners to 2n^2 edges), but the answer is easier to come to if we simply look at the general situation of each corner and each edge instead.

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In a fully-tiled situation, a corner sits at the intersection of 4 tiles while an edge sits at the intersection of 2 tiles.  There we have it then!  For an infinitely tiled surface, you will need a corner for every 4 tiles, but an edge every 2 – there will then be a ratio of edges to intersections/corners of 2:1.  Thus, this must be a design constraint of the cross-pieces (which agrees with why my second thought experiment worked unlike the first).

While this does not give us a way to define all possible cross piece shapes (one could imagine a complicated cross-piece shape with 74 edges and 37 intersections but does not tessellate), it does allow us to check if a cross piece shape could potentially tessellate.

Avatar: The Last Airbender Power Levels

When physics and “superpowers” reconcile (or not), it’s all in great fun.

A show that I like for various reasons, but that has also shown a variety of great feats is “Avatar: The Last Airbender” and the subsequent “Avatar: The Legend of Korra.”

One of the most extreme examples of bending happens when Ghazan melts the Northern Air Temple Mountain:

Given: an approximate energy of 900kJ/kg to melt rock[1] and an average density of 2.7g/cm^3 for rock [2].

[1] ( http://home.earthlink.net/~jimlux/lava.htm )

[2] ( http://en.m.wikipedia.org/wiki/Continental_crust )

Approximate: The temple has a footprint of about 500m x 500m = 250,000m^2.  The mountain has rather sheer faces so let’s give it a cylindrical profile and say about 400m vertically is melted.  That’s 100,000,000m^3 of rock.

Based on our givens, that is 270,000,000,000kg of rock or 243,000,000,000,000kJ of energy, 243 petaJoules.

Gazan outputs the energy equivalent of the Tsar Bomba, the largest nuclear device ever tested.

Or enough to power all the electricity in New York City for 6,136 days, i.e. 17 years.

If the energy all came from within him through conventional biology, he burned 5.8×10^13 kilocalories (or just capital “C” Calories – of which we are recommended to eat 2000 a day by food packaging), that’s 29,000,000,000 days or 79 million years of meals.  If he was converting mass directly to energy, then via Einstein, he lost 2.7kg of pure mass (not bad!).

One could argue that characters are tapping into energy sources external to themselves when bending, nevertheless, it is interesting to know just how much of it they must be accessing.

See this Google Doc for further calculations with Zaheer, Avatar Kyoshi, etc.

Google Doc with More Details

Geometry 12/2/2015

Here’s another puzzle that had me occupied all of my bike ride home yesterday:

While stopped at an intersection, I noticed how the windows on the building across the street allowed me to see oncoming traffic from the side from farther away than I had turned my head.  And so I wondered, what relation and how much of a reduction in neck strain does such a situation promote?

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Clearly you need to turn your head only b degrees instead of a degrees, but that relationship can be anywhere from a negligible difference when the target is right against the mirror to a near complete reduction if the mirror was at infinity.  What’s the actual relationship though?

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The target is at some x and y position away from us the viewer, and in the case of a building where the reflection is always ideally positioned, it only matters the y’ distance of the building from the target (the image will be the same distance behind the mirror).

With all of three of these values, we can find the angle of the triangles formed in the diagram by arctan(y/x) and arctan((y+2y’)/x).

The angle I must turn to look at a car then is (90º – arctan(y/x)), while I only need to turn (90º – arctan((y+2y’)/x)).

Wolfram alpha can help us with the rest of the work here in visualizing this relationship, and we can choose to look at all three as variables or arbitrarily set certain one’s as constants.

For example, if we are interested in how the ratio of the reduction behaves as a function of the distance of the mirror, see the following graph.  The ratio of the image in the mirror angle to the object angle is y and the distance of the mirror from the object is x in this case.  (as expected, mirror farther away means you have to turn your head less)

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Or, if we are interested in how the ratio of the reduction behaves as a function of the x position of the object (the farther away it is, the closer the angle is between the mirror image and it):

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Or we can vary both the position of the object and the mirror at the same time (reflected in the top right quadrant of the below graph since both values can only be positive):

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ADDITIONAL:

By the way, if you were wondering about the ideal positioning of a mirror to see an object at your side, the relationship is governed by (y+y’)/(x-x’) = y’ / x’ where x and y are the position of the desired target object to your side and x’ and y’ are the position of the mirror relative to that object.

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x’ and y’ have an expected asymptotic relationship where they start right on top of the target to halfway between you and the target in x while infinitely far away in y. Per Wolfram Alpha below (setting y=1 and x=2).

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Paracord Shock Load

Here’s a question that I’ve wondered about:  What’s actually a safe load to put on paracord in case of it slipping and dynamic loading?  Paracord should be a shock cord since it was originally for parachutes, but it’s always rated by a static load weight.

Here’s the best I’ve gathered (internet, please correct me if any of this is wrong):

An approximate equation for shock load is:  shock load = weight * ((fall distance / elongation) + 1)

Properly rated paracord should elongate at minimum 30% (more is better so let’s use 30% as a worst case).

If are you dropping something with the cord, the fall distance is the cord length or 100%.  The equation reduces to weight * (1/0.3 + 1) or weight * (13/3).  A weight being dropped on paracord can generate a shock force of about 4.333 times it’s weight.

So going in reverse, if we take the static load of paracord as it’s net breaking strength, do not load it with more than the inverse of that factor in weight, 3/13 or 23% of it’s rated strength.