Hands-On Spherical Trigonometry

Spherical Trigonometry is used as a gateway to abstract mathematics because it can be taught as the study of the  geometry of  an intrinsically curved surface, without reference to its embedding in a “flat” 3-d space, as we experience it. So this is the preferred and usual method of teaching.

Still, I found myself a bit unsatisfied when I recently looked up, for study, the Spherical Law of Sines, which stands in direct analogy to the Law of Sines of plane trigonometry, with one important variation. That is, the sides, a and b, as referenced in the latter, become sin a and sin b in the former … very mysterious!

I found that the proofs tended to be algebraic in nature, depending on various “trigonometric identities” and I wondered if I couldn’t come up with something a little more “concrete” using constructive methods of traditional spatial ( Euclidian ) geometry.

So, doodling around, I found that I had great difficulty visualising 3-d constructions by reliance on 2-d diagrams, i.e. drawn on paper.

So I decided, well, I’ll make a model out of wood.

Of course, that requires a little doing. The obvious method is to cut out a sector of a sphere along central planes, but that isn’t exactly easy, and it wastes most of the sphere ( even if you can make several this way from one sphere. )

Also, wooden craft or carving balls are expensive, and usually rather small. So I decided to cut one from a block. I could buy a bag of 8 or 10 suitable blocks for a few dollars, and I could use two of the flat sides for the plane surfaces of the defining central planes. This would give me a nice RIGHT spherical triangle, while minimizing the carving effort.

So, here is the completed result, along with another block showing the preliminary mark-up:

BTW, forming the curved surface wasn’t as laborious as I thought it would be. I could get reasonably close by cutting easily defined tangent planes to the surface, and then sandpaper made short work. Of course, it’s not perfect!

Napier’s Rules and the Spherical Law of Sines

Well, I had already noticed that the Spherical Law of Sines could be reduced to one of Napier’s Rules, in this case (R2) as named in the Wikipedia article on Spherical Trigonometry.

(R2)      sin a    =   sin A   sin c

In the image below, angle A is at the base of the spherical triangle at the top of the photo, and measures the dihedral angle at bottom right. So it’s a simple application of “opposite over hypotenuse” to see that   sin A = sin a / sin c , which is (R2) above.

Note also that sin a and sin c are defined by the central angles, a and c, which measure the sides, a and c,  of the spherical triangle.

To get the Spherical Law of Sines all that is required is to place next to it another right spherical triangle sharing side a, along with the perpendicular plane face subtending it, to form a “general” triangle.

The shared altitude, sin a, then plays the same role in the proof as the shared altitude used in the proof of the Plane Law of Sines.

… and that’s all I have to say about that.

The Hot-wire Foam cutter and the Spherical jig

Now for the really fun part:

This is a hot-wire foam cutter equipped with a “spherical jig” of my own devising. As you can see it holds a 3″ foam sphere to be cut in half ( as adjusted . )  By cutting the same sphere in half  “three  ways”, I can form a spherical triangle along with its subtending solid central angle, just like the carved wooden model above.

Of course, I automatically get 8 such triangles. In the general case these will be 4 distinct triangles, along with their mirror inverses.

A Spherical Zoo

Here’s a “special” case of a such a production, contrived to produce a “large” equilateral triangle ( of course two of them ) and whatever else “falls out” :

The “large” equilateral triangles, inverses of each other, but in this case identical, are in the center.

Then six identical isosceles triangles are arranged on the sides. These form an equatorial ring between the two polar equilateral caps.

I first thought that this accounted for all possible isoceles triangles, but these are just a subset. They are the isoceles triangles which have the length of each ( equal )  side as the supplement of the base. That is, the base plus one side equals a 180 degree arc.

Just part of the arrangement that I think of as the Spherical Zoo.

Some “Nuts and Bolts”

I had been pondering how to organize this Zoo, and I was thinking in terms of the edges of the triangles of a partitioned sphere. This seems natural, since a “small” triangle can have arbitrarily short edges, just like plane triangles.

However, the triangles of a partition can also be specified by their ( corner ) angles, and this has the advantage that we can use the “spherical excess” rule, which gives the area of a spherical triangle in terms of the “excess” above 180 degrees, or “pi”. In fact, the area, in units of R squared, regarded as “Unity” for a given sphere, is exactly this value, expressed in radians.

So, using this scheme, I came up with a simple “map” of all the possible divisions of the sphere into triangles, in terms of A, B, and C, the angles of any one of the 8 triangles in such a division.

Well, with regard to units, we can have our cake and eat it too by specifying these angles in terms of pi/180 radians, which corresponds to “degrees”, but reminds us that the area is measured in “steradians”, of which there are 4 pi to be accounted for among the 8 triangles of a division, using the spherical excess rule. Note that this is nothing to do with “square degrees” … which we won’t touch.

In fact, since our “degree” is pi/180 radians, the 4 pi steradians  of the sphere are measured by 720 degrees, or 720 X pi/180. I explain all this so that we can do examples with angles, and areas, expressed in small integers. That this is possible is remarkable in itself!

… so then my “map” :

… The angles A, B, and C , with values 0 thru 180, as indicated, will specify a valid spherical triangle whenever they are the coordinates of a point lying inside the tetrahedron drawn in red.

Very simple!

… to be continued



The Point of Collapse

There is an extant video, from a traffic camera, I believe, that shows the moment of the onset of the the FIU bridge collapse of March 15, 2018.

I found a video showing the video on a screen with some people watching, and it is in a loop, which makes it easier to examine. It shows not only the moment, but the location of the onset, consistently with reports that the collapse “began at one end.”

Here is a two-frame gif from the video, showing the moments just before and after the onset:

I think this view is quite revealing, as it shows that the collapse was initiated by a break in the roof, meaning that it was bearing a critical load, or in crude terms, “It was the only thing holding up the bridge.”

This all makes sense from a simple point of view, considering the departure of the installation procedure from the initial design.

Here is a photograph of the bridge just before the installation, when it was being maneuvered into place. It is a very clear view, and answers many questions I had about the initial design … i.e. Had it been altered or even abandoned?

A comparison with an available diagram of the “cable-stay” design shows that all the design elements on the span itself are in place, and the installation allows for  the addition of the tower, and a second shorter span, as shown in the design:

A careful comparison of these shows that the configuration of the struts between the deck and the roof matches up very well, within the limits allowed by the images, of course.

Also, note the very prominent attachment points along the center of the roof. These “points” of attachment are very large and complex in appearance. I read that these attachment elements are not cables, per se, but “pipes”, but as per the design, they would have functioned as cables, that is, provided support under tension.

Compression vs. Tension

Here is a simple schematic analysis of an idealized truss, showing that in this configuration, the top and end pieces are under compression, and the bottom deck is under tension:

The red arrows show the forces on the nodes ( red dots ) due to the struts and deck, as per the requirement that the sum of the forces at each node must be zero.

So, the strut along the top is pushing outwards, and by reaction, it must be under compression. Conversely for the deck.

This logic applies exactly to the FIU bridge, and to me it accounts very well for the chain of events.

Well, I don’t mean to get into a particular analysis. I’ll just say that in my mind, “The Emperor has no clothes.”


In search of “Bloodstone Hill”

The Mars Curiosity Mission Update for Sols 1954-1956 includes the stated goal of … an RMI mosaic of “Bloodstone Hill.”

RMI stands for “remote micro-imager” and refers to the use of the ChemCam camera for simple image acquisition, without sectroscopy. It has a long focal length, so it makes a fair “spot zoom”.

Then the Mission Update for Sols 1955-1957 is titled,

Sol 1957-1958: Onward towards gray patch

and features this image: … so I thought for a long time that this was “gray patch”. I don’t want to put anything on the Curiosity Team. The principal goal is not to explain everything to ME. If I have to work a little bit for it, so be it.

The text does say,  inter alia, “… We’ll also take a Mastcam mosaic of “Bloodstone Hill,” another target from the weekend plan that warranted further investigation – this area is featured in the black and white RMI image above. ” OK.

I made the RMI mosaic from sol 1955 ( including the above image, ) and here it is:


I thought the circular framing would interfere with the mosaic, but the long focal length and narrow FOV kept the stitching very nice.

So where is it? What are we looking at? For the promised Mast Cam mosaic, I only found two images in the very annoying rastered grey rendition. Here they are side by side with a “tetch” of Gaussian blur, and some contrast enhancement:

OK! Great context! … except I still couldn’t place this in a broader scene. Note there is no skyline! This makes it difficult.

Well, I made another mosaic from the Right Navcam 

Do you see it? I think it’s very hard to spot.  Look at the top center of the rightmost pane, and the elongated whitish feature is the feature at the center of the Mast Cam mosaic above. If you click the image and enlarge it, you will be convinced.



Mars rock circle in tabloids

A Mars Curiosity image of a “rock circle” has appeared in the British tabloids in the last few days,  touted by “alien hunters” to be an artifact of some kind .

The NY Post gave secondary coverage and carefully stated that it was “recently found” among the images by said “hunters.”

From the setting, I could see that it did not match the recent terrain being covered ( we’re up to sol 1734, ) and I did spend a little time looking back over the route to find a likely spot. I was pretty lucky, although I will credit my intuition, as I found it as a Mastcam image from sol 528

In fact, this image was evidently not a chance acquisition, since they immediately acquired this followup image, also on sol 528

Here we get a more detailed view, and it looks somewhat more “natural”, I think, with noticeable irregularities. Then, on sol 529, they actually “moved in for a look” and acquired this image

Here you can start to get an idea of its formation. It seems to be sort of a disk with the center scoured low, like a lozenge. Of course, this is just one small feature! From what I understand, the landscape as we see it is formed almost exclusively by very slow erosion from the dust storms.

BTW, the images were acquired on 2014-01-30 13:53:37 UTC,  2014-01-30 13:54:26 UTC, and  2014-01-31 15:46:03 UTC, over three years ago.

Also, here is the section of the path showing the location

The “dog-doo”

This is the name which occurred to me when I saw this feature, and true to my habit, I’m sticking with it. Here it is in a sol 1700 Mastcam shot, which I already included in a panorama linked in “Ozymandias”, below.

It does seem to be a distinctive feature. The darker blobby looking rock on the top doesn’t seem to fit with the more predominant thinly layered rock, which are easily seen as eroded remnants of dust-formed strata.

Well, at this point ( sol 1700 ) I had high hopes that Curiosity would make a bee-line for it and do a major examination, but these met with frustration.

Coincidentally, I read at about this time, an ONION article on the “Morbid Curiosity Rover”. Of course this is a pun, and nothing to do with “morbid curiosity”, but a facetious evaluation of  Curiosity’s “morbid” temperament, as note that the official MSL accounts allow for Curiosity to speak in the first person. The Onion found that Curiosity was uncooperative, and even unresponsive, and given to obsessive lingering and drilling activities.

I thought this to be a wry comment on the incompatibility of the inscrutable science campaign with the preferences of those of us more concerned with sight-seeing.

So these thoughts came to the fore as Curiosity roved closer and closer to the “dog-doo”, but seemed to deliberately eschew imagery of it. I thought, and still think, that it was partly because it was up-slope from Curiosity, and the imagery had a tendency to pan on a level plane, so that we were treated to extensive views of “where it had been” but the looming features ahead, including the “dog-doo”, were chopped off at the ankles. ( If you don’t believe me, check it out! )

Well, there were a few Navcam shots, but this feature was obviously not a target, and at closest approach, Curiosity veered to the east, and focused on other things, this to my ultimate frustration, I thought.

But then, there was a late adder to the sol 1726 SUBFRAME products from the Mastcam. These are monochromatic, and have a funky grid superimposed, but OTOH, are highly detailed, and feature a magnificent view of the “dog-doo” at the extreme right.

Accordingly, I offer this truncated version of what is approximately a 180 degree panorama, consisting of  2 of the 20 component images:

In Paint.net, I used just a “tetch” of Gaussian blur ( a minimal setting of 1 ) to alleviate the crosshatching of the “grid”, and gave it a sepia tint, which I modified with a slight reddening of the hue.

“You’re in the shop!”

Where Is Curiosity?

The MSL website helpfully provides maps to show the progress of Curiosity, but lately, circumstances have led to some frustration.

Curiosity has reached the southern limit of the displayed range, and it is heading south, so it is hard to see what is in store. In addition, the large scale inset is placed so that it obscures significant adjacent topography … as shown here …

Never fear! I have the Hiview software and image files, which I downloaded some time ago, and these include the imagery that is the basis for these maps. So here I display a  small scale ( … N.B. a small scale displays a large area … )  image with the approximate boundary of the sol 1707 inset map marked.

You’ll note that the “foothills” of Mt. Sharp are well distant to the south. The features there were evident in the panoramic views from the landing site, as see my  Sol3 panorama  post from Dec 5,2012. ( This was my first Mars Curiosity post. )

Curiosity is more than halfway there from the landing site, but I’m not sure what the schedule is, or the planned path.

“Hills peep o’er hills, and alps on alps arise!”



The latest vistas from Curiosity put me in mind of Shelley’s sonnet.

“Look on these works, ye mighty, and despair.”

Of course, these are not the ruins of some civilization, but purely inanimate landscape. Yet, everything takes a form through a very leisurely evolution, compared to earth, and there it all sits, like some incredible junkyard.

Here is a simple panorama from the mastcam images of sol 1698.

… click to enlarge!

… Here’s the same scene on sol 1700, except for a forward displacement of Curiosity’s POV.

The very prominent knoll in the left foreground on sol 1700 can easily be located in the left middle ground on sol 1698.

The prominent features from the center to the lower right in the sol 1698 view are no longer in view on sol 1700.

Here’s a sol 1705 Navcam view which can be matched up with the sol 1700 Mastcam view, above.

Curiosity has advanced to a position in the sol 1700 view, just in front of the 3 large “rocks” near the center of that view. This formation appears in the sol 1705 view on the left center. It can be identified by the form of the sand, or dust, drifted against them.



The Martian Tree Stump

The “tree stump” on Mars was in the news this month, and it is indeed an interesting formation, of which there are countless examples in the Curiosity image gallery.  I don’t mind such fanciful descriptions, since it does bring attention to these images, and in fact my attention has flagged badly in the last few years, after having followed them assiduously in the early days.

I still have my PTGui panorama maker, and my “curiosity”, so I followed up on this one.

Since “You’re in the shop” I’ll share a minor glitch which vexed me greatly just yesterday, but today’s a new day and all is well.

I composed the following panorama which contains the frame from the Mast Camera that was publicized as showing a “tree stump”

Of course, it’s missing a piece! This does show that the panorama is composed of 8 separate ( overlapping ) images. But it is certainly a detraction. I spent maybe an hour trying to figure out what was wrong before I went ahead and let PTGui figure it out.

If you “click to enlarge”, then click the “+” to magnify, you will see the whole thing in the resolution of the “stump” panel that was in the news.

Well, then, today I found the following in the Raw image gallery:

So there we have it. ( This is an excerpt with the header note pasted on. ) So of course it was a simple matter to patch in and form the full panorama:

Much better! But what about this “stump” ? I looked for other images, and found the following from sol 1648, whereas this was sol 1647:

This is a panorama of 5 images from the mast cam, in color this time, as they usually are. Curiosity had moved “overnight” and you can see the large “fin” , which is to the right of the “stump” in both images, but this time the view of the “stump” shows lateral extension, indicating that the sol 1647 image was “edge on”.

Never fear, at center bottom we have a cylindrical post with some kind of runic marking! The poetry of Mars is never dead.

40 Eridani

40 Eridani is the designation of “a star” ( actually a 3 star system ) in the constellation Eridanus. One of these 3 stars is a White Dwarf, and famously the most easily accessible of this type to amateur observation.

All this became known to me in a recent conversation, and I noted, via my Starry Night software, that the current viewing window was drawing to a close this month of March, so I determined to get a photograph of it, as I have found that photography shows a lot that is undetectable at the eyepiece.

I use prime focus projection using my Nikon D5100 camera body mounted with an adapter on my Orion StarMax 127 mm, with a 1540 mm focal length. In this configuration it’s essentially a very long focal length camera lens, with an upright image through the view finder.

I have a “clocked” mount and I take a maximum 30″ exposure using a remote shutter release. This is pretty low end stuff compared to the deep sky shots you see with much more sophisticated amateur equipment and processing.

I can get good results, though, and in this case, the picture came out well.

40 Eridani

The bright yellowish star at lower right with the smaller white companion is 40 Eridani and the companion is the white dwarf. If you “click to enlarge”, you can easily see the small reddish close companion of the white dwarf, which I did not really expect to resolve, so of course I’m quite pleased with the picture.

Here’s an enlarged excerpt, just to celebrate.

40 Eridani close-up

There is a picture of 40 Eridani in Volume Two of the classic Burnham’s Celestial Handbook, called there Omicron 2, as “omicron” was the sequential designation, after alpha, beta, etc. of a pair of stars in the constellation Eridanus ( The River. ) Omicron 1 is not physically associated with the Omicron 2 system, aka 40 Eridani.

Just to complete the confusion, my Starry Night software shows Omicron 1 and Omicron 2 as “Beid” and “Keid”, which are “traditional star names”, although I read that the tradition in this case was officially adopted in 2016 ! Of course, Burnham’s does not use these names.

Anyway, the picture there is from an exposure at Lowell Observatory, of Pluto fame, taken around the same time as that discovery.

I took my photo and “pushed” the brightness and contrast to the maximum to bring out the background stars, then did a match-up with the photo illustration from Burnham’s. Here is a “blink comparison” using the Free Online Animated Gif Maker:

… click to run the animation.

It’s a little confusing at first glance, but hold your gaze on the stars above 40 Eridani ( in this orientation ) and you will see that they match almost perfectly. However, 40 Eridani “jumps up and down” .

I wasn’t looking for this and it threw me for a loop, but in reality, this is a record of the actual “proper motion” of this triple star system, relative to our solar system. In fact Burnham’s describes this motion and has a pair of illustrations showing the position circa 1934 and 1965.

I pasted these into one and added my observation ( in blue ) …

proper motion of Eridani 40

The blue imagery is a replication to scale taken from my high contrast image used in the blink comparison.

More on Proper Motion

The term “proper motion” in my experience, refers to the apparent angular movement of an object against an ideal inertial reference system centered on the sun. However, one might think that it ought to include the “third component” of the relative velocity, i.e. what is termed “radial velocity”.

Nevertheless, I think historical usage has treated these separately. This caused me some confusion in reference to a statement in Burnham’s that the “true space velocity is about 62 miles per second.”

I took this at face value at first, and assumed that the value must include the component of “radial velocity”, later mentioned. However, I don’t believe this is the case. The discussion was of “proper motion” as traditionally defined, so this “true velocity” was simply a conversion from angular to linear velocity along the apparent path in the sky.

The determination of radial velocity is a whole different question, since it involves physical analysis of the received light, and not any apparent motion of the object.

The question is mooted once we have the “proper motion” expressed in arcseconds per year, along with the “radial velocity” expressed in km/sec, or any such units for that matter.

In fact we may note that 186,000 = 62 X 3000, and I think this indicates that the value of 62 miles per second was being given as a “round number” for the transverse linear speed of 40 Eridani … i.e. c/3000 .

… but all this is a distraction! I just want to get an idea of the past and future apparent motion of 40 Eridani, and this can be based on a given specification of the proper ( angular ) motion, and the radial velocity.

With the understanding that the linear approximation won’t hold up forever, we see that we have a simple freshman physics problem of relative velocity between two objects, in this case the sun and 40 Eridani.

The trajectory of 40E wrt the sun is then a “great circle” in the sky, and wrt this path it is determined by the general equation for any such “fly by”, within a scale factor only for time.

The “zero point” of this trajectory is the point of closest approach, which provides the distance scale, and the time scale of the angular displacement is determined by this distance divided by the speed of the object.

That only leaves the determination of the point of closest approach of 40E to the sun, expressed as an angular displacement along its apparent path from its current position. I hope this diagram explains everything!

Here is my estimation of the apparent path in the sky of 40 Eridani over the next 19000 yrs, in accordance with the diagram above, and overlayed on a view from my Starry Night software:

The Incredible Cuboctahedron

On what basis I don’t recall, I was led into the contemplation of the generalized process of “cuboctization” which produces the cuboctahedron from the octahedron. This process is namely the connection of the midpoints of “neighboring” edges, to produce a new figure with vertices at these midpoints, and with the constructed edges.

It’s important to note that this figure, being composed of vertices and edges, is not necessarily a geometric solid, since the new vertices may not lie in a plane, to form a face. Nevertheless, we may proceed.

Of course the founding idea is the production from the octahedron


of the cuboctahedron, which may be regarded as a truncation of the octahedron, but in this case can be regarded as being produced in the manner described.
So we may proceed in this manner, and to arrive at the nub of the discussion in short order, we may present “cubocta5”, the result of applying the algorithm of edge midpoint connection four more times.
But here we have run aground. You can see that the four expected quadrilateral faces next to the green quadrilateral face in the center have been “bent”. Their defining vertices did not lie in a plane. We may remind ourselves that our procedure did not have any reference to faces at all, but only vertices and edges, so that we were only lucky up to this point to produce flat, or planar, four sided faces.

… but it’s close! So, what to do? Can we “flatten” these faces somehow, and produce a true cubocta5 polyhedron?

Never fear! A way was found. By retreating to the cubocta2, and adjusting the position of the vertices, we produced this version

All on a hunch, if you will. But that’s not half the story! At any rate, three iterations of our edge production algorithm produce the desired result. A truly remarkable figure with 192 vertices, 384 edges, and 194 faces. 8 of these faces are small triangles at the corners of a cube, and the remainder are quadrilaterals of varying proportions, but all perfectly flat.

If this figure “exists in the literature”, we would like to know, but at any rate, this version of it is original with us.

Some Numerology

It’s additionally remarkable that the entire figure can be specified in terms of small integers for the coordinates of the vertices, and there are only six distinct vertices required to produce the entire figure under octahedral symmetry. Here they are with the number of replications:

(24) 10 6 0
(24) 12 2 0
(48) 11 4 2
(24) 8 8 2
(48) 9 6 4
(24) 7 7 6

The 48 replications are the 6 permutations of the values times the 8 versions of each obtained by taking +/- each value, in combination.

since +/- 0 = 0, there are only 4 sign combinations for 10 6 0 and 12 2 0, and in the case of 8 8 2 and 7 7 6 there are only 3 distinct permutations, times the 8 sign combinations. Of course these add to 192.