Monday, May 15, 2017

Professor Quibb's Picks – 2017

My personal prediction for the 2017 North Atlantic Hurricane season (written May 15, 2017) is as follows:

15 cyclones attaining tropical depression status*,
15 cyclones attaining tropical storm status*,
6 cyclones attaining hurricane status, and
3 cyclones attaining major hurricane status.
*Note: Tropical Storm Arlene formed on April 19, long before the official start of the season on June 1 and before I made these predictions.

This prediction calls for a nearly average Atlantic hurricane season, with predictions slightly exceeding historical averages in all categories.

In contrast to 2016, the conditions for the 2017 season are fairly common and uncertainty is relatively low. The first condition taken into account is the state of the El Niño Southern Oscillation Index (or ENSO index), a measure of sea surface temperature anomalies in the Pacific Ocean that has a tendency to affect Atlantic hurricane activity. After the index took a brief dip into negative territory this past winter, the index has returned to nearly zero, or "neutral." As shown in the figure below from the International Research Institute for Climate and Society, a modest increase is expected over the coming months.

As a result, the conditions prevailing for the hurricane season are likely to be neutral or weakly El Niño. Since El Niño tends to suppress Atlantic activity and cause cyclones to, on average, take more easterly tracks, this factor would suggest a quieter hurricane season.

Sea surface temperatures, meanwhile, are a bit above average across the Atlantic basin, but the anomalies are not as great in magnitude as they have been over the past few years. The warmest areas are currently the Caribbean and the tropical portion of the Atlantic farther east. Parts of the Gulf of Mexico, meanwhile, are slightly cooler than average. Further warming of the current higher-than-normal areas is likely over the next few months, so these might be conducive to cyclonogensis. The tropical Atlantic has also been quite moist, as has the Caribbean, supporting the development of hurricanes. The Gulf of Mexico, in contrast, has been persistently dry. Finally, with the developing El Niño, increasing wind shear is likely across the Atlantic, especially at higher latitudes and near the United States. Such shear is hostile to tropical systems, so I predict limited activity near the U.S. east coast, despite a pocket of warm water there.

My estimated risks for different parts of the Atlantic basin are as follows (with 1 indicating very low risk, 5 very high, and 3 average):

U.S. East Coast: 3
The presence of an El Niño would tend to reduce risk, as stated above. However, seasonal forecasts indicate that high temperatures will prevail near the coast for most of the summer, resulting in higher oceanic heat content. Look for quick forming and quick hitting systems - long-lived hurricanes are likely to miss the coast this year.

Yucatan Peninsula and Central America: 3
Signs in the adjacent Caribbean Sea point to elevated tropical activity this year: warm waters, moist air, and limited wind shear. However, steering ridges will have a difficult time setting up along the Caribbean Islands to the north, preventing developing storms from tracking due westward for the most part and instead allowing them to gain latitude. A combination of these two opposing factors leads to the "average" designation for this region.

Caribbean Islands: 4
Complementing the previous point, the Caribbean Islands will be in the more likely path of tropical cyclones. Coupled with the fact that the tropical Atlantic is warm, there is significant risk for landfalling tropical storms and hurricanes this year.

Gulf of Mexico: 1
Atmospheric conditions about the Gulf are already dry and strong upper-level winds are moving across the region. With an increasing ENSO index, this state of affairs is likely to continue indefinitely. Combined with the slightly cooler waters, these signs indicate a very low risk for the Gulf coast.

Overall, the 2017 season is expected to be near or just slightly above average, but with a lower than average risk to landmasses (most storms should curve out to sea). While the confidence in this forecast is somewhat higher than last year, everyone in hurricane-prone areas should still take due precautions as hurricane season approaches. Dangerous storms may still occur in quiet seasons. Sources:,

Saturday, May 13, 2017

Tropical Storm Arlene (2017)

Storm Active: April 19-21

During mid-April, a non-tropical low over the central Atlantic well east of Bermuda was producing a large area of tropical storm force winds and scattered thunderstorm activity. As the system drifted eastward, it more more organized, and began to show signs of subtropical development by April 18. Though convection remained mainly confined to the southeast quadrant by the next morning, the low had acquired enough organization to be classified Subtropical Depression One. At that time, the cyclone was moving north-northeast at a moderate clip as it interacted with an extratropical low.

Any cover the center of circulation had managed to develop that day was quickly stripped away by increasing wind shear by early on April 20. The system made a comeback later that morning, however, and in fact became more symmetric, resulting in its reclassification as a tropical depression. It turned toward west-northwest that afternoon and unexpectedly strengthened into Tropical Storm Arlene, only the second known tropical storm to form in April in the Atlantic. Further, its central pressure dropped to 993 mb, the lowest for a tropical system ever recorded in the month of April. Arlene's unusual run ended the next day as it became extratropical and was quickly absorbed by a larger system.

The above image shows Tropical Storm Arlene near its peak intensity over the open Atlantic.

Arlene did not approach any landmasses during its short lifetime. However, it was notable in that it was only the second Atlantic tropical storm known to form in April, after Ana in 2003.

Friday, May 12, 2017

Hurricane Names List – 2017

The name list for tropical cyclones forming the North Atlantic basin for the year 2017 is as follows:


This list is the same as used in the 2011 season, with the exception of Irma, which replaced the retired name Irene.

Sunday, April 16, 2017


OSIRIS-REx is a NASA sample return mission to the near-Earth asteroid 101955 Bennu. It aims to collect a sample from an asteroid whose composition could reveal a great deal about the beginning of the Solar System and the formation and evolution of the Earth. The first asteroid sample return mission was Hayabusa, developed by the Japanese Aerospace Exploration Agency (JAXA). This probe returned about 1,500 microscopic grains from the asteroid 25143 Itokawa. OSIRIS-REx, however, was designed to obtain at least 60 grams of material in the form of macroscopic samples. In addition, Bennu differs enormously from Itokawa in that it is carbonaceous while the latter is siliceous. Further, there is evidence that it is rich in organic and volatile compounds. Bennu is also of interest because its orbit takes it very close to Earth. It was measured to have a small cumulative probability of 0.037% of striking the Earth sometime in the 22nd century. This is due to the present uncertainty as to whether Bennu with pass through a gravitational "keyhole" in its 2135 flyby of Earth that would set it on a collision course. This mission will allow more precise predictions of its trajectory.

Bennu's orbit is slightly larger than Earth's but also more elliptical. As a result, it crosses inside Earth's orbit with every revolution.

The spacecrafts's name stands for Origins, Spectral Interpretation, Resource Identification, Security, Regolith Explorer. This unwieldy acronym contains the four main goals of the mission: to return a sample to Earth that will elucidate the Solar System's origins, to map the asteroid with spectroscopy to learn about its composition and formation, to investigate whether near-Earth asteroids such as Bennu could provide materials as resources for human development, and to discover what impact threat Bennu poses, if any. The word "regolith" describes the layer of loose material at the surface of an asteroid, from which OSIRIS-REx will obtain a sample.

On September 8, 2016, the OSIRIS-REx mission began with a launch from Cape Canaveral. After arriving at Bennu in 2018, it will map the surface to select a target for the 2019 sample acquisition. After leaving the asteroid in 2021, the spacecraft will then return the sample to Earth in 2023.


Sunday, March 26, 2017

More Evidence for Planet Nine

For the first post in this series, which explains the motivation for the Planet Nine hypothesis, click here.

The previous post touched on some ways in which the orbits of certain outer Solar System objects are similar. These may be quickly summarized in the following way: both the arguments and longitudes of the objects' perihelia are unusually clustered around certain values.

The above image shows numerous relevant parameters concerning the position of an orbit. In the case of orbits in the Solar System, the plane of reference is the plane of the Earth's orbit and the Sun, also known as the ecliptic. The reference direction often used for heliocentric objects is called the First Point of Aries, defined as the position of Earth's vernal equinox and so named for its location within the constellation Aries. The ones with which we are concerned here are the argument of periapsis ω (this is the general name for argument of perihelion to include non-heliocentric objects) and the longitude of the ascending node Ω. The sum of these two angles is called the longitude of perihelion because it measures the angle between the perihelion and the reference direction. In summary, the similarity in the arguments of perihelion indicates that the members of the relevant population of objects have similar orientations with respect to the plane of the Solar System, while the similarity in the longitudes indicates a clustering of these orbits in space.

A 2016 paper by Konstantin Batygin and Michael E. Brown ran a statistical analysis of these parameters for the six most extreme known trans-Neptunian (beyond Neptune) objects. Since they were discovered by a number of distinct observational surveys, the possibility of observational bias was dismissed. The analysis found that the clustering of the objects had only a 0.007% probability of occurring by chance. This suggested that another explanation was in fact required for the phenomenon. Further simulations suggested that a Planet Nine could account for the observations, provided that it have the required heft: at least around 10 Earth masses (or, equivalently, 5000 Pluto masses). In comparison, all the previously known trans-Neptunian objects put together weighed much less than a single Earth mass.

Shortly afterward, more evidence for Planet Nine was discovered, using data from a surprising source: the Cassini space probe. Launched in 1997, this Saturn orbiter allowed the calculation of the position of Saturn over time to unprecedented precision. These were compared to an extremely precise gravitational model of the Solar System known as INPOP, which accounts for the gravitational influence of the Sun, the planets, and many asteroids. The model then outputs planetary ephemerides, namely positions of the planets at given times. A paper published in February 2016 by Agnès Fienga et al. experimented with adding a Planet Nine at different positions to the INPOP. If the residuals (differences in Saturn's position between the predictions of INPOP and the real measurements from Cassini) are increased, this rules out the existence of Planet Nine in this position. However, if they are decreased, then this is evidence in support of Planet Nine, since it would partially explain the observed discrepancy.

The results of the paper are summarized in the diagram above. They showed that Planet Nine of 10 Earth masses and a semi-major axis of 700 AU was ruled out by Cassini's data to be in the red zones (this increased the residuals). The pink zones correspond to areas that would be ruled out by further inclusion of Cassini's data (the paper only used the measurements through 2014). The green zone, however, is where a Planet Nine would decrease residuals, making the INPOP model a more accurate picture of the Solar System. Therefore, the paper found this to be the most likely zone to find Planet Nine (with the single most likely position indicated). The addition of a Planet Nine in the farther regions of its orbit would not produce significant perturbations, and thus this is labeled "uncertainty zone".

Further analysis fine-tuned the estimates of mass, eccentricity, semi-major axis, and other parameters for the supposed Planet Nine. With an array of increasingly large telescopes at their disposal, astronomers will soon be able to settle the Planet Nine hypothesis one way or the other, bringing new insight into the current structure and the formation of our Solar System.


Sunday, March 5, 2017

The Planet Nine Hypothesis

Beginning in the 1990s, advances in astronomy allowed the detection of many extrasolar planets, adding thousands of the number known within two decades. However, apart from the reclassification of Pluto as a dwarf planet in 2006, the population of true planets in our Solar System did not change. Many, many other smaller objects were discovered, though.

Many of these smaller objects lay within the asteroid belt between Mars and Jupiter, or in the Kuiper Belt, just beyond Neptune's orbit. Eris, Haumea, and Makemake are other dwarf planets whose perihelia (closest approaches to the Sun) bring them within the Kuiper Belt, 30 to 50 astronomical units (AU) from the Sun. However, an unusual object was discovered in 2003 whose orbital properties were quite different.

The object was later named Sedna and measures a little less than half the diameter of Pluto. Though the best images of it by telescopes are only a few pixels wide, it is clearly of a reddish color, nearly as red as Mars. The perihelion of this object was, at the time, the largest known in the Solar System, at 76 AU. However, it also has an extremely elongated orbit, bringing it to an aphelion (farthest point) of 936 AU! This orbit is shown in red above, compared to the orbits of the outer planets and Pluto (in pink). About a decade later, another object, provisionally designated 2012 VP113, was discovered with comparable orbital parameters, except with a slightly farther perihelion of 80 AU and an aphelion of 438 AU. The scarcity of known objects of this type is not only a consequence of their distance, however.

This scatterplot, published in a paper by astronomers Chadwick A. Trujillo and Scott S. Shephard, shows the perihelia and eccentricities (a measure of the "elongatedness" of an elliptical orbit; a perfect circle has an eccentricity of 0) of various objects outside Neptune's orbit. Curiously, there is a clear drop-off at around 50 AU, with only a few known objects beyond. Notably, there is also a gap between 55 and 75 AU. This gap is not only an artifact of our telescopes being insufficiently powerful: Sedna and 2012 VP113 were detected farther out, so if there were objects in this gap they should have been easier to find. The high eccentricity of Sedna and 2012 VP113, as well as the existence of this gap, aroused suspicion that a massive object may have gravitationally perturbed the trajectories of objects in this region, illustrated in the image below.

The same paper indicated another unusual feature of the population of these farthest known objects.

The horizontal direction indicates the semi-major axis of each object (yet another measure of the size of an orbit; however, it is closely related to the two discussed previously: it is simply the average of the perihelion and the aphelion). The vertical variable on the scatterplot is the argument of perihelion, which is simply the angular position around the orbit of the orbit's perihelion (relative to where it crosses the plane of the Solar System). All known objects whose semi-major axes exceed 150 AU have arguments of perihelion all clustered roughly around 0°. In the eight-planet Solar System model, this should not be the case: gravitational perturbations from the gas giants would randomize the arguments of perihelion over millions of years. However, a large planetary body orbiting well beyond the known planets could constrain the arguments of perihelion. This led to the hypothesis of a new planet, nicknamed Planet Nine.

The above image shows the orbits of many of the same objects represented by dots to the right of the black line in the scatterplot. Note how in addition to the clustering trend noted above, the perihelia are also all on the same side of the Sun. The figure also shows where Planet Nine would possibly orbit given the positioning of those objects. The story of the Planet Nine hypothesis continues in the next post.


Sunday, February 12, 2017


Rainbows are among the most recognizable of atmospheric phenomena. They appear in situations in which there are water droplets in the air during a period of sunshine. As a result, they commonly occur after rainstorms. Before exploring the properties of rainbows, we cover atmospheric optics in the absence of water droplets. This situation is dominated by Rayleigh scattering, which makes our sky blue.

Rayleigh scattering of the sun's rays occurs when sunlight strikes air molecules. Higher frequencies of light (green, blue, violet) are more readily scattered than lower ones (red, orange, yellow) so when we look at the sky away from the Sun, most of what we see is scattered blue light. The interaction of sunlight with much larger water droplets is categorically different. Instead of scattering, light traveling from air to water (or for that matter, across the boundary of any two different media) is refracted.

This means that the angle of the light ray to the normal (the perpendicular to the boundary between media) changes as it passes from one to another. The origin of this effect is the fact that light travels at different speeds through different media. The extent to which this occurs for different substances is measured by a medium's index of refraction, often denoted n. If two media have indices of refraction n1 and n2 then the angles of the light rays to the normal within each (denoted θ1 and θ2) are given by Snell's Law:

n1sinθ1 = n2sinθ2

For air and water, the indices take values nair = 1.000293 and nwater = 1.330. Snell's Law then yields the fact that light rays bend toward the normal as they pass from air to water and do the opposite upon exiting. However, these values of the indices of refraction are for a specific wavelength of light (actually a standard color of yellow light emitted from excited atoms of sodium with a wavelength of 589.3 nm). The degree of refraction varies slightly across the visible wavelengths, leading to the separation of colors that we observe as a rainbow. The small droplets of water in the atmosphere are roughly spheres, leading to the kind of refraction illustrated below:

Note that the angles by which the light rays are refracted depends on where it hits the drop (the redness of the lines has no significance in this image) since the boundary between water and air is spherical, rather than flat. Each of the rays shown undergoes a single internal reflection before emerging from the water droplet, though some light just passes through, and some is internally reflected multiple times (more on this later). However, the maximum angle between the incoming and outgoing rays are different for different colors of light: in particular, they are greater for longer wavelengths than shorter. Therefore, at the very highest angles, the colors are separated.

At one end of the spectrum, violet light has a maximum angle of 40° from the incoming light ray, while in the longest visible wavelengths, red light has a maximum angle of 42° (left). As a result, for a fixed observer, red light will appear to come from a certain angle in the sky, while violet will appear to come from another (right). Orange, yellow, green, blue, and indigo will appear in between. The result is what we see as a rainbow.

Several properties of rainbows follow directly from this understanding. The first is that all (primary) rainbows are of the same angular size in the sky, namely 42° in radius. A rainbow therefore does not have a fixed position and appears the same size to every observer, meaning that every observer in fact sees their own rainbow. Also, the center of the rainbow's circular arc must be opposite to the position of the Sun in the sky. This point is called the anti-solar point and must always be below the horizon (since the Sun is above). As a result, the higher the Sun is in the sky, the lower the (primary rainbow). If the Sun is more than 42° above the horizon, it cannot be seen at all. This is why rainbows are typically seen early in the morning or later in the afternoon. In addition, though the maximum angle is 40-42° for different colors of light, some light (of all colors) is reflected from raindrops at smaller angles, making the sky just inside the rainbow noticeably brighter. This effect is apparent in the image above.

Though most light reflected within the raindrop undergoes only a single internal reflection, some is in fact reflected more than once, leading to what are known as higher-order rainbows, notably the secondary rainbow.

The colors of the secondary rainbow are reversed since an additional reflection inside the drop reverses the color spread. Further, it is situated at 52°, outside the primary rainbow, and is considerably fainter.

The secondary rainbow is sometimes too faint to be visible, but it is always there. In fact, light can reflect internally even more, producing higher-order rainbows. However, three reflections sends the light on a path at about 43° inclined from its original trajectory, meaning that it would form a circle of this radius around the Sun. Due to its faintness and proximity to the Sun, it is very difficult to photograph, but photographs have recently captured this phenomenon (see below).

Thus, a simple application of atmospheric optics may explain the rainbow, the beauty of which has captivated humanity since antiquity.


Sunday, January 22, 2017

Solar Sails

Solar sailing is a method of propulsion in space that utilizes solar radiation to accelerate a spacecraft, reducing the amount of fuel required for interplanetary missions.

The key to solar sailing is that light, though it has no mass, does have momentum! At first, this seems contradictory; the typical (Newtonian) definition of momentum that one first learns is that momentum equals mass times velocity, or p = mv (p denotes momentum). The mass m is simply a number indicating the quantity of matter in a given object, while p and v are vector quantities, having both magnitude and direction.

However, this definition of momentum is only approximate. Einstein's theory of special relativity holds that momentum, energy, and mass are all different aspects of a single quantity. The famous mass-energy equivalence E = mc2 (c is the speed of light) captures part of this relation. However, this equation is actually a special form of a more general expression for energy:

where p is momentum and m0 is the rest mass of an object (objects which are moving have additional mass and therefore additional energy by the mass energy relation). Photons, the particles of light, travel at the speed of light and are in fact never at rest. However, since objects with a nonzero rest mass can never reach the speed of light, it makes sense to classify photons as massless. Since m0 = 0, the equation reduces to E = cp, or p = E/c. Furthermore, light has energy, so it must have momentum. Different frequencies of light have different energies so photons of greater frequencies (such as X-ray or gamma ray photons) have correspondingly greater momentum.

Considering ordinary molecules for a moment, the macroscopic phenomenon of pressure (for example air pressure) emerges from individual collisions of particles with a surface such as the surface of a balloon. The average force that air molecules colliding with a surface exert is the pressure on that surface. Moreover, each of these collisions involves a transfer of momentum: a particle bouncing from a surface reverses the direction of its momentum vector so by the conservation of momentum the deflecting object also experiences a change in momentum. A similar momentum transfer occurs when light impacts a surface, creating what it known as radiation pressure.

The reason we do not feel radiation pressure whenever we enter sunlight is simply because this pressure is minute relative to the other forces we feel, dwarfed even by the force of a single tissue resting on a surface. The atmospheric pressure at sea level, around 100 pascals (Pa), is over ten billion times greater than the radiation pressure on a perfectly reflecting surface in direct sunlight on Earth (around 10 μPa = 10-5 Pa). Note that this phenomenon is distinct from what is called the solar wind, a term which refers to the stream of particles with mass constantly emanating from the Sun. These particles also exert a pressure when they collide with objects in space, but it is over a thousand times smaller than even the minute radiation pressure. Despite the apparent insignificance of radiation pressure, as in the case of ion propulsion, even small forces add to significant acceleration in space over time.

The concept of using radiation pressure as a means of propulsion is the foundation of the solar sail. Its design is simple: a large sheet of lightweight, reflective material surrounds the spacecraft payload (as in the artist's conception above). Notably, it is desirable for the sail material to reflect rather than absorb photons because this increases the acceleration of the sail.

The concept of a solar sail dates back to shortly after Maxwell's theory of electromagnetism was established in the 1860's in the works of Jules Verne. However, its first applications in spaceflight occurred almost 150 years later. Radiation pressure was used to save fuel in minor maneuvers on the MESSENGER mission and to compensate for a loss of maneuverability in the Kepler space telescope. However, the first true solar sail was IKAROS (Interplanetary Kite-craft Accelerated by Radiation of the Sun), a spacecraft launched by the Japanese Aerospace Exploration Agency (JAXA) in 2010 to demonstrate the technology.

IKAROS's solar sail measured 20 meters across the diagonal with a reflective film only 0.0075 mm thick that incorporated 0.025 mm thick solar cells to power the telemetry and steering instruments. The orange panels around the edges of the sail steered the craft by altering their reflectance with liquid crystal reflectors. For example, if one side of the sail were made more reflective then the opposite sides, the radiation forces would differ across the sail, causing it to rotate.

Launched on May 21, 2010, the IKAROS payload weighed only 310 kg and its cylindrical body measured on 1.6 meters in diameter and 0.8 meters in height. After reaching space, it followed the above procedure to release the sail (click to enlarge). By taking advantage of the centrifugal forces on four "tip masses" at each corner of the sail, the continually rotating apparatus can expand to full diameter and remain there without any rigid structure supporting the sail. The mission was a full success, demonstrating telemetry, propulsion, navigation, and attitude control for a solar sail.

Over the following years, NASA and the Planetary Society launched their own solar sails into Earth orbit for further testing demonstration of the technology, but IKAROS remained more significant as the first interplanetary solar sail. Once in space, craft employing solar sails do not have to carry any additional fuel, greatly reducing the amount of weight necessary for interplanetary missions. These sails may soon realize their potential as an inexpensive and efficient means of exploring the Solar System.


Sunday, January 1, 2017

Voronoi Diagrams and Metrics

In mathematics and visual art, a Voronoi diagram is a type of partition on a surface (usually a plane). Such a diagram is determined from some set of points (called "seeds") on the surface and a notion of distance on the surface by assigning each point a "cell," namely the region in the plane within which the given seed is closer than any other seed. The diagrams are named for the Ukrainian mathematician Georgy Voronoy.

Our first example of a Voronoi diagram consists of only two seeds (the black dots) and two cells, where the line connecting the two seeds is also shown. The maroon region contains the points in the plane closest to the left-hand seed, and the blue region the right. The divider between the two regions bisects the line between the two seeds (since the midpoint is by definition equidistant from the two endpoints) and is in particular the perpendicular bisector of this line. We now present a more complicated example.

In this image, the dots again represent the seeds, while the differently colored regions are the cells of the diagram. The inner region is bounded by a polygon (specifically a pentagon) whose sides are perpendicular bisectors of the lines connecting each of the outer seeds to the center seed. Note also that the central region is finite, since the center seed is surrounded by other seeds, while the other regions extend outward forever. Finally, each point at which three regions meet is the circumcenter of the triangle formed by three nearby seeds. The image below illustrates this fact for our example with six seeds.

Three of the seeds have been connected to form a triangle (white). The circumcenter of the triangle is the center of the circle containing the triangle's three vertices (black). By the definition of a circle, the circumcenter (red) is equidistant from the three seeds and is therefore the point at which the three neighboring regions meet.

Further, regular patterns of seeds produce correspondingly regular patterns of the cells. For example, a repeating square lattice of points produces a repeating pattern of square cells, as shown below.

The reader may experiment with different seed placements using the interactive feature found here. There are many ways to generalize the Voronoi Diagram concept beyond the two-dimensional plane. For example, it is possible to construct three-dimensional Voronoi diagrams, again using points as seeds, except that space will now be divided into three-dimensional cells instead of two.

The above image shows a number of seeds scattered in three-dimensional space and a single cell corresponding to the seed at the center. The lines connecting the center seed to the surrounding ones are also shown. Instead of a polygon, the cell is a polyhedron, bounded by faces which are sections of the planes that form the perpendicular bisectors of the line segments connecting the seeds.

In mathematics, Voronoi diagrams are useful for visualizing the notion of a metric. Metrics are generalizations of the familiar concept of distance to a number of different spaces in addition to the normal Euclidean plane and space (which we have worked with so far). For example, consider the surface of a sphere, such as the Earth. Typically, we define the distance between two points to be the length of the straight line connecting them (which in Euclidean space is the shortest path between the points). However, given two points on the Earth (a sphere), the line connecting them might go through the interior. When we speak of "distance" on the sphere, we want the shortest path along the surface between the two given points, or in other words the fastest travel route from one to the other!

The shortest distance between the points A and B above on the sphere is not the latitude line that they share (though this would be the straight path between them on the 2D map projection) but the arc of a circle passing through the sphere's center. These circles are known as great circles. The distance between two points on a sphere is defined to be the length of the great circle arc connecting them. This is also why planes take what appear to be inefficient paths on two-dimensional maps: they are in fact following a great circle (see below).

Having defined a metric for the sphere, we may choose some collection of points on it and create Voronoi diagrams, just as before. The diagram below takes major airports around the world as seeds and constructs a Voronoi diagram on the Earth's surface (which, of course, is nearly a sphere).

Voronoi diagrams also have a number of applications outside mathematics in settings where understanding distances from a fixed set of sources is important. They are used in modeling the spread of disease, the growth of forests, cell development, the distribution of minerals in the Earth's crust, and rainfall maps, among other things. They are a beautiful visual tool for comprehending the relative positions of points in a given space.