Storm Active: June 19-21

A tropical wave moved over the eastern Atlantic from Africa during the first week of June. However, it being so early in the hurricane season, the system had no opportunity to develop further for nearly two weeks, when it began to produce thunderstorm activity near the coast of Nicaragua on June 15. The wave continued generally west-northwestward for the next few days, during which time land interaction inhibited organization. On June 18, it emerged into the Bay of Campeche, and a low pressure center formed in association with the wave. During the next day or so, moderate wind shear affected the system, hindering convective banding. However, the center of circulation became better defined on June 19 as shear gradually lessened, while increasing shower activity showed hints of banding in the northern semicircle. As a result, the system was designated Tropical Depression Four.

After a brief lull in convective activity that evening, a large area of deep convection blossomed overnight, albeit not all that organized about the center of circulation. By the morning of June 20, reconnaissance aircraft discovered gale force winds in the system, prompting an upgrade to Tropical Storm Danielle. This was the earliest naming of a fourth storm in the history of Atlantic hurricane seasons, beating out the record set by Tropical Storm Debby in 2012 by 3 days. Heavy rain began over the Mexican coast on the west side of the Bay of Campeche and continued as Danielle moved west-northwestward toward land. Danielle reached its peak intensity of 45 mph winds and a minimum pressure of 1007 mb before making landfall in Mexico that evening. The storm brought 8-12 inches of rain to a large area of Mexico as it quickly weakened over the mountainous terrain of Mexico. It dissipated early on June 21.

During its brief period as a tropical system, Danielle brought heavy rains to eastern Mexico.

Forming in the central Bay of Campeche, Danielle did not have time to strengthen significantly before it moved westward over land.

## Sunday, June 19, 2016

## Sunday, June 5, 2016

### Tropical Storm Colin (2016)

Storm Active: June 5-7

Around June 3, a large area of disturbed weather formed over the northwestern Caribbean Sea. The next morning, a broad low-pressure center developed in association with the system just east of the Yucatan Peninsula. Moving over land that day, it was unable to organize further and convection remained over water well to the east. On June 5, however, the center emerged into the Gulf of Mexico. Shortly after, the low had acquired enough organization to be classified Tropical Depression Three. Due to shear out of the west, the depression's thunderstorm activity, though significant, was located in a north-to-south linear band over 100 miles east of the circulation center. Despite this disorganization, hurricane hunter aircraft discovered winds to tropical storm force that afternoon, prompting an upgrade to Tropical Storm Colin. The formation of the year's third named storm on June 5 broke the record for earliest third tropical storm, set on June 12, 1887.

Colin exhibited a very curious structure for a tropical cyclone. Overnight and into June 6, it was apparent that there were at least two distinct low-level centers widely separated from one another, both of which lay outside the intense area of convection to the east. The size of the system and its disheveled state were prohibitive to significant strengthening. Despite this, the impacts of the storm remained: by June 6, with Colin still over water, heavy rains spread throughout much of Florida, Georgia, and even South Carolina. The system consolidated a bit that day into a single circulation, albeit with several small-scale gyres. Meanwhile, it was accelerating rapidly toward the northeast. That evening, the center of Colin made landfall in the Big Bend region of Florida (although most of the rainfall had moved off to the east by this time).

Though at its peak intensity as a tropical system of 50 mph, the system was more disorganized than ever as it sped off to the northeast, quickly emerging off the coast of the Carolinas early on June 7. At this point, Colin was rapidly losing tropical characteristics, and it become post-tropical near the Outer Banks later that morning. By the afternoon, the system had moved away from land, still intensifying as a post-tropical system. It continued to speed toward the far northern Atlantic before being absorbed a few days later.

Colin remained disorganized during its brief stint as a tropical system. The above satellite image, taken June 6, shows multiple vortices exposed to the west of the convective canopy.

Colin made landfall in Florida on June 6 before quickly moving out to sea.

Around June 3, a large area of disturbed weather formed over the northwestern Caribbean Sea. The next morning, a broad low-pressure center developed in association with the system just east of the Yucatan Peninsula. Moving over land that day, it was unable to organize further and convection remained over water well to the east. On June 5, however, the center emerged into the Gulf of Mexico. Shortly after, the low had acquired enough organization to be classified Tropical Depression Three. Due to shear out of the west, the depression's thunderstorm activity, though significant, was located in a north-to-south linear band over 100 miles east of the circulation center. Despite this disorganization, hurricane hunter aircraft discovered winds to tropical storm force that afternoon, prompting an upgrade to Tropical Storm Colin. The formation of the year's third named storm on June 5 broke the record for earliest third tropical storm, set on June 12, 1887.

Colin exhibited a very curious structure for a tropical cyclone. Overnight and into June 6, it was apparent that there were at least two distinct low-level centers widely separated from one another, both of which lay outside the intense area of convection to the east. The size of the system and its disheveled state were prohibitive to significant strengthening. Despite this, the impacts of the storm remained: by June 6, with Colin still over water, heavy rains spread throughout much of Florida, Georgia, and even South Carolina. The system consolidated a bit that day into a single circulation, albeit with several small-scale gyres. Meanwhile, it was accelerating rapidly toward the northeast. That evening, the center of Colin made landfall in the Big Bend region of Florida (although most of the rainfall had moved off to the east by this time).

Though at its peak intensity as a tropical system of 50 mph, the system was more disorganized than ever as it sped off to the northeast, quickly emerging off the coast of the Carolinas early on June 7. At this point, Colin was rapidly losing tropical characteristics, and it become post-tropical near the Outer Banks later that morning. By the afternoon, the system had moved away from land, still intensifying as a post-tropical system. It continued to speed toward the far northern Atlantic before being absorbed a few days later.

Colin remained disorganized during its brief stint as a tropical system. The above satellite image, taken June 6, shows multiple vortices exposed to the west of the convective canopy.

Colin made landfall in Florida on June 6 before quickly moving out to sea.

Labels:
2016 Storms

## Friday, May 27, 2016

### Tropical Storm Bonnie (2016)

Storm Active: May 27-30, June 2-4

Around May 24, a broad area of scattered showers and thunderstorms began to develop in association with a low-pressure trough situated to the north of Hispaniola. Over the next few days, convection gradually became more concentrated near the low-pressure center as it deepened and moved generally toward the west-northwest. By the afternoon of May 27, the circulation had become well-defined with a curved band of strong thunderstorms just to the north and west of the center. This resulted in the designation of the system as Tropical Depression Two.

Overnight, the system continued west-northwestward toward a tongue of warmer Gulf Stream waters off of the coastline of South Carolina. Even as it encountered higher sea surface temperatures, however, significant wind shear out of the south kept the center exposed through the day of May 28. By that morning, rain bands had begun to affect the U.S. coastline. The convection deepened somewhat that afternoon and the circulation became better defined, so the system was named Tropical Storm Bonnie. Since Bonnie was named before the official start of the season on June 1, 2016 became the first year since 2012 to have multiple preseason storms. A few hours later, the storm reached its peak intensity of 45 mph winds and hesitated slightly in its forward motion.

Shortly afterward, however, a large burst of wind shear out of the south ripped the existing convection away from Bonnie, weakening it overnight just before its landfall in South Carolina as a tropical depression during the morning of May 29. Despite the decay of the system, heavy rainfall continued for a large region of the southeast U.S. through the day. Shortly afterward, the cyclone took its anticipated turn toward the north east over land. By the morning of May 30, the system had weakened sufficiently that it no longer met the criteria of being a tropical cyclone. Bonnie's remnants continued to generate shower activity as the center moved slowly northeast back over the Atlantic Ocean on May 31.

In fact, the system became more organized back over water, slowly redeveloping convective bands and increased sustained winds. By the morning of June 2, it had regained tropical depression status near the Outer Banks. That day, the Bonnie strengthened slightly over the warm Gulf Stream waters as it moved away from land, but remained slightly below tropical storm strength. Meanwhile, it began to accelerate eastward under the influence of a subtropical ridge. Overnight, the system seemed to be on the wane as it moved over cooler water and convection diminished, but a resurgence during the afternoon of June 3 prompted once again upgrading Bonnie to a tropical storm. However, very cold water finally took its toll on the storm on June 4: it lost all significant shower activity, weakening to a tropical depression, and shortly thereafter, a remnant low. The remnants moved east-southeast for another couple of days before dissipation.

The above image shows Bonnie shortly before landfall in South Carolina.

Bonnie's slow motion near the U.S. coastline as both a tropical and a non-tropical system brought torrential rains to the Carolinas.

Around May 24, a broad area of scattered showers and thunderstorms began to develop in association with a low-pressure trough situated to the north of Hispaniola. Over the next few days, convection gradually became more concentrated near the low-pressure center as it deepened and moved generally toward the west-northwest. By the afternoon of May 27, the circulation had become well-defined with a curved band of strong thunderstorms just to the north and west of the center. This resulted in the designation of the system as Tropical Depression Two.

Overnight, the system continued west-northwestward toward a tongue of warmer Gulf Stream waters off of the coastline of South Carolina. Even as it encountered higher sea surface temperatures, however, significant wind shear out of the south kept the center exposed through the day of May 28. By that morning, rain bands had begun to affect the U.S. coastline. The convection deepened somewhat that afternoon and the circulation became better defined, so the system was named Tropical Storm Bonnie. Since Bonnie was named before the official start of the season on June 1, 2016 became the first year since 2012 to have multiple preseason storms. A few hours later, the storm reached its peak intensity of 45 mph winds and hesitated slightly in its forward motion.

Shortly afterward, however, a large burst of wind shear out of the south ripped the existing convection away from Bonnie, weakening it overnight just before its landfall in South Carolina as a tropical depression during the morning of May 29. Despite the decay of the system, heavy rainfall continued for a large region of the southeast U.S. through the day. Shortly afterward, the cyclone took its anticipated turn toward the north east over land. By the morning of May 30, the system had weakened sufficiently that it no longer met the criteria of being a tropical cyclone. Bonnie's remnants continued to generate shower activity as the center moved slowly northeast back over the Atlantic Ocean on May 31.

In fact, the system became more organized back over water, slowly redeveloping convective bands and increased sustained winds. By the morning of June 2, it had regained tropical depression status near the Outer Banks. That day, the Bonnie strengthened slightly over the warm Gulf Stream waters as it moved away from land, but remained slightly below tropical storm strength. Meanwhile, it began to accelerate eastward under the influence of a subtropical ridge. Overnight, the system seemed to be on the wane as it moved over cooler water and convection diminished, but a resurgence during the afternoon of June 3 prompted once again upgrading Bonnie to a tropical storm. However, very cold water finally took its toll on the storm on June 4: it lost all significant shower activity, weakening to a tropical depression, and shortly thereafter, a remnant low. The remnants moved east-southeast for another couple of days before dissipation.

The above image shows Bonnie shortly before landfall in South Carolina.

Bonnie's slow motion near the U.S. coastline as both a tropical and a non-tropical system brought torrential rains to the Carolinas.

Labels:
2016 Storms

## Wednesday, May 18, 2016

### Professor Quibb's Picks – 2016

My personal prediction for the 2015 North Atlantic Hurricane Season (written May 18, 2015) is as follows:

14 cyclones attaining tropical depression status*,

13 cyclones attaining tropical storm status*,

7 cyclones attaining hurricane status*, and

3 cyclones attaining major hurricane status.

*Note: Hurricane Alex formed on January 13, 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 just barely exceeding historical averages in all categories.

The picture for the 2016 Atlantic hurricane season is unusually murky, due to several uncertainties regarding significant factors that influence tropical cyclone formation. First, the 2015-16 El Ninò event has continued to unfold, ranking in the top 3 historically in both intensity and duration. Positive sea surface temperature anomalies have persisted into May in the equatorial Pacific, indicating the continuation of the event. The chart below compares El Ninò events since 1950.

The 2015-16 event (black line) is probably most comparable to the 1997-98 event in its qualities, so if this trend were to repeat, the 2016 season would end with the ENSO in a negative phase. However, it has occurred that El Ninò events persist to the end of the second year, or that they become roughly neutral. A neutral ENSO (El Ninò Southern Oscillation) index, all else held equal, would lead to an average hurricane season, and a negative index to a more active season. The latest predictions indicate that neutral conditions will in fact prevail during the season's peak in September and October, but there is a great deal of uncertainty.

Second, the Atlantic Multi-Decadal Oscillation (AMO) (an empirically observed trend in tropical cyclone activity that has decades-long period) appears to be wrapping up the positive phase that led to busier hurricane seasons during the 2000's and early 2010's. However, this trend is harder to predict than the ENSO, and while some meteorological experts believe that it is now entering its negative phase, it is difficult to know for certain. The combination of these two factors yield an expectation of an average season, but with an unusually high probability of deviance from this prediction.

Finally, we examine a few more proximate factors to cyclone formation in the Atlantic. Current mean sea surface temperatures, as with all global temperatures, are anomalously high relative to historical data. However, temperatures in the Gulf of Mexico and along the U.S. Eastern seaboard are lower relative to average than the southern Caribbean and central tropical Atlantic. These latter areas may therefore be especially favorable to cyclonogenesis. Normally, preseason wind shear tendencies would also be relevant to my forecast, but due to the possible rapid changes in the ENSO index, these observations would have little predictive power.

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: 2

Neither the jet stream nor the negative anomaly in sea surface temperatures is as pronounced in this region as in 2015. Nevertheless, wind shear may still inhibit development in this region, leading to a lower risk of landfalls.

Yucatan Peninsula and Central America: 4

The southern Caribbean has some of the most anomalously warm temperatures in the Atlantic, and could fuel tropical cyclones that traverse it. After upper-level winds subside about midway through the season, there is potential for dangerous hurricanes to develop in this region.

Caribbean Islands: 3

The Caribbean Islands are at about average risk this year, with moderately warm temperatures and a diminishing El Ninò that will lead to a fair, but not exceedingly high likelihood of westward-tracking cyclones. Expect 2-3 tropical storms, at least one of which is of hurricane strength, to affect the islands.

Gulf of Mexico: 2

The Gulf remains rather safe this year, continuing the trend from the previous two seasons. Rather low temperatures will limit the potential for significantly damaging landfalls.

Overall, the 2016 season is expected to be around average, but there is an unusually low degree of confidence in this forecast due to expected shifts in climate throughout the year. Regardless, everyone should take sufficient preparedness measures, since dangerous storms can occur even in quiet seasons.

Sources: https://www.wunderground.com/blog/JeffMasters/first-look-at-2016-hurricane-season-unusually-big-question-marks, https://weather.com/storms/hurricane/news/2016-hurricane-season-forecast-atlantic-colorado-state-csu, http://www.esrl.noaa.gov/psd/enso/mei/, http://www.ospo.noaa.gov/Products/ocean/sst/anomaly/

14 cyclones attaining tropical depression status*,

13 cyclones attaining tropical storm status*,

7 cyclones attaining hurricane status*, and

3 cyclones attaining major hurricane status.

*Note: Hurricane Alex formed on January 13, 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 just barely exceeding historical averages in all categories.

The picture for the 2016 Atlantic hurricane season is unusually murky, due to several uncertainties regarding significant factors that influence tropical cyclone formation. First, the 2015-16 El Ninò event has continued to unfold, ranking in the top 3 historically in both intensity and duration. Positive sea surface temperature anomalies have persisted into May in the equatorial Pacific, indicating the continuation of the event. The chart below compares El Ninò events since 1950.

The 2015-16 event (black line) is probably most comparable to the 1997-98 event in its qualities, so if this trend were to repeat, the 2016 season would end with the ENSO in a negative phase. However, it has occurred that El Ninò events persist to the end of the second year, or that they become roughly neutral. A neutral ENSO (El Ninò Southern Oscillation) index, all else held equal, would lead to an average hurricane season, and a negative index to a more active season. The latest predictions indicate that neutral conditions will in fact prevail during the season's peak in September and October, but there is a great deal of uncertainty.

Second, the Atlantic Multi-Decadal Oscillation (AMO) (an empirically observed trend in tropical cyclone activity that has decades-long period) appears to be wrapping up the positive phase that led to busier hurricane seasons during the 2000's and early 2010's. However, this trend is harder to predict than the ENSO, and while some meteorological experts believe that it is now entering its negative phase, it is difficult to know for certain. The combination of these two factors yield an expectation of an average season, but with an unusually high probability of deviance from this prediction.

Finally, we examine a few more proximate factors to cyclone formation in the Atlantic. Current mean sea surface temperatures, as with all global temperatures, are anomalously high relative to historical data. However, temperatures in the Gulf of Mexico and along the U.S. Eastern seaboard are lower relative to average than the southern Caribbean and central tropical Atlantic. These latter areas may therefore be especially favorable to cyclonogenesis. Normally, preseason wind shear tendencies would also be relevant to my forecast, but due to the possible rapid changes in the ENSO index, these observations would have little predictive power.

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: 2

Neither the jet stream nor the negative anomaly in sea surface temperatures is as pronounced in this region as in 2015. Nevertheless, wind shear may still inhibit development in this region, leading to a lower risk of landfalls.

Yucatan Peninsula and Central America: 4

The southern Caribbean has some of the most anomalously warm temperatures in the Atlantic, and could fuel tropical cyclones that traverse it. After upper-level winds subside about midway through the season, there is potential for dangerous hurricanes to develop in this region.

Caribbean Islands: 3

The Caribbean Islands are at about average risk this year, with moderately warm temperatures and a diminishing El Ninò that will lead to a fair, but not exceedingly high likelihood of westward-tracking cyclones. Expect 2-3 tropical storms, at least one of which is of hurricane strength, to affect the islands.

Gulf of Mexico: 2

The Gulf remains rather safe this year, continuing the trend from the previous two seasons. Rather low temperatures will limit the potential for significantly damaging landfalls.

Overall, the 2016 season is expected to be around average, but there is an unusually low degree of confidence in this forecast due to expected shifts in climate throughout the year. Regardless, everyone should take sufficient preparedness measures, since dangerous storms can occur even in quiet seasons.

Sources: https://www.wunderground.com/blog/JeffMasters/first-look-at-2016-hurricane-season-unusually-big-question-marks, https://weather.com/storms/hurricane/news/2016-hurricane-season-forecast-atlantic-colorado-state-csu, http://www.esrl.noaa.gov/psd/enso/mei/, http://www.ospo.noaa.gov/Products/ocean/sst/anomaly/

Labels:
Hurricane Stats

## Saturday, May 14, 2016

### Hurricane Names List – 2016

For the North Atlantic Basin, the list for naming tropical cyclones in 2016 is as follows:

Alex

Bonnie

Colin

Danielle

Earl

Fiona

Gaston

Hermine

Ian

Julia

Karl

Lisa

Matthew

Nicole

Otto

Paula

Richard

Shary

Tobias

Virginie

Walter

This list is the same as that for the 2010 season, with the exception of Ian and Tobias, which replaced the retired names Igor and Tomas, respectively.

Alex

Bonnie

Colin

Danielle

Earl

Fiona

Gaston

Hermine

Ian

Julia

Karl

Lisa

Matthew

Nicole

Otto

Paula

Richard

Shary

Tobias

Virginie

Walter

This list is the same as that for the 2010 season, with the exception of Ian and Tobias, which replaced the retired names Igor and Tomas, respectively.

Labels:
Hurricane Stats

## Saturday, April 16, 2016

### ExoMars Mission

**ExoMars**, or Exobiology on Mars, is a mission jointly run by the European Space Agency (ESA) and the Russian Federal Space Agency (Roscosmos) to investigate possible traces of life on the planet Mars. The mission includes two launches: one in 2016 and one in 2018, with the first delivering an orbiter and a lander to Mars and the second the ExoMars rover.

The first launch took place on March 14, 2016 in Kazakhstan using a Russian-built launch vehicle. Both the Trace Gas Orbiter (TGO) and the Entry, Descent, and Landing Demonstrator Module (EDM) will arrive in the Martian system in October 2016.

The primary mission of the TGO, as the name suggests, is to refine our measurements of the scarcer components of the Martian atmosphere, including methane and water vapor. From an orbit about 250 miles above the surface of the red planet, the orbiter will obtain information orders of magnitude more accurate than any previous results. Methane in particular is generated by specific geological and organic processes. While the Trace Gas Orbiter would not be able to identify the cause of gaseous emissions by itself, it can pinpoint the sources geographically, aiding in the selection of the ExoMars rover landing site. The orbiter itself was constructed by the ESA while the Russian agency contributed several of its instruments.

Meanwhile the EDM lander (also called Schiaparelli after the Italian astronomer Giovanni Schiaparelli) will demonstrate crucial techniques for landing on the Martian surface shortly after the first spacecraft arrives at Mars. Weighing over 1300 pounds, the lander requires a controlled landing to reach the Martian surface safely, just like the

*Curiosity*rover. The probe will use a heat shield and parachutes to slow its descent and a liquid propulsion braking system to control its final touchdown on Mars. In addition, the static lander will carry instruments to record the landing. Communicating through the TGO, the lander will transmit its data after the fact to guide future landings.

While Schiaparelli's main purpose is the demonstration of landing technologies, it also carries a science payload that will operate for roughly 2-8 (Earth) days after its arrival on the surface. The onboard instruments suite called DREAMS will provide standard weather measurements such as humidity, pressure, wind speed, and temperature. In addition, a camera onboard will capture images of the landing itself. Finally, the lander will measure atmospheric transparency and search for electric fields on the red planet's surface, the first such measurement of its kind.

The second launch will occur sometime in the latter half of 2018, carrying the European-built ExoMars rover and a surface platform on which it will land, contributed by Roscosmos. The spacecraft will arrive at Mars in early 2019 at a landing site chosen with help from the 2016 mission's data. The same technology demonstrated in the first landing will allow the second module to perform a soft touchdown on the surface of Mars. After landing, the surface platform will deploy ramps, off of which the rover will exit to begin its exploration of the surface.

The rover's mission will last at least six months. Its primary mission will be to search for organic substances on the Martian surface. Since the harsh conditions of the surface may have obliterated traces of chemicals, the ExoMars rover will have the ability to bore holes as deep as two meters to obtain better preserved samples. After collecting samples, the rover will transfer them to its onboard laboratory for chemical analysis. With its careful site selection and dedicated exobiology instruments, the ExoMars mission has perhaps the best opportunity yet of discovering definitive biosignatures on Mars. It also would accomplish the technological objective of honing the ability to make soft, precision landings on the red planet. Finally, the mission paves the way for the holy grail of Martian exploration: returning a sample from the red planet back to Earth. Sources: http://exploration.esa.int/mars/, http://exploration.esa.int/mars/47852-entry-descent-and-landing-demonstrator-module/

Labels:
Space Missions

## Saturday, March 26, 2016

### The Projective Plane: An Algebraic Exploration II

This is the third post in a series discussing the projective plane. For the first, see here.

The previous post explained how certain types of polynomials, namely the homogeneous polynomials, define curves in the projective plane called projective varieties. This post will explicate the relation between projective varieties and affine varieties (typical curves in the plane) and indicate how projective varieties are in a way the extensions of affine varieties to include their points at infinity.

First, we consider how projective varieties naturally give rise to normal affine plane curves. Consider the projective variety defined by the equation

Since we obtain the affine plane by setting

There is also an algebraic process that does the reverse by taking a polynomial

Now we may apply these algebraic tools to solve the problems introduced in the last post that cannot be solved visually. First, regarding the hyperbola, algebra confirms our intuition. To see this, take the equation

Next, recall that the points at infinity in the projective plane are those for which

The cubic

This image shows the cubic curve in the affine plane as well as its projection (via lines through the origin, the center of the sphere) onto the surface of the sphere. It differs slightly from our earlier sphere representation since the plane is below and not above the sphere, but this makes little difference. At the bottom of the sphere, the origin of the plane touches the sphere (which is a point on the curve). At first, the path veers away from the

The projective plane has very elegant geometric properties (every two lines in the plane intersect in exactly one point, for example) and gives us a sturdy mathematical grounding for the slippery concept of behavior "at infinity." Generalizations of this concept are crucial in the study of polynomial curves and their corresponding equations.

Sources: http://voltage.typepad.com/.a/6a00e55375ef1c8833014e610f8df7970c-pi

The previous post explained how certain types of polynomials, namely the homogeneous polynomials, define curves in the projective plane called projective varieties. This post will explicate the relation between projective varieties and affine varieties (typical curves in the plane) and indicate how projective varieties are in a way the extensions of affine varieties to include their points at infinity.

First, we consider how projective varieties naturally give rise to normal affine plane curves. Consider the projective variety defined by the equation

*F*(*x*:*y*:*z*) = 0, where*F*is a homogeneous polynomial. In the first post of this series, we saw that the plane*z*= 1 in three-dimensional space can represent the subset of the projective plane that corresponds to the normal affine plane (i.e., without the points at infinity). We repeat the image from the first post for convenience, where each line through the origin (a point of projective space) is represented by the point at which it intersects the plane.Since we obtain the affine plane by setting

*z*= 1, it seems reasonable that we should be able to "collapse" projective varieties algebraically by setting*z*= 1 in the equation*F*(*x*:*y*:*z*) = 0. This is indeed the case, since substituting*z*= 1 yields a polynomial in only*two*variables:*f*(*x*,*y*) =*F*(*x*,*y*,1). For example, if*F*(*x*:*y*:*z*) =*x*^{2}*y*+ 2*y**z*^{2}- 5*z*^{3}(note that*F*is homogeneous and therefore defines a projective variety), then*f*(*x*,*y*) =*F*(*x*,*y*,1) =*x*^{2}*y*+ 2*y**1^{2}- 5*1^{3}=*x*^{2}*y*+ 2*y*- 5. The projective variety*F*(*x*,*y*,*z*) = 0 therefore corresponds to an affine variety*f*(*x*,*y*), as desired.There is also an algebraic process that does the reverse by taking a polynomial

*f*(*x*,*y*) and producing a corresponding homogeneous polynomial in three variables,*F*(*x*:*y*:*z*). The process works as follows:- Add up the powers of
*x*and*y*in each term of*f*and let*n*be the greatest degree that appears - Multiply each term by
*z*^{n-k}, where*k*is the degree of the term (this ensures that the resulting polynomial is homogeneous)

*f*(*x*,*y*) =*x*^{2}*y*+ 2*y*- 5 from above and apply the new process. The degrees of the three terms are 3, 1, and 0, respectively (-5 = -5*x*^{0}*y*^{0}). The maximum of these is 3, so we multiply the first term by*z*^{3-3}=*z*^{0}= 1, the second by*z*^{3-1}=*z*^{2}, and the third by*z*^{3-0}=*z*^{3}. The resulting polynomial is*F*(*x*:*y*:*z*) =*x*^{2}*y*+ 2*y**z*^{2}- 5*z*^{3}, exactly the same as the original polynomial! It is easy to verify that these two processes are mutually inverse in general, except in certain special cases such as when*F*is only a function of*z*.Now we may apply these algebraic tools to solve the problems introduced in the last post that cannot be solved visually. First, regarding the hyperbola, algebra confirms our intuition. To see this, take the equation

*xy*- 1 = 0 and transform it into the corresponding projective variety. The result is easily calculated as*xy*-*z*^{2}= 0. The asymptotes*x*= 0 and*y*= 0 to this hyperbola (see image in previous post) are unchanged by the process since they only have one term, clearly of maximal degree.Next, recall that the points at infinity in the projective plane are those for which

*z*= 0 in the homogeneous coordinates (*x*:*y*:*z*). This can be seen in the above visualization, where only points of the form (*a*:*b*:1) belong to the affine subset of the projective plane. Now any (*x*:*y*:*z*) can be scaled to this form by multiplying each component by 1/*z*(remember, only the ratio of the coordinates matters), but*only*when*z*is nonzero. Therefore, we substitute*z*= 0 and solve the equations to see which points at infinity each curve intersects. For the hyperbola, this gives*xy*= 0, so*x*= 0 or*y*= 0. Therefore, the two points at infinity the hyperbola intersects are (0:1:0) and (1:0:0). Other coordinate triples satisfying*xy*= 0 such as (3:0:0) differ only by a scale factor from one of the two solutions above and therefore define the same point in the projective plane. It follows that (0:1:0) and (1:0:0) are the only solutions. But the asymptotes*x*= 0 and*y*= 0 hit exactly the same points, (0:1:0) and (1:0:0), respectively! This confirms our intuition: a hyperbola and its asymptotes really do intersect at infinity.The cubic

*y*-*x*^{3}= 0 has no asymptote, but clearly goes off to infinity in some manner. We may use our algebraic tools to investigate the function's behavior in the projective plane. The highest degree term is*x*^{3}, of degree 3, so we must multiply the other term (namely the expression*y*, of degree 1) by*z*^{3-1}=*z*^{2}. The projective variety corresponding to the cubic is therefore defined by the equation*y**z*^{2}-*x*^{3}= 0. Substituting*z*= 0 yields*x*^{3}= 0, which has the single point (0:1:0) in the projective plane as a solution (since*z*is already set to 0). Note that even though the cubic goes to infinity in both the positive and negative directions, it meets only one point at infinity because opposite directions are identified (see the representation of the projective plane in a sphere in the first post). This indicates that the projective variety induced by the cubic meets that induced by the*y*-axis with equation*x*= 0 at infinity. Indeed, this makes some intuitive sense: as*x*becomes very large, it becomes insignificant relative to*y*=*x*^{3}and therefore the point (*x*,*y*) is "close" to the*y*-axis*x*= 0 (this can also be seen by zooming out a graph of the cubic - the graph eventually becomes nearly indistinguishable from the*y*-axis). We can also visualize the projective variety*y**z*^{2}-*x*^{3}= 0 that extends the cubic on the sphere (see below).This image shows the cubic curve in the affine plane as well as its projection (via lines through the origin, the center of the sphere) onto the surface of the sphere. It differs slightly from our earlier sphere representation since the plane is below and not above the sphere, but this makes little difference. At the bottom of the sphere, the origin of the plane touches the sphere (which is a point on the curve). At first, the path veers away from the

*y*-axis (the grid line from top to bottom through the origin), but notice how when the curve approaches the equator of the sphere (infinity), it comes back to hover above the*y*-axis. Images like these help to interpret the results of our algebraic manipulations.The projective plane has very elegant geometric properties (every two lines in the plane intersect in exactly one point, for example) and gives us a sturdy mathematical grounding for the slippery concept of behavior "at infinity." Generalizations of this concept are crucial in the study of polynomial curves and their corresponding equations.

Sources: http://voltage.typepad.com/.a/6a00e55375ef1c8833014e610f8df7970c-pi

Labels:
Mathematics

## Saturday, March 5, 2016

### The Projective Plane: An Algebraic Exploration I

This is the second post in a series on projective space. For the first, see here.

The idea of "adding points at infinity" to the plane introduces new behavior to the study of the intersections of lines and curves. Since there are different points of infinity for each direction in the affine plane (as discussed in the last post) and parallel lines intersect at infinity, it is reasonable to suppose that certain lines and curves also intersect at infinity (see below).

For example, the hyperbola above is given by the equation

Answering this question is difficult in our as yet fuzzy picture of the structure of the projective plane. However, the algebraic definition of the projective plane provides the tools necessary for solving this and many other related problems. Introducing this machinery is the purpose of this post.

Lines, parabolas, cubics, hyperbolas and many other curves in the plane may be expressed in the following algebraic form:

The previous post introduced the projective plane as the set of lines through the origin in three-dimensional space. It then illustrated two different ways in which certain representatives may be chosen from the lines to get a "picture" of the projective plane in three-dimensional space. We show that for equations of certain forms, it does not matter which representative we choose from a given line through the origin. First, let

The line above has equation

Next we consider

The crucial fact used in the proof (click to enlarge) is that the exponents of each term (the

It follows that a homogeneous polynomial in three coordinates has a solution set of points (a curve) in the projective plane. These solution sets are the projective varieties. The next post (coming soon) continues to fill in the algebraic picture of the projective plane and relates affine varieties to projective ones, ultimately answering the questions posed at the beginning of this post.

Sources: http://intmstat.com/plane-analytic-geometry/xyis1.gif, http://www.s-cool.co.uk/assets/learn_its/gcse/maths/graphs/algebraic-graphs/g-mat-graph-dia04.gif

The idea of "adding points at infinity" to the plane introduces new behavior to the study of the intersections of lines and curves. Since there are different points of infinity for each direction in the affine plane (as discussed in the last post) and parallel lines intersect at infinity, it is reasonable to suppose that certain lines and curves also intersect at infinity (see below).

For example, the hyperbola above is given by the equation

*xy*= 1. Away from the origin, the two branches of this curve approach the*x*- and*y*-axes, defined by the equations*y*= 0 and*x*= 0, respectively. Since the distance between the curve and these lines (called**asymptotes**of the curve) approaches 0 far from the origin, it makes sense to suppose that the hyperbola intersects with these asymptotes at infinity. On the other hand, for other curves that clearly "go off to infinity" like the cubic curve*y*=*x*^{3}shown below, there is no asymptote. What point at infinity, if any, does the cubic intersect?Answering this question is difficult in our as yet fuzzy picture of the structure of the projective plane. However, the algebraic definition of the projective plane provides the tools necessary for solving this and many other related problems. Introducing this machinery is the purpose of this post.

Lines, parabolas, cubics, hyperbolas and many other curves in the plane may be expressed in the following algebraic form:

*f*(*x*,*y*) = 0, where*f*is a polynomial function of*x*and*y*. This means that it is a sum of terms of the form*a***x*where^{i}y^{j}*i*and*j*are nonnegative integers and*a*is a constant coefficient. For example, the equation for the hyperbola written above may be written*xy*- 1 = 0 and the equation for the cubic*y*-*x*^{3}= 0. Any curve defined by an equation of this form is known as an*affine variety*.The previous post introduced the projective plane as the set of lines through the origin in three-dimensional space. It then illustrated two different ways in which certain representatives may be chosen from the lines to get a "picture" of the projective plane in three-dimensional space. We show that for equations of certain forms, it does not matter which representative we choose from a given line through the origin. First, let

*P*= (*x*,*y*,*z*) be a point distinct from the origin in three-dimensional space (that is, at least one of*x*,*y*, and*z*is nonzero). Then since any two distinct points determine a line,*P*determines a unique line*L*through the origin. The point (*ax*,*ay*,*az*) is then on*L*for any constant*a*and*every*point on*L*is of this form. In other words, only the*ratio*of the coordinates to one another is required to determine on which line through the origin a given point lies. This fact can more easily be seen in two dimensions, as in the figure below.The line above has equation

*y*= 2/3**x*. It passes through the origin and has slope 2/3, so*any*point (*x*,*y*) for which*y*/*x*= 2/3 is on the line (as demonstrated by the construction of a suitable triangle, as above). With this fact in mind, we introduce the concept of**homogeneous coordinates**. Homogeneous coordinates (*x*:*y*:*z*), where at least one is nonzero, define a point of the projective plane, with the understanding that only the ratio of*x*,*y*, and*z*matters. Thus (1:2:3) = (3:6:9), for example. With these identifications in mind, every point in the projective plane may be assigned homogeneous coordinates (though in many equivalent ways).Next we consider

**projective varieties**, i.e. certain types of curves in the projective plane. As before, they are defined as the set of points satisfying a certain polynomial equation, but in three variables instead of two:*F*(*x*,*y*,*z*) = 0. However, in light of the equivalence between points with different*x*-*y*-*z*coordinates, we must consider only polynomial equations that have the same points of the projective plane as solutions for*any*coordinate representation of the given points. These are called**homogeneous polynomials**. A polynomial is homogeneous if each one of its terms, or**monomials**, is of the same degree, meaning that the sum of the exponents in each term are the same. For example,*F*(*x*,*y*,*z*) =*x*^{2}*y**z*^{3}is trivially homogeneous of degree 6 because it has only one term and the sum of its powers are 2 + 1 + 3 = 6.*F*(*x*,*y*,*z*) =*x**y*^{2}+*z*^{3}is homogeneous of degree 3 because the sum of the exponents of the*x**y*^{2}term is 1 + 2 = 3 and is obviously also 3 for the second term,*z*^{3}. The crucial property of homogeneous polynomials is that if*F*(*x*,*y*,*z*) = 0, then*F*(*ax*,*ay*,*az*) = 0 for any constant*a*:The crucial fact used in the proof (click to enlarge) is that the exponents of each term (the

*p*'s,*q*'s, and*r*'s) must always add up to the same degree*n*. The*a*^{n}term can then be factored out, confirming that*F*(*x*,*y*,*z*) = 0 always implies*F*(*ax*,*ay*,*az*) = 0. This means that for any point in the projective plane, a homogeneous polynomial that is zero on one representative is zero on all. Conversely, if*F*(*ax*,*ay*,*az*) = 0, then the same proof (using 1/*a*) shows that*F*(*x*,*y*,*z*) = 0 so long as*a*is not zero. All this manipulation distills down to the following crucial statement: it is meaningful to say that a homogeneous polynomial is zero at a point in the projective plane since any representative gives the same result. We can thus denote projective varieties by the equation*F*(*x*:*y*:*z*) = 0 in homogeneous coordinates.It follows that a homogeneous polynomial in three coordinates has a solution set of points (a curve) in the projective plane. These solution sets are the projective varieties. The next post (coming soon) continues to fill in the algebraic picture of the projective plane and relates affine varieties to projective ones, ultimately answering the questions posed at the beginning of this post.

Sources: http://intmstat.com/plane-analytic-geometry/xyis1.gif, http://www.s-cool.co.uk/assets/learn_its/gcse/maths/graphs/algebraic-graphs/g-mat-graph-dia04.gif

Labels:
Mathematics

## Saturday, February 20, 2016

### The Detection of Gravitational Waves

For an introduction to gravitational waves, see here.

Before 2016, a nobel prize had already been rewarded for an observation that was consistent with, and seemed to confirm, the existence of gravitational waves. In 1974, Russell Hulse and Joesph Taylor discovered a very compact binary system of objects at a distance of 21,000 light years, consisting of two neutron stars orbiting one another. One of the bodies was also a pulsar, meaning that the radiation beams emitted from its poles periodically point toward Earth as it rotates. Since the rotation rate of a neutron star changes only very slowly over time, pulsars are fairly precise clocks. However, Hulse and Taylor detected that the pulses did not reach Earth precisely on time, but varied slightly from the expected arrival time. They were sometimes sooner, sometimes later in a regular pattern, indicating that the pulsar in question was in fact part of a binary system.

The above diagram depicts the binary system consisting of pulsar B1913+16 and its companion, another neutron star. No radiation from the companion has been observed on Earth, indicating that its poles oriented away from us. However, its presence can be inferred from the fact that the pulsar moves farther and closer to Earth in a short, regular period, indicating an orbit. The difference in arrival times is about 3 seconds, indicating that the orbit is about 3 light-seconds across. Further, the orbital period is 7.75 hours.

This discovery provided an excellent opportunity to confirm the predictions of general relativity: such a compact system with rapidly orbiting masses would radiate fairly large quantities of gravitational radiation. However, direct detection was well beyond 1970's technology. Instead, Taylor observed the pulsar system over a number of decades, and found the following:

Since the discovery of the pulsar, its orbital period had been decreasing very slowly, though steadily and measurably, by about 35 seconds over a timespan of 30 years. This is very little relative to the total period of 7.75 hours, but the data matched the predictions of general relativity almost precisely: as energy was lost to gravitational waves, the neutron stars gradually spiral inward toward one other as their orbits becoming shorter and shorter. This remarkable confirmation of a prediction of relativity won Hulse and Taylor the Noble Prize in physics in 1993.

And there the matter sat. Though detectors grew more and more advanced, no direct detections of gravitational waves were made for over 20 years. This all changed in 2015.

On September 14, 2015, at 09:50:45 UTC, shortly after LIGO (the Laser Interferometer Gravitational-Wave Observatory) resumed activity following an upgrade, the two detectors in Washington State and Louisiana picked up a transient gravitational wave signal, the first ever observed by humankind. The announcement of the discovery was made several months later, on February 11, 2016.

The above image shows the signals recorded at Hanford, Washington (left) and Livingston, Louisiana (right). The signals are also superimposed on the right to demonstrate their similarity. The horizontal axis is time, measured relative to 09:50:45 UTC on that day. The reader may notice that the event was distinguishable from the surrounding noise in the detector for only about 0.05 seconds (the third row charts the residual noise after the theoretical waveform in the second row is subtracted out). The final row shows the rapid increase in gravitational wave amplitude during the event and the subsequent silence. The vertical dimension in the first several rows is the relative strain on the detectors, or the amount by which the different arms of LIGO were stretched or compressed by the ripples in spacetime. The scale for these axes measures strain by parts in 10

The theoretical wave form above was a simulation of the event that generated the gravitational waves: the final in-spiraling and ultimate merging of two black holes. The increasing frequency and amplitude of the signals corresponds to the final moments of the collapsing system as the two black holes orbit faster and faster and tighter and tighter around one another before finally combining. Further, the signals at the two detectors were separated by 6.9 ms, smaller than the light travel time between the sites of 10 ms. The delay between the arrival times allows the direction of the source to be identified.

This image shows the region in the sky from which the signals likely originated. The colors indicate the confidence that the source lay within the indicated region: purple is the 90% confidence region and yellow the 50% confidence region. The uncertainty arises from the fact that there were two detectors, and not the three required for a full triangulation.

In addition to the location of the source, the analysis of the waveform yields more. The distance of the system was roughly 1.2 billion light-years, meaning that the merger that we are just now observing occurred over a billion years ago. The two black holes had respective masses of about 36 and 29 solar masses, while the final black hole after the merger weighed in at 62 solar masses. This corresponds to a loss of about 3 solar masses, which was all converted into energy released as gravitational waves as the holes merged. The magnitude of this cataclysm can scarcely be overstated: at its peak, the rate of energy release was an estimated 3.6x10

In addition to being a resounding confirmation of general relativity, the observation was the first truly direct detection of black holes: the fact that such massive objects came within hundreds of kilometers of one another indicates that they had extremely high densities, densities only possible in black holes. But while significant, cosmologists were already nearly certain that both gravitational waves and black holes existed. However, this discovery marks the opening of a brand new field of astronomy. Gravitational waves, which pass unimpeded through nearly anything over nearly any distance, allow us to "hear" cosmic events that we could not have detected before. In theory, these waves could allow us to observe the earliest stages of the universe, before it became transparent to electromagnetic radiation. In 2016, 100 years after Einstein predicted gravitational waves, we took the first step towards seeing the universe in a new way.

Sources: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1989ApJ...345..434T&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf, https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.061102, http://resources.edb.gov.hk/physics/articlePic/InterestingTopics/BinaryStars_pic04E.gif

Before 2016, a nobel prize had already been rewarded for an observation that was consistent with, and seemed to confirm, the existence of gravitational waves. In 1974, Russell Hulse and Joesph Taylor discovered a very compact binary system of objects at a distance of 21,000 light years, consisting of two neutron stars orbiting one another. One of the bodies was also a pulsar, meaning that the radiation beams emitted from its poles periodically point toward Earth as it rotates. Since the rotation rate of a neutron star changes only very slowly over time, pulsars are fairly precise clocks. However, Hulse and Taylor detected that the pulses did not reach Earth precisely on time, but varied slightly from the expected arrival time. They were sometimes sooner, sometimes later in a regular pattern, indicating that the pulsar in question was in fact part of a binary system.

The above diagram depicts the binary system consisting of pulsar B1913+16 and its companion, another neutron star. No radiation from the companion has been observed on Earth, indicating that its poles oriented away from us. However, its presence can be inferred from the fact that the pulsar moves farther and closer to Earth in a short, regular period, indicating an orbit. The difference in arrival times is about 3 seconds, indicating that the orbit is about 3 light-seconds across. Further, the orbital period is 7.75 hours.

This discovery provided an excellent opportunity to confirm the predictions of general relativity: such a compact system with rapidly orbiting masses would radiate fairly large quantities of gravitational radiation. However, direct detection was well beyond 1970's technology. Instead, Taylor observed the pulsar system over a number of decades, and found the following:

Since the discovery of the pulsar, its orbital period had been decreasing very slowly, though steadily and measurably, by about 35 seconds over a timespan of 30 years. This is very little relative to the total period of 7.75 hours, but the data matched the predictions of general relativity almost precisely: as energy was lost to gravitational waves, the neutron stars gradually spiral inward toward one other as their orbits becoming shorter and shorter. This remarkable confirmation of a prediction of relativity won Hulse and Taylor the Noble Prize in physics in 1993.

And there the matter sat. Though detectors grew more and more advanced, no direct detections of gravitational waves were made for over 20 years. This all changed in 2015.

On September 14, 2015, at 09:50:45 UTC, shortly after LIGO (the Laser Interferometer Gravitational-Wave Observatory) resumed activity following an upgrade, the two detectors in Washington State and Louisiana picked up a transient gravitational wave signal, the first ever observed by humankind. The announcement of the discovery was made several months later, on February 11, 2016.

The above image shows the signals recorded at Hanford, Washington (left) and Livingston, Louisiana (right). The signals are also superimposed on the right to demonstrate their similarity. The horizontal axis is time, measured relative to 09:50:45 UTC on that day. The reader may notice that the event was distinguishable from the surrounding noise in the detector for only about 0.05 seconds (the third row charts the residual noise after the theoretical waveform in the second row is subtracted out). The final row shows the rapid increase in gravitational wave amplitude during the event and the subsequent silence. The vertical dimension in the first several rows is the relative strain on the detectors, or the amount by which the different arms of LIGO were stretched or compressed by the ripples in spacetime. The scale for these axes measures strain by parts in 10

^{-21}. This corresponds to extraordinarily minute changes in length: the 4 kilometer arms of the LIGO detector changed by only about 10^{-18}meters, only about one thousandth the diameter of a proton!The theoretical wave form above was a simulation of the event that generated the gravitational waves: the final in-spiraling and ultimate merging of two black holes. The increasing frequency and amplitude of the signals corresponds to the final moments of the collapsing system as the two black holes orbit faster and faster and tighter and tighter around one another before finally combining. Further, the signals at the two detectors were separated by 6.9 ms, smaller than the light travel time between the sites of 10 ms. The delay between the arrival times allows the direction of the source to be identified.

This image shows the region in the sky from which the signals likely originated. The colors indicate the confidence that the source lay within the indicated region: purple is the 90% confidence region and yellow the 50% confidence region. The uncertainty arises from the fact that there were two detectors, and not the three required for a full triangulation.

In addition to the location of the source, the analysis of the waveform yields more. The distance of the system was roughly 1.2 billion light-years, meaning that the merger that we are just now observing occurred over a billion years ago. The two black holes had respective masses of about 36 and 29 solar masses, while the final black hole after the merger weighed in at 62 solar masses. This corresponds to a loss of about 3 solar masses, which was all converted into energy released as gravitational waves as the holes merged. The magnitude of this cataclysm can scarcely be overstated: at its peak, the rate of energy release was an estimated 3.6x10

^{49}W, greater than the radiation emitted from all stars in the observable universe combined!In addition to being a resounding confirmation of general relativity, the observation was the first truly direct detection of black holes: the fact that such massive objects came within hundreds of kilometers of one another indicates that they had extremely high densities, densities only possible in black holes. But while significant, cosmologists were already nearly certain that both gravitational waves and black holes existed. However, this discovery marks the opening of a brand new field of astronomy. Gravitational waves, which pass unimpeded through nearly anything over nearly any distance, allow us to "hear" cosmic events that we could not have detected before. In theory, these waves could allow us to observe the earliest stages of the universe, before it became transparent to electromagnetic radiation. In 2016, 100 years after Einstein predicted gravitational waves, we took the first step towards seeing the universe in a new way.

Sources: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1989ApJ...345..434T&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf, https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.061102, http://resources.edb.gov.hk/physics/articlePic/InterestingTopics/BinaryStars_pic04E.gif

Labels:
Astronomy and Physics,
Black Holes,
Forces,
Universe

## Friday, February 12, 2016

### The Projective Plane: A Visual Introduction

The statement "any two distinct lines intersect in a point" is

From this, we naturally comes the intuition that "parallel lines intersect at infinity." Certainly this tidies up our intersection statement because it provides a way for even parallel lines to intersect. But what does "at infinity" mean? Is there really a "point" there? The notion of

We focus on the (real)

The first method of visualization illustrates how the projective plane is related to the ordinary plane (sometimes called the

However, not every line through the origin intersects the plane

The above visualization illustrates the connection between the projective plane and the affine plane. It also indicates that there are many points at infinity, one for each line through the origin "lying flat" in the

Using a sphere (or a hemisphere, to be more precise) to represent the projective plane is just as legitimate as using a plane: all that matters is that there is one point for each line through the origin. It does not matter

In fact, nearly every person is intimately familiar with this representation of projective space! Imagine that it is a clear night and you go out to look at the stars. You catch sight of the familiar constellation Orion, the hunter. The stars marking Orion's shoulders are Betelgeuse and Bellatrix, which we perceive to be neighboring stars that connect to form the figure of Orion. In fact, however, Betelgeuse is between two and three times as distant as Bellatrix. When we look up at the sky, we do not perceive the true three-dimensional space but points of light etched into the inner surface of the celestial sphere passing overhead. Stars in similar directions, regardless of their distances, are projected onto nearby points. This is why the result of treating all points along a line through the origin as equivalent is known as projective space.

It is clear, however, that every line through the origin intersects the sphere at exactly

The above image shows how the affine plane (and our first visualization) relate to our second visualization of the projective plane as (part of) a sphere. Lines through the origin (

Finally, the sphere illustrates how the projective plane solves the motivating problem of parallel lines than began this post.

Two parallel lines in the plane correspond to precisely the same lines in our first visualization, which indeed embeds a "copy" of the affine plane in three-dimensional space. When these parallel line are transferred to the sphere in the same manner that the point was above (remember: each transferred point represents a line through the origin and "transferring" a point is merely choosing a different representative), the figure above is the result. However, it is evident that the resulting arcs on the sphere intersect at the equator (green circle) and we know the equator contains the points at infinity! Though there appear to be two intersections, recall that points diametrically opposite from one another are on the same line through the center, so that these points are identified as one in the projective plane. We have our desired result: two parallel lines intersect in exactly one point.

The next post provides an algebraic description of the projective plane and explores more of its properties.

Sources:

*almost*true in normal plane geometry. The exception, of course, is the case of two parallel lines. However, from real experience we know from the rules of perspective that two parallel lines "converge" very far away, even if we know that they in fact maintain the same distance apart.From this, we naturally comes the intuition that "parallel lines intersect at infinity." Certainly this tidies up our intersection statement because it provides a way for even parallel lines to intersect. But what does "at infinity" mean? Is there really a "point" there? The notion of

**projective space**makes these ideas explicit and rigorous.We focus on the (real)

**projective plane**, the extension of the normal plane to include these "points at infinity" where parallel lines intersect. The set of points in the projective plane is defined, somewhat enigmatically, as "the set of lines through the origin in three-dimensional space." Defining each point to be a line in a different space seems extremely confusing at first, but there are multiple ways to visualize this concept.The first method of visualization illustrates how the projective plane is related to the ordinary plane (sometimes called the

**affine plane**). Consider three-dimensional space with ordinary coordinates*x*,*y*, and*z*as shown. The plane labeled*z*= 1 contains all points for which the*z*coordinate is 1, namely all those of the form (*x*,*y*,1). Clearly this plane is just like the ordinary two-dimensional plane (under the correspondence (*x*,*y*,1) → (*x*,*y*)), only embedded in three dimensions, like a flat sheet of paper in our world (but infinite). The dotted line shown that passes through the origin intersects the plane at the particular point (*a*,*b*,1). Remembering that the projective plane is meant to be an extension of the affine plane, we identify the dotted line with the point where it intersects the plane. Clearly, for each point in the*z*= 1 plane, there exists*exactly one*line through the origin and the given point. This shows how the ordinary plane is a subset of projective space (the set of lines through the origin)!However, not every line through the origin intersects the plane

*z*= 1. For instance, the*x*-axis, the*y*-axis (both shown), and any other line in the plane of these two axes only contain points for which the*z*coordinate is 0 and can never intersect the plane*z*= 1 (to be clear, the three-dimensional space considered here does not have*its own*points at infinity!). Therefore, these lines cannot correspond to points on the ordinary plane. These special lines, in fact, are the points at infinity in the projective plane.The above visualization illustrates the connection between the projective plane and the affine plane. It also indicates that there are many points at infinity, one for each line through the origin "lying flat" in the

*xy*-plane. However, it fails to indicate how points at infinity are truly the intersections of parallel lines. For this, we use another visualization that chooses different representative points.Using a sphere (or a hemisphere, to be more precise) to represent the projective plane is just as legitimate as using a plane: all that matters is that there is one point for each line through the origin. It does not matter

*which*points we choose.In fact, nearly every person is intimately familiar with this representation of projective space! Imagine that it is a clear night and you go out to look at the stars. You catch sight of the familiar constellation Orion, the hunter. The stars marking Orion's shoulders are Betelgeuse and Bellatrix, which we perceive to be neighboring stars that connect to form the figure of Orion. In fact, however, Betelgeuse is between two and three times as distant as Bellatrix. When we look up at the sky, we do not perceive the true three-dimensional space but points of light etched into the inner surface of the celestial sphere passing overhead. Stars in similar directions, regardless of their distances, are projected onto nearby points. This is why the result of treating all points along a line through the origin as equivalent is known as projective space.

It is clear, however, that every line through the origin intersects the sphere at exactly

*two*points, while there can only be one representative for a point of projective space. Thus, by convention we consider only intersections with the upper hemisphere (just as in our example of the night sky - one cannot see stars looking downward!). This leaves only the "horizontal" lines intersecting the equator of the sphere twice. For these, we choose the points of intersection for positive*y*-values (the area colored dark green above) and finally the*x*-axis is represented by the dark red point of positive*x*. The projective plane is therefore the union of the yellow upper hemisphere, the dark green semicircle, and the dark red point. The latter two parts are the points at infinity.The above image shows how the affine plane (and our first visualization) relate to our second visualization of the projective plane as (part of) a sphere. Lines through the origin (

*O*) and a point in the upper hemisphere intersect the plane to form a one-to-one correspondence. As we would expect, points at infinity correspond to lines through the sphere's equator that are parallel to the plane and are therefore not part of our original affine plane.Finally, the sphere illustrates how the projective plane solves the motivating problem of parallel lines than began this post.

Two parallel lines in the plane correspond to precisely the same lines in our first visualization, which indeed embeds a "copy" of the affine plane in three-dimensional space. When these parallel line are transferred to the sphere in the same manner that the point was above (remember: each transferred point represents a line through the origin and "transferring" a point is merely choosing a different representative), the figure above is the result. However, it is evident that the resulting arcs on the sphere intersect at the equator (green circle) and we know the equator contains the points at infinity! Though there appear to be two intersections, recall that points diametrically opposite from one another are on the same line through the center, so that these points are identified as one in the projective plane. We have our desired result: two parallel lines intersect in exactly one point.

The next post provides an algebraic description of the projective plane and explores more of its properties.

Sources:

*Algebraic Curves: An Introduction to Algebraic Geometry*by William Fulton, https://www.math.toronto.edu/mathnet/questionCorner/qc_hlimgs1/image87.gif, http://jwilson.coe.uga.edu/EMAT6680Fa11/Chun/Final1/4.png, http://courses.cs.washington.edu/courses/cse557/98wi/readings/xforms/diagram/homogeneous.gif, http://earthsky.org/astronomy-essentials/how-far-is-betelgeusehttp://en.wikipedia.org/wiki/Projective_space
Labels:
Mathematics

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