{"id":486214,"date":"2024-09-01T14:40:00","date_gmt":"2024-09-01T12:40:00","guid":{"rendered":"https:\/\/innovationorigins.com\/?p=486214"},"modified":"2024-09-01T14:40:00","modified_gmt":"2024-09-01T12:40:00","slug":"munich-and-eindhoven-mathematicians-debunk-gps-assumptions","status":"publish","type":"post","link":"https:\/\/ioplus.nl\/archive\/en\/munich-and-eindhoven-mathematicians-debunk-gps-assumptions\/","title":{"rendered":"Munich and Eindhoven mathematicians debunk GPS assumptions"},"content":{"rendered":"\n

The summer holidays are ending, which for many concludes with a long drive home and reliance on GPS devices to get safely home. But every now and then, GPS devices can suggest strange directions or get briefly confused about your location. But until now, no one knew for sure when the satellites were in a good enough position for the GPS system to give reliable direction. TU\/e\u2019s Mireille Boutin and her co-worker Gregor Kemper at the Technical University of Munich have turned to mathematics to help determine when your GPS system has enough information to determine your location accurately.<\/p>\n\n\n\n

\u201cIn 200 meters, turn right.\u201d This is a typical instruction that many have heard from their global positioning system (GPS).<\/p>\n\n\n\n

Without a doubt, advancements in GPS technologies and mobile navigation apps have helped GPS play a major role in modern car journeys.<\/p>\n\n\n\n

But, strictly adhering to instructions from GPS devices can lead to undesirable situations. Less serious might be turning left instead of right, while more serious could be driving your car into a harbor \u2013 just as two tourists did in Hawaii in 2023. The latter incident is very much an exception to the rule, and one might wonder: \u2018How often does this happen and why?\u2019<\/p>\n\n\n\n

GPS and your visibility<\/h2>\n\n\n\n
\"Mireille
Mireille Boutin, \u00a9 TU\/e<\/figcaption><\/figure>\n\n\n\n

\u201cThe core of the GPS system was developed in the mid-1960s. At the time, the theory behind it did not provide any guarantee that the location given would be correct,\u201d says Mireille Boutin<\/a>, professor at the Department of Mathematics and Computer Science.<\/p>\n\n\n\n

It won\u2019t come as a surprise then to learn that calculating an object\u2019s position on Earth relies on some nifty mathematics. And they haven\u2019t changed much since the early days. These are at the core of the GPS system we all use. And it deserved an update.<\/p>\n\n\n\n

So, along with her colleague Gregor Kemper at the Technical University of Munich, Boutin turned to mathematics to expand on the theory behind the GPS system, and their finding has recently been published in the journal Advances in Applied Mathematics<\/em>.<\/p>\n\n\n\n

How does GPS work?<\/h2>\n\n\n\n

Before revealing Boutin and Kemper\u2019s big finding, just how does GPS work?<\/p>\n\n\n\n

Global positioning is all about determining the position of a device on Earth using signals sent by satellites. A signal sent by a satellite carries two key pieces of information \u2013 the position of the satellite in space and the time at which the position was sent by the satellite. By the way, the time is recorded by a very precise clock on board the satellite, which is usually an atomic clock.<\/p>\n\n\n\n

Thanks to the atomic clock, satellites send very accurate times, but the big issue lies with the accuracy of the clock in the user\u2019s device \u2013 whether it\u2019s a GPS navigation device, a smartphone, or a running watch.  <\/p>\n\n\n\n

\u201cIn effect, GPS combines precise and imprecise information to figure out where a device is located,\u201d says Boutin. \u201cGPS might be widely used, but we could not find any theoretical basis to guarantee that the position obtained from the satellite signals is unique and accurate.\u201d<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Google says \u2018Four\u2019<\/h2>\n\n\n\n

If you do a quick Google search for the minimum number of satellites needed for navigation with GPS, multiple sources report that you need at least four satellites.<\/p>\n\n\n\n

But the question is not just how many satellites you can see, but also what arrangements can they form?  For some arrangements, determining the user position is impossible. But what arrangements exactly? That\u2019s what the researchers wanted to find out.<\/p>\n\n\n\n

\u201cWe found conjectures in scientific papers that seem to be widely accepted, but we could not find any rigorous argument to support them anywhere. Therefore, we thought that, as mathematicians, we might be able to fill that knowledge gap,\u201d Boutin says.<\/p>\n\n\n\n

Linear equations are easier to solve<\/h2>\n\n\n\n

To solve the problem, Boutin and Kemper simplified the GPS problem to what works best in practice: equations that are linear in terms of the unknown variables.<\/p>\n\n\n\n

 \u201cA set of linear equations is the simplest form of equations we could hope for. To be honest, we were surprised that this simple set of linear equations for the GPS problem wasn\u2019t already known!\u201d Boutin adds.<\/p>\n\n\n\n

The problem of uniqueness<\/strong><\/h3>\n\n\n\n

With their linear equations ready, Boutin and Kemper then looked closely at the solutions to the equations, paying special attention as to whether the equations gave a unique solution.<\/p>\n\n\n\n

\u201cA unique solution implies that the only solution to the equations is the actual position of the user,\u201d notes Boutin.<\/p>\n\n\n\n

If there is more than one solution to the equations, then only one is correct \u2013 that is, the true user position \u2013 but the GPS system would not know which one to pick and might return the wrong one.<\/p>\n\n\n\n

Four may not be enough<\/h2>\n\n\n\n

The researchers found that nonunique solutions can emerge when the satellites lie in a special structure known as a \u2018hyperboloid of revolution of two sheets\u2019.<\/p>\n\n\n\n

\u201cIt doesn\u2019t matter how many satellites send a signal \u2013 if they all lie on one of these hyperboloids then it\u2019s possible that the equations can have two solutions, so the one chosen by the GPS could be wrong,\u201d says Boutin.<\/p>\n\n\n\n

But what about the claim that you need at least four satellites to determine your position? \u201cHaving four satellites can work, but the solution is not always unique,\u201d points out Boutin.<\/p>\n\n\n\n

Why mathematics matters<\/h2>\n\n\n\n

For Boutin, this work demonstrates the power and application of mathematics.<\/p>\n\n\n\n

\u201cI personally love the fact that mathematics is a very powerful tool with lots of practical applications,\u201d says Boutin. \u201cI think people who are not mathematicians may not see the connections so easily, and so it is always nice to find clear and compelling examples of everyday problems where mathematics can make a difference.\u201d<\/p>\n\n\n\n

Central to Boutin and Kemper\u2019s research is the field of algebraic geometry in which abstract algebraic methods are used to solve geometrical, real-world problems.<\/p>\n\n\n\n

\u201cAlgebraic geometry is an area of mathematics that is considered very abstract. I find it nice to be reminded that any piece of mathematics, however abstract it might be, may turn out to have practical applications at some point,\u201d says Boutin.<\/p>\n\n\n\n

Full article information<\/h2>\n\n\n\n

\u2018Global positioning: The uniqueness question and a new solution method\u2019<\/a>, Mireille Boutin and Gregor Kemper, Advances in Applied Mathematics, (2024).<\/p>\n","protected":false},"excerpt":{"rendered":"

The summer holidays are ending, which for many concludes with a long drive home and reliance on GPS devices to get safely home. But every now and then, GPS devices can suggest strange directions or get briefly confused about your location. But until now, no one knew for sure when the satellites were in a […]<\/p>\n","protected":false},"author":2589,"featured_media":503964,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"advgb_blocks_editor_width":"","advgb_blocks_columns_visual_guide":"","footnotes":""},"categories":[84026],"tags":[21785],"location":[6763],"article_type":[43139],"serie":[],"archives":[],"internal_archives":[],"reboot-archive":[82795],"class_list":["post-486214","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data","tag-tu-eindhoven","location-netherlands","article_type-features","reboot-archive-data"],"blocksy_meta":[],"acf":{"subtitle":"In new research, TU\/e\u2019s Mireille Boutin has obtained major results that help improve GPS technologies using unconventional mathematical techniques.","text_display_homepage":false},"author_meta":{"display_name":"Team IO","author_link":"https:\/\/ioplus.nl\/archive\/author\/erikdevries\/"},"featured_img":"https:\/\/ioplus.nl\/archive\/wp-content\/uploads\/2018\/08\/GPS-verboden-Luik.jpg","coauthors":[],"tax_additional":{"categories":{"linked":["DATA+<\/a>"],"unlinked":["DATA+<\/span>"]},"tags":{"linked":["TU Eindhoven<\/a>"],"unlinked":["TU Eindhoven<\/span>"]}},"comment_count":"0","relative_dates":{"created":"Posted 2 years ago","modified":"Updated 2 years ago"},"absolute_dates":{"created":"Posted on September 1, 2024","modified":"Updated on September 1, 2024"},"absolute_dates_time":{"created":"Posted on September 1, 2024 2:40 pm","modified":"Updated on September 1, 2024 2:40 pm"},"featured_img_caption":"","series_order":"","_links":{"self":[{"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/posts\/486214","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/users\/2589"}],"replies":[{"embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/comments?post=486214"}],"version-history":[{"count":0,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/posts\/486214\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/media\/503964"}],"wp:attachment":[{"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/media?parent=486214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/categories?post=486214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/tags?post=486214"},{"taxonomy":"location","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/location?post=486214"},{"taxonomy":"article_type","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/article_type?post=486214"},{"taxonomy":"serie","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/serie?post=486214"},{"taxonomy":"archives","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/archives?post=486214"},{"taxonomy":"internal_archives","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/internal_archives?post=486214"},{"taxonomy":"reboot-archive","embeddable":true,"href":"https:\/\/ioplus.nl\/archive\/wp-json\/wp\/v2\/reboot-archive?post=486214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}