![]() ![]() If we venture back to 2014, the last time the Open was held at Royal Liverpool saw Rory win his only Open crown. His near miss in Los Angeles aside, this could be the year the man from the West Coast finally lifts the Claret Jug, the trophy for the tournament. The Open Championshipįor now, Rickie will be focusing on the last major of the year, the British Open. Maybe the bogey at 18 late Saturday evening was the precursor to the eventual happenings on Sunday. Quickly after, Clark complained they couldn’t see shots due to the lateness in California. On Saturday at roughly 11:00 PM Eastern time, Rickie and tourney champion, Wyndham Clark, completed their round. His birdie at 8 caused a stir at the Los Angeles Country Club as Fowler made it to -8, but the four bogeys and one birdie on the back saw him drop to a tie for 5th place at five under par.įowler and others had a legitimate beef with the tee times for the entire tournament. ![]() The crowd tried to will him back into the lead. His first seven holes on the final round were completed in +3, and he dropped quickly off the top of the leaderboard. A HISTORIC OPENING ROUND! #USOpen #GoRickie /LKPHwmgOCUįowler followed up his amazing Thursday with a 68 and 70 on Friday and Saturday, respectively. OPEN HISTORY, WITH 10 BIRDIES! RICKIE FOWLER TIES MAJOR CHAMPIONSHIP HISTORY WITH A 62! RICKIE FOWLER IS CURRENTLY ALONE AT THE TOP WITH A ROUND 1 -8! ✅☢️□☢️ He carded a 62 in a US Open! Never been done, and yet Xander Schauffele completed the same thing just 22 minutes later! Rickie started off doing things this weekend that NO GOLFER has ever done. Being able to go to something that has a little bit more loft, that can definitely help with the side-to-side dispersion, because once you go down to loft, you can get the ones that kind of squirt right or knuckle left. The LS is a fairly low-spinning head, which has been great for me. A post shared by actually went up about a half-degree (in loft) from where I was.
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![]() For the first day of the month, you use the ordinal number “ 1 st ”.For the day of the month, you use the cardinal numbers 2 to 31.Here's the basic format to use when saying the date in French: Just like the days of the week, the first letters of the months of the year ( les mois de l'année ) are also not capitalized in French. Try to practice pronouncing the words by using the pronunciation guide in the third column. Unlike in English, the first letters of the days of the week ( les jours de la semaine ) aren't being capitalized in French. The days of the week and months of the year On est le combien aujourd'hui? What's today's date?Ģ. Quelle est la date aujourd'hui? What's the date today? Let's start with how to ask for the date. It is also a very basic thing to learn when studying the French language. This nifty little guide will save you a lot of confusion when planning appointments and travel arrangements in French-speaking countries. So we've compiled here the most important things for you to know. No French lesson on telling time is complete without discussing days and months. Let's wrap up the lesson on time with some very useful videos for you to watch: Try to practice the pronunciation included in the last column. Here are some extra French phrases for you to take note of. Additional French phrases related to time Some variations in expression between regular French and 'street' or French slang.Ħ.To say 'half past' you add et demie to the end of the time phrase.įor example: Neuf heures et demie (or half past nine)įor example: I l est minuit.When using the specific number of minutes, there is no need to use “et”. “et” can only be used when saying quarter past or half past.“moins” is the blanket term used to say “minutes before” an hour.įor example: it's ten to eleven il est onze heures moins dix (It is eleven hours minus ten).It's a quarter to five: il est cinq heures moins le quart (Or literally: It is five hours minus quarter) Here are a couple of examples on how to say it: The terms “moins” (which literally means minus), and “et” (which means and) are being used. Just like in English, expressions such as “quarter to.” or “quarter past.” are also used in French.The common expressions used in telling time If you are going to write down minutes only, use mn.ĥ. In English, we use a colon ( : ) to separate hours from minutes, such as 11:45. Note: une heure is without s because of its singular form. This format applies to both 12-hour clock and 24-hour clock. The format goes like this: Il est (insert number of hours) heure(s) (insert number of minutes). Say: Quelle heure est-il? It is pronounced as kell eurh eh teel? Meaning, what's the time? Or what time is it? Here are the important things you need to know. So let's dive to the nitty-gritty right away. Whether you are simply learning the language so you could use it on your upcoming travel to a French-speaking country, or you're planning to learn it all the way, telling the time in French helps you avoid confusion in setting up meetings, appointments, or in keeping track of the time. Knowing how to properly ask for and tell the time in French is one of the most basic things you need to know. If not, you can brush up on your French numbers by checking this link - The Easy Guide to French Numbers. These are all just a big challenge though, and we're going to breeze through it all with simplified rules and easy steps by using this guide.īefore we proceed to tell how to say time in French, make sure you are already familiar with French numbers and how they work. Third, expressions are heavily used which could also add to the complexity. The word “heure” which is used in telling time is pronounced in one smooth flow with the letter before it, spoken in the same breath as the number it follows. Second, the “liaisons” in the French intonation or that flowing pronunciation make it all sound so confusing. In English, the 12-hour clock is more often used, and the military time is usually reserved for, well, the military. So okay, who am I kidding? It's actually A LOT more complicated, in fact.įirst, French uses military time or the 24-hour clock for public use or official purposes, while the 12-hour clock is mostly used for personal consumption. Telling time in French could be a tad bit more complicated than in English. ![]() What's the difference between a small and large SUV? Popular examples include the Volvo XC40 and the Volkswagen T-Roc. However, other than an increased ride height and commanding driving position, they're normally no better equipped for off-road use than a hatchback or saloon model. While there are no hard and fast rules (sizes and shapes can vary greatly between manufacturers), small SUVs are normally slightly larger and have a conventional SUV shape. Examples include the Honda Jazz Crosstar and the Audi Q2. ![]() What's the difference between a compact and small SUV?Ĭompact SUVs are typically the smallest models available, and are normally little more than regular small hatchbacks given mildly raised suspension and beefed-up body styling to give a more rough-and-tumble look. If you’re not already a member, join Which? to unlock our recommendations below and all of our expert impartial reviews. Which? members can log in to see the small SUVs and crossovers we recommend. The small petrol, diesel or hybrid SUVs we've highlighted in the table below strike a good balance of passenger and load space, ride comfort, fuel economy and reliability, so you can buy with confidence. But for the most part, it's best to think of a crossover as a small, road-biased SUV. The term is also often used to describe medium-sized SUVs that do without the off-road hardware of proper 4x4s, as well as high-rise MPVs, and other models that blend features of two traditional types of car. It's typically used to describe smaller SUVs, which are a crossover between a conventional small hatchback and a full-sized 4x4. The term 'crossover' is a fairly recent addition to the new-car lexicon. Scroll down the page for the very best small SUVs and crossovers we've tested, including new EV and hybrid models. In fact, one of the most popular models we’ve tested falls short against its rivals due to high CO emissions at motorway speeds – something the official tests don’t cover. Pick the wrong one and you could end up driving a car with mediocre fuel economy, middling driving experience, little safety equipment and so-so passenger space. However, despite cars in this class typically following a similar recipe, our tests continually find models that don’t meet expectations. They can also typically be powered with smaller engines than larger full-sized off-roaders, so you can have the tough aesthetic and road presence of a 4x4 without the higher running costs of a full-fat SUV. The best small SUVs and crossovers give you a commanding high-rise view of the road, a practical and spacious interior, and large door openings. ![]() If you have an animal you can no longer care for, contact Fort Wayne Animal Care & Control so that it can be properly screened before being rehomed.ĭon't treat this subreddit like OKCupid, either. If it's a story which is likely to be uniquely interesting to people in this subreddit, or likely to encourage a lot of comments, by all means post, but we will be culling news articles from the front page when they start dominating and pushing more thoughtful discussion off of the page.ĭon't offer a free pet on this subreddit.
First, she unbuttoned her bra and Julia Fox’s nude boobs jumped out. And you will definitely drool looking at how she undresses in front of the webcam on the orders of the guy. KANYE West’s new girlfriend Julia Fox went totally nude for her role as a web cam girl in the racy indie film PVT Chat before her romance with the rapper. ![]() ![]() Last year, hot brunette Julia Fox starred nude in the PVT CHAT drama. The influence of celebrities and movies extends far beyond the entertainment industry, with many actors and actresses using their platforms to bring attention. Leaked Julia Fox Nude & Blowjob, Masturbating In PVT CHAT. Nude Video Celebs Julia Fox Nude PVT Chat. We have a free collection of nude celebs and movie sex scenes which include naked celebs, lesbian, boobs, underwear and butt pics, hot scenes from movies and series, nude and real sex celeb videos. Julia Fox Photography, free sex galleries julia fox nude photos thefappening, julia fox nude photos uncovered. From classics to modern blockbusters, movies or even social media nude scenes have the power to captivate audiences and take them to new worlds. AZNude has a global mission to organize celebrity nudity from television and make it universally free, accessible, and usable. PVT Chat premiered at the Fantasia International Film Festival on August 21, 2020. It stars Peter Vack as an online gambler who becomes obsessed with a cam girl played by Julia Fox. There are countless nude celebrities that have become famous and everyone looks for them, with the actors and actresses who starred in them becoming household names. PVT Chat is a 2020 erotic drama film, written and directed by Ben Hozie. Many celebrities have used their fame to launch successful careers in film, while others have used their acting skills to catapult themselves to superstardom about times the nude or even sex scenes can make the actors… be more famous like the videos on this website □. From the early days of Hollywood to the present, actors and actresses have become some of the most recognizable and beloved in the world. Julia Fox nude and explicit scenes from PVT CHAT (2020) drama. Here the best scenes are Julia Fox nude - PVT CHAT (2020) Explicit Sex Movie Scene (and all masturbation scene). in case you don’t like or not tolerant to nude and famous women. You are browsing the web-site, which contains photos and videos of nude celebrities. Julia Fox Nude Scenes From PVT Chat on Facet Porn and you can also watch other webistes NUDEJIHAD and CELEBPORNERĬelebrities and movies have always been intertwined and in all movies or series there all always a nude or sex Scene in popular culture. Julia Fox in Pvt Chat (2020) Julia Fox in Pvt Chat (2020) The age of the celebrity during this appearance is being counted automatically and might be approximated. So to keep it safe and secret, she tells him she lives in San Francisco, but that illusion is dashed when Jack sees her walking down the street in Chinatown NYC, just a couple blocks from his apartment! Now Jack becomes what ever cam girl fears, a stalker who knows where she lives, knows she has a real life boyfriend, and still craves her attention.□ More about this video □ Julia Fox Nude Scenes From PVT Chat When Jack becomes a return customer to Scarlet, they both begin to let their collective guards down a bit as they seemingly have plenty in common, outside of him pretending to be her "human ashtray". Scarlet is a pro at dominatrix style web cam shows for her adoring fans, where she dons a leather body suit and smokes cigarettes while doling out some humiliation for the gents who are "tipping" enough. Julia Fox has hot masturbating scenes and shows the big bare boobs. ![]() Enter his favorite Cam Girl, Scarlet ( Julia Fox). Julia Fox is nude in the movie PVT Chat which was released in 2020. So Jack does what any lonely person does, he seeks out the company of others, and the only way he knows how to open himself up is by talking with complete strangers online. His landlord doesn't like him and wants him out, his roommate killed himself, and he doesn't really have anyone to connect with on a personal level, outside of the online gamblers he makes his living off of. To put it lightly, Jack ( Peter Vack), is a loner in a city full of people who don't want to interact with him on any level. ![]() It won the prestigious Advanced+ Malware Protection Award from AV-Comparatives in March 2022 and the Advanced+ Performance Award in April 2022.Norton secured 100% virus detection in tests from SeaLabs and tests from AV-TEST.We should also note that Norton AntiVirus backs up its security claims with impressive results: The sheer variety of protection on offer means that you can pick and choose the ones that you need for your specific lifestyle. Norton 360 comes with a ton of security features. To make the comparison simple, we’ve laid out each package side by side and indicated which features they include. If you’re on the fence, the table below might help you decide which subscription plan to choose. While it’s very expensive, it offers some incredible features, such as bank and credit card alerts as well as fictitious identity monitoring. Norton 360 with LifeLock Ultimate Plus is a good option for those who require additional privacy, data protection, and investment-related features. While Norton Secure VPN isn’t the best VPN, it’s a better option than being unprotected. You’ll also get some “free” cloud storage and access to the password manager. It has the benefit of including a VPN in case you don’t already have one. The Norton 360 Standard plan should suit most users. At $49.99 for the first year, it can be a little pricey for first-time users. On the other hand, Norton 360 Deluxe packs some useful additional apps that create a fairly comprehensive security suite. Norton AntiVirus Plus is the most basic of the bunch and offers the minimum security features we’d typically see in an antivirus. Norton offers a fairly wide selection of packages at different price points. Norton AntiVirus Specifications and Subscriptions Otherwise, check out our full Norton AntiVirus review for a detailed breakdown. You can discover the latest offers on Norton AntiVirus using the button below. Though a little on the expensive side, we felt the price was fair when considering the litany of features that Norton 360 offers. ![]() Any problems that arise can be resolved swiftly by contacting Norton’s customer service, which is available 24/7 via live chat.įor our Norton 360 review, we tried out Norton 360 with LifeLock Select, which is priced at $99.99 for the first year. Whether you’re on desktop or mobile, Norton makes it easy to use the full suite of security features, as well as any additional apps you choose. Each app adds further value to the Norton 360 AntiVirus Suite.īoth beginners and advanced users will be able to navigate the well-designed user interface. These include a VPN (Virtual Private Network), a password manager, and a cloud backup service. Not stopping there, in the past three decades, Norton has delivered a host of fine-tuned security apps. It has a track record of consistent and constant protection against various security threats, as reflected in near-perfect scores across different test metrics. We routinely test new and existing antivirus solutions, so we can confidently say that the Norton 360 Security Suite is the best antivirus software currently available on the market. ![]() Of course, they are still great and relevant but the Layout Builder is a built-in solution that is created with a focus on user-friendliness. The module is meant to replace the developers’ favorite page building tools such as the Paragraphs, the Panels, the Panelizer, or the Display Suite contributed modules. By content pages we do not mean specifically content nodes - the module is used for all kinds of structured, or fieldable Drupal entities. Welcome to take a look at how the Layout Builder works! What is the Layout Builder module in Drupal 8 and 9?ĭrupal’s Layout Builder module is a drag-and-drop core tool for creating the layout of content pages on Drupal websites. Since then, it has become stable and enriched with more capabilities for both editors and developers. The Layout Builder jumped into the Drupal core in version 8.5 as an experimental module. Open up package.Today, we will discuss one of the powerful tools that transformed authoring experiences in Drupal 8 and 9. In the sub-theme create another file called and add the following to it: let mix = require('laravel-mix') Ĥ. Then install Laravel Mix with the following command: npm install laravel-mix cross-env -save-devģ. Just follow the prompts and you should end up with a package.json file.Ģ. Go into your sub-theme and run the following command: npm init -y We’ll use Laravel Mix to compile Sass which I find the easiest to set up.īefore you begin, make sure you download and install Node.js and you can run the npm command.ġ. You’ll need to add the correct blocks into the right regions or the site will look broken. Once the sub-theme has been generated go and install and set as default.Ĭonfigure the settings like the image below then click on “Save configuration”. The only way I was able to get this library was by installing Drush using Composer, composer require drush/drush. NOTE: In the video, this step didn’t work because I got a fatal error because I didn’t have the Symfon圜omponentFilesystemFilesystem library. Enter in details into the form, then click on the Create link. Click on Settings, and scroll to the bottom and expand the “Subtheme” section.ģ. We’re going to look at using it via the settings form.Ģ. The Bootstrap 4 theme allows you to create a sub-theme in two ways using a script or via the settings form. lando init -source cwd -recipe drupal8 -name bs-build -webroot web -full Download Modules and ThemesĬomposer require drupal/ds drupal/field_group drupal/bootstrap4 drush/drush Install Modulesĭrush en ds field_group media_library layout_builder Sub-theme Generate Sub-theme In this video, we used Lando for our local environment. composer create-project drupal/recommended-project FOLDER_NAME -no-interaction Set Up Lando Create Card Row Section in Layout Builderįirst go download Drupal using Composer.Enable Layout Builder “Basic page” Content Type.Get a copy of the built site from GitHub. So here are the show notes for the video. ![]() We’re not using paragraphs in this video. We built the three-card components using a custom block type and layout builder. In this first video, we’re focusing on the three-card components below the homepage carousel. To help with the build process we’re going to follow a template and in each live stream session, we’ll build a component from the template.Īnd the plan is to have a semi-complete website after a few live streams. Drupal Live Site Build (Part 3) – Create Bootstrap Grid using Views and Display Related Contentīelow are the show notes for the first video in a series on building a Drupal site using layout builder and Bootstrap.Drupal Live Site Build (Part 2) – Create Layout Builder Section, Bootstrap Carousel.Drupal Live Site Build (Part 1) – Project Set Up, Build Bootstrap Card Component using Layout Builder.01:48:30 Add custom styling for card component.01:20:32 Switch on Field Templates in Display Suite.56:33 Configure Layout Builder on “Basic page” content type.52:28 Create card block type and attach fields.Don’t forget to subscribe to our YouTube channel to stay up-to-date. This could be every 1-2 hours for some toddlers! If your one year old is nursed, rocked, bounced, or helped to sleep in any way at bedtime, he’s going to need your direct help each time he wakes at night. (We all wake throughout the night, it’s completely normal.) The way that your child falls asleep at bedtime is the way he needs to fall back asleep every time he wakes in the night. ![]() If your 1 year old sleeps 12 hours total everyday, but has frequent meltdowns or is clingy or fussy, then try to add in more sleep. Your toddler’s moods are the best way to gauge if he’s getting enough sleep each day. (More on 1 year old nap schedules below.) This can be divided between 2 naps or taken as one long midday nap. Your 1 year old needs 2-3 hours of napping each day. Speak to your pediatrician if you suspect your toddler may still need night feeds.) (The only exception is toddlers with weight gain or growth concerns. Ideally none! Toddlers can be expected to sleep through the night without feeding. Your 1 year old needs 10-12 hours sleep each night. Research shows that a consistent and age-appropriate bedtime leads to longer sleep, less aggression and better attention in children. Here is what you can expect your one year old’s sleep to look like:Īn age-appropriate bedtime for your 1 year old is 6.30 – 8 pm. If you want to get your one year old sleeping well, it helps to have a goal and know what you’re working towards. And it guides you through the most common sleep problems that your one year old may encounter (like waking early, fighting bedtime, the 18 month sleep regression, and more!) This article will show you how to get your one year old sleeping great, day and night. On the contrary, they can persist for quite a long time unless you step in and actively show your one year old how to sleep well. This doesn’t mean that sleep problems magically disappear on your child’s first birthday. I’ve found that 6-12 months old is usually the most disturbed period of sleep for babies (due to development, teething and a whole lot more.) But the good news is you’re past it! Watching my toddler become more comfortable on her feet, learn to communicate better and grow more independent melts my mama heart. “How can I stop my toddler waking at night?”. ![]() “What’s the best way to handle separation anxiety?”.“What’s the 18 month sleep regression?”.“My toddler is fighting bedtime? What do I do?”.“When should we transition to a toddler bed?”.“How do I transition my toddler to one nap?”.“My toddler’s waking at 4-5 am everyday. The papers can be made with so many fun templates of your favorite animals and the star war heroes too. You can make as many as you want to gift around too. The papers are folded into a mini triangle shape and then placed at the corner of the book page to look just so adorable. Origami Triangle Corner BookmarksĪdd some beauty and appeal to your books with these elegantly fancy paper bookmarks. See the details here thatkidscraftsite 10. ![]() The papers are folded into the hearts and then glued to the colorfully painted Popsicle sticks to be the cutest bookmarks ever. If your kids are at the beginning of the paper art then making these cute DIY origami heart bookmarks would be a great practice. This lovely and chirpy shark looks just so fun and would make the mood of kids and adults in just minutes. DIY Origami Shark Bookmarkĭo you have some blue, white and black origami paper lying around? If yes then making this lovely DIY origami shark bookmark is the best thing to make out of these papers. So grab some colorful paper and permanent marker. Like Dream, Marry me, Girl power and more. The colorful heart paper origami bookmarks can say a lot to the person you are going to gift to. These heart-shaped origami bookmarks are the best gifts on casual days with so many fun uses. Using some colorful papers and some of the white you can easily create these lovely monster bookmarks to use with so much fun and excitement during the reading hours. Monsters can be cute too and these origami monster bookmarks are the real proof. You can use plain paper or the papers with designs and patterns per your desires. With a few folds and twists to the paper, you can create a triangular-shaped bookmark to use it on the corners of your book page. Simple Origami BookmarksĪ square piece of the paper can make the best of the projects for you like this lovely bookmark. These monster eye and teeth origami bookmarks are super easy to make. ![]() With ideas like this easy DIY origami bookmark, you can enjoy folding the paper and create such an interesting and useful thing out of it. Paper folding is one of the finest arts and here is how you can have acute fun with it too. Also, this bat corner bookmark would be an exciting thing for the little readers too. By making this absolutely stunning DIY bat corner bookmarks you can create a fun factor in your book reading hobby. Origami paper is so trending on the internet these days and you can make the most out of this paper art too. These would make a great gift idea for the kids around. You can print them out on the paper with a bunch of your favorite animals and then use them as the cute corner bookmarks. You would fall in love with the book reading more when you have these gorgeous woodland animals’ origami bookmarks. With a few folds and twists, you can easily make this pretty origami bookmark and use it as the best gift too. ![]() Make Your Own Origami Bookmarkīookmarks are the fun things to make on your own and the fun is doubled when you can make them out of the origami paper. These origami bookmarks would make a great gift as the party favors too. Another thing you can do with these lovely and lively bookmarks is turning them into a beautiful conversation with some lovely sayings. The kids and adults would equally adore these wildlife origami bookmarks at the corners of their book pages.įor the shapes of the bookmarks, you can fold the papers in a mini triangle or the heart design. ![]() The origami bookmarks with animal and bird theme templates are just so damn cool and cute. Paper folding would be so much fun with these origami bookmarks, taking you on a journey of exciting and creative paper art.Įven though you can make these little cute origami bookmarks with the plain but colorful pages but the creative ones are just so stunning. After looking at these absolutely gorgeous, colorful, cute and versatile DIY bookmarks you are never going to see the bookmarks as the page reminders only. The origami bookmark lets you remember the page you left reading till the next time but now they are much more than that. If you are a craft lover and an avid reader then you are surely going to fall in love with these 15 DIY origami bookmarks. ![]() Weitere Aspekte der neuen diskreten Orthogonalitätzbedingung werden in der Einleitung behandelt. Wir klassifizieren und integrieren die zugehörige Laguerre-geometrische Verallgemeinerung der Schachbrettinkreisnetze. Inkreisnetzte sind durch zwei Folgen von Geraden in der Ebene mit der Kombinatorik des Quadratgitters gegeben, so dass alle elementaren Vierecke einen Inkreis besitzen. Wir geben eine Vielzahl von konkreten Beispielen an, insebesondere eine Parametrisierung durch Jacobi elliptische Funktionen.Įin besonderes Beispiel von diskreten konfokalen Koordinaten im zwei-dimensionalen Fall ist durch Inkreisnetze gegeben. Verschiedene Folgen entsprechen verschiedenen diskreten Parametrisierungen. Es wird gezeigt, dass diese diskreten konfokalen Koordinaten durch eine äquivalente geometrische Konstruktion durch Polarität in einer Folge von klassischen konfokalen Quadriken charakterisiert ist. ![]() Diese Definition führt zu einer größeren Klasse von Netzen als im ersten Ansatz und beinhaltet beliebege Umparametriesierungen. Wir übertragen diese beiden Eigenschaften auf eine weitere Definition diskreter konfokaler Koordinaten, wieder unter Verwendung der genannten neuen diskreten Orthogonalitätsbedingung. Die Koordinatenfunktionen sind explizit durch gamma-Funktionen gegeben.įür den zweiten Ansatz zeigen wir zunächst, dass klassische konfokale Koordiatensysteme bis auf Umparametrisierung entlang der Koordinatenlinien durch Orthogonalität und die Faktorsierbarkeit bereits charakterisiert sind. Die konstruierten Lösungen sind diskrete Koenigs-Netze und durch eine neue diskrete Orthogonalitätsbedingung gekennzeichnet. In dieser kumulativen Dissertation werden zwei Ansätze zur Diskretisierung konfokaler Koordinaten sowie der Zusammenhang zu Schachbrettinkreisnetzen behandelt.ĭer erste Ansatz begründet sich auf einer integrablen Diskretisierung der Euler Poisson-Darboux-Gleichung. Konfokale Quadriken bilden ein Beispiel für orthogonale Koordinatensysteme. These include discrete Lamé coefficients, discrete focal nets, discrete parallel nets, and discrete isothermicity, as well as the relation to pairs of circular and conical nets. Thus, thirdly, we classify and integrate the class of checkerboard incircular nets, which constitute the Laguerre geometric generalization of incircular nets.įurther aspects of the novel discrete orthogonality constraint are studied in the introduction of this thesis. We give several explicit examples, including parametrizations in terms of Jacobi elliptic functions.Ī particular example of discrete confocal coordinates in the two-dimensional case is closely related to incircular nets, that is, congruences of straight lines in the plane with the combinatorics of the square grid such that each elementary quadrilateral admits an incircle. ![]() Different sequences correspond to different discrete parametrizations. We show that these discrete confocal coordinate systems may equivalently be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. In comparison to the first approach, this definition results in a broader class of nets capturing arbitrary reparametrizations also in the discrete case. We use these two properties to propose another discretization of confocal coordinates, while again employing the aforementioned discrete orthogonality constraint. Secondly, we show that classical confocal coordinates and their reparametrizations along coordinate lines are characterized by orthogonality and the factorization property. The coordinate functions of these discrete confocal coordinates are explicitly given in terms of gamma functions. The constructed solutions are discrete Koenigs nets and feature a novel discrete orthogonality constraint defined on pairs of dual discrete nets, as well as a corresponding discrete isothermicity condition. In this cumulative thesis we propose two approaches to the discretization of confocal coordinates, and study the closely related checkerboard incircular nets.įirst, we propose a discretization based on factorizable solutions to an integrable discretization of the Euler-Poisson-Darboux equation. Confocal quadrics constitute a special example of orthogonal coordinate systems. |