![]() Use variables and strings when programming in Python 2 ![]() Get the Screenshot Editing Feature from Android Pie on Any Phone Get Android 9.0 Pie's New Ringtones & Notifications on Any Phone Get Android Pie's Quick Settings & Notification Shade on Any Phone - No Root Needed Having Connection Issues on Android Pie? Turn Off 'Turn on Wi-Fi Automatically' Galaxy Devices Are Getting Better Edge LightingĮnable Samsung's New Dark Theme on Your Galaxy in Android PieĮnable Android 9.0 Pie's New Manual Dark Theme on Your Google Pixel Get Android 9.0 Pie's Volume Slider on Any Phone & Control Media Volume by Default Galaxy Devices Are Getting All-New Home Screen Icons Install Android 9.0 Pie Beta on Your Google Pixel or Pixel 2 Right NowĪdd Crossfade Transitions Between Video Clips in Adobe Premiere Clip Get the Pixel's Boot Animation on Any Android Samsung's Phone App Looks So Much Better on One UI Get started using a Verizon Droid 2 Google Android smartphoneĬheck spelling and grammar in MS Word 2010Įnable the New Multitasking Swipe Gestures in Android 9.0 Pie The Best-Looking Android App Animations of 2013Ĥ4 Cool New Features & Changes in Android 9.0 Pie Get New Transition Animations on Your Nexus 5 Google Added an iPhone-Style Text Magnifier to Android 9.0 PieĪdd Fade-Ins, Fade-Outs & Fade-Through Transitions to iMovie Projects on Your iPhoneĥ Legitimately Hidden Features in Android PieĪdd Fade Ins & Fade Outs to Videos in Adobe Premiere Clip 19 Harry Potter Spells Your Android Phone Can Cast Using Google AssistantĬover image and screenshots by Dallas Thomas/Gadget Hacks Related.Who needs a wand? Unlock your magical powers and transform yourself from a Muggle into a Wizard or Witch just by using your Android phone. Follow WonderHowTo on Facebook, Twitter, Pinterest, and Flipboard.Follow Gadget Hacks on Facebook, Twitter, YouTube, and Flipboard.Nonetheless, the new animations have a Material Design feel to them, so they're a welcome addition!ĭon't Miss: The Volume Buttons in Android 9.0 Pie Actually Control Media by Default Now This doesn't apply to home screen animations, which are still the same in the first developer preview build of Android Pie. Of course, some third-party apps will employ their own animations, but the new 9.0 Pie style is present in Google apps such as Chrome. These animations are present in menus everywhere, virtually any time you move from one activity to another within the same app. When you go back, the animation plays in reverse (from top to bottom).
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![]() It’s normal to ask for some sort of deposit or incremental payments (e.g., 30% at the start, 30% at the halfway point, and 40% after handoff) especially for longer projects. It’s always useful to write how long the project usually takes. Customers usually ask for quotes for projects that take more than one day. Prices for materials and workers may change over time, so you should always include an expiration date for your estimate. For example, if you’re working on building renovations, the local city council may require an inspection of your plans and materials. The number of hours required to complete a task.This section usually makes up the bulk of the estimate. Other Invoice Templates Proforma Invoice Template USA Invoice Template Billable Hours Consultant Invoice Template Construction Invoice Template USA. Try to be as descriptive as possible with these. Line items: breakdown the different parts of the project.Standard info: date, estimate number (for organizing and future reference), company info.Here’s a break down of what you need to include in your estimates: The templates are the same as the estimate templates here, except they say ‘quote’ in various places. Check out our dedicated quote template page. Whether you prefer to give out quotes or estimates, we’ve got you covered. They both contain the same sections and information. As a document, quotes and estimates are essentially the same. On the other hand, construction companies and programmers give out estimates. For example, graphic designers and photographers typically give out quotes. Which to use depends on your industry and the type of work you do. Templates can also calculate totals, taxes, etc. Many contain basic fields like date, invoice number, customer name and address, etc. Templates allow you to reuse the same format repeatedly, ensuring consistency. On the other hand, with an estimate, it’s understood that the end price may differ as the scope of the project becomes clearer. Well-designed invoice templates make creating and tracking invoices much easier. You can even add rows/columns for additional information. Feel free to customize the template according to your business needs. Since it’s designed with light colors, you can save ink/toner when printing it out. Even if you end up putting more time in than you expected. The invoice template PDF is the perfect version to use for those looking for a simple US invoice. Your customers may perceive that you are sticking to that price when you give out a quote. One difference is that quotes are often used for fixed-cost projects, whereas estimates are more of an educated guess. Notes: Include any additional info your customer should know, including terms of service and payment terms (for example, payments are due 30 days after the invoice has been issued).Quotes, pricing quotes, or sales quotes are very similar to estimates.Total: Outline the total amount due from the customer, after tax.This is legally required to provide on invoices, and your rate may differ depending on where you run your business. Tax: Indicate the tax rate applied to the subtotal.Subtotal: Add up the subtotal of your goods or services, before tax has been applied.For each line item, include a brief description, quantity, individual unit price, and total price. Line Item: Add individual line items for each unique good or service you provided.Use any of these free Excel and fillable PDF invoice templates to tailor billing to your business needs so that you get paid on time. Customize It: The first step is to customize the blank invoice template by editing the header with the name of your business, inserting a logo, etc. It is meant to be printed and filled out by hand, so it contains no formulas. Dates: Include the date when your invoice has been issued and the date when payment is due. This basic invoice template comes complete with billing and shipping addresses sections and an invoice number and date for easy tracking. This printable blank invoice template uses an ink-friendly design to save printing costs.Our Sample Templates are Perfect for Freelance and Corporate Services, and Business Products, Too With Designs and Formats Readily Available, No Need to Start From a Blank Word Document. For example, if you're sending your very first customer their first invoice, the invoice number could be 001-001. Keep Track of Bills and Payments with s Free Invoice Templates in Word. You can format this based on sequence and customer. Invoice Number: Include a unique invoice number to help you track down this invoice in the future.Customer Details: Under "Bill To", add your customer's name, address, and contact information.Company Details: Add your company name, address, phone number, and logo to the top-right corner.Title and Description: Name the project and briefly describe what type of work your client is being invoiced for. ![]() Whatever the case may be, they can also be used to cook your rice when you are camping. Some people like to bring charcoal grills of their own when they go camping because charcoal grilling produces a distinctly delicious flavor. Many campgrounds and picnic areas near camping locations come with little charcoal grills. ![]() Click here if you want to read that article. I also wrote an article detailing 13 unique ways to boil water while you are camping. Cast iron pots work best for this, and it may take a long time to cook this way, but it is fun.īasically, for cooking rice you mostly need to find a way to boil water in most circumstances. You can simply cook rice in a pot on top of the campfire grill. ![]() The grills can usually be adjusted to hover over the campfire at varying distances, which allows you to cook using the grill in a variety of ways. If you are lucky enough to be camping with a fire ring that comes with a grill attached to it then your cooking possibilities are so much larger than just a campfire alone. Another method is to simply place a pot with water inside the campfire and use the boiling water to cook instant rice. This works best with fast-cooking rice since the inner part of the campfire is often the warmest part. You can place a cast iron put inside of your campfire in order to cook your rice. Many kinds of food can be prepared with your campfire including rice if you are creative and diligent enough. Your campfire is a versatile and unique method of preparing your food when you are camping. Nevertheless, you can boil water to add to instant rice or cook rice fully using this method, and it just looks cool! Please be safe if you are going to try cooking this way since heavy pots full of boiling water can be dangerous when they are hung using an inadequate structure. This method of cooking is more advanced since you will have to build something sturdy enough and secure enough to hold a heavy pot or kettle full of water. Rice can be cooked over the campfire by hanging a kettle or pot with a handle. If you need some ideas for what kind of camping stove to get then you can check out this one on Amazon where I review the camping stoves that I personally use and enjoy. If you are more into backpacking and more wilderness camping where you are camped further away from your vehicle then a small backpacking camping stove may be more ideal based on its size and weight. If it is just you and someone else or just you solo camping most of the time then a single burner and more compact camping stove can be ideal for carrying less weight and preparing mostly one-pot meals. If you are just primarily car camping very close to your vehicle and you are preparing a large amount of food for a family or group then your camping stove should be more freestanding and have multiple burners. The type of camping stove and fuel that it uses doesn’t really matter too much, but the stove needs to be the appropriate size for you, your family, and your camping style. I think everyone that goes camping should have their own camping stove because of its versatility. You can also cook rice the old-fashioned way with rice and water in a regular pot. You can use boil in the bag rice in a pot of water. You can boil water in a pot or kettle and then add that boiling water to instant rice. You can prepare rice in several different ways using your camping stove. It’s safe, it’s quick, it’s affordable, and camping stoves can easily make some delicious camping meals. The method that is the easiest and most recommended for most campers that need to cook rice is to just use a camping stove. Read on to learn more about cooking rice when camping.īy the way, If you are in the market for a new camping stove, then you should click here to see the one I recommend on Amazon. How to cook rice while camping (11 Examples):Ĭooking rice is easy when you are camping, and rice can provide a lot of energy for your camping activities. It is a staple among campers since rice is low-cost, plentiful, non-perishable, and easy to prepare. ![]() Rice is one of the most versatile foods there is. ![]() To understand sound refraction better, let’s consider an example. The speed of sound is influenced by factors such as temperature, humidity, and the composition of the medium. This bending occurs due to the variation in the speed of sound in different mediums. When sound waves encounter a change in the properties of the medium they are traveling through, such as a change in temperature or density, they can be refracted or bent. In this article, we will explore the explanation of sound refraction and the factors that influence it. It refers to the bending of sound waves as they pass from one medium to another, resulting in changes in their direction and speed. Sound refraction is a fascinating phenomenon that occurs when sound waves travel through different mediums or encounter changes in temperature, pressure, or density. Refraction in Forests Sound waves bending and scattering as they pass through trees and foliage. Refraction in Buildings Sound waves bending and reflecting off buildings, causing echoes in urban environments. Refraction in Mountains Sound waves bending and echoing in mountainous regions due to the uneven terrain. Underwater Refraction Sound waves bending as they travel through water, affecting marine communication. ![]() Key Takeaways: Example Description Atmospheric Refraction Sound waves bending due to changes in temperature and humidity in the atmosphere. ![]() In this article, we will explore some examples of sound refraction and understand how it affects our perception of sound. This can lead to the bending or deviation of sound waves, creating unique auditory experiences. Refraction of sound occurs when sound waves change direction as they pass through mediums with varying densities. However, sound can also undergo refraction, resulting in interesting phenomena. When we think of refraction, we often associate it with light bending as it passes through different mediums. ![]() On the other hand, Actor-Observer Bias covers both attributions of others’ and one’s own behaviors. ![]() One’s own behaviors are irrelevant in this case.įor example, if someone trips and falls, we might call them clumsy or careless. The major difference lies between these two biases in the parties they cover.įundamental Attribution Error is strictly about attribution of others’ behaviors. Differences Between Fundamental Attribution Error and Actor-Observer Bias Various studies have indicated that both fundamental attribution error and actor-observer bias is more prevalent when the outcomes are negative. In both cases, others’ behaviors are blamed on their internal dispositions or their personality.Īnother similarity here is the manner in which the disposition takes place. The first similarity we can point is that both these biases focus on the attributions for others’ behaviors. But, before we dive into separating them apart, let’s look at few obvious similarities. These biases seem quite similar and yet there are few clear differences. Similarities Between Fundamental Attribution Error and Actor-Observer Bias It talks about the difference in perspective due to our habitual need to prioritize ourselves. The Actor-Observer bias is best explained as a tendency to attribute other people’s behavior to internal causes while attributing our own actions to external causes.Īctor-Observer bias discusses attributions for others’ behaviors as well as our own behaviors. Read more about Fundamental Attribution Error. It is strictly about attributions for others’ behaviors. So, fundamental attribution error is only focused on other people’s behavior. ![]() According to Monopoly’s actual rules, the player who collected your last dollar gets to take all of your properties. You turn in your properties to the bank when you go bankrupt.Ĭapitalism is much more cutthroat than that. The true benefit of the “Free Parking” space is that you won’t have to fork over your hard-earned money to some evil land baron if you end up there. We love playing with this rule, but if you’re playing by the book, you don’t get anything from landing on “Free Parking.” All those taxes and fines should go right back where they came from: the bank. Whoever lands on “Free Parking” collects the whole payout. ![]() Lots of Monopoly stalwarts pay taxes and fines into a collective pot in the center of the board. In the official rules, you have to take that card. Some players insist that you choose whether you want to collect a card when you land on the “Chance” square. It’s not part of the official rulebook, however, nor should it be The Landlord’s Game was designed with a boring endgame, and Monopoly essentially keeps this “feature” intact. Go ahead and play that way if you want-it makes the game go faster, and you’ll probably get into fewer arguments with your in-laws. Somewhere back in the mists of time, a mysterious innovator made up the Mercy Rule, which holds that a player who first reaches a predetermined personal wealth wins automatically. Monopoly can sometimes take all night to play. While you’re grappling with that, we’ve got bad news about the win conditions… 2. In the stark, capitalistic world of Monopoly, there’s no such thing as free money. That’s just a house rule someone made up a long time ago, and it stuck. That money is supposed to represent the wages you earn by the sweat of your brow, remember? Why would landing right on the space pay double? If you’re lucky enough to land right on the “Go” space, you can collect $400. ![]() Here are the “rules” that you won’t find in the official Monopoly rule book. Workers, who actually create value, should simply be able to collect their $200 and be on their way around the board.Īnyway, it’s not surprising that a game with such a secret history has led to a profusion of house rules and niche tweaks so widespread that very few people play the game with orthodox strictness these days. That line is direct from the thinking of Henry George, the 19th-century anti-monopolist who favored taxing land-owners for the value of their holdings. “Labor upon Mother Earth produces wages,” she wrote on her original board. If you want any more proof that Magie invented Monopoly as a clever bit of left-wing propaganda, consider the original text of what has become the “Go” space. People are still playing Monopoly more than 100 years later. “It might well have been called the ‘Game of Life,'” she continued, “as it contains all the elements of success and failure in the real world, and the object is the same as the human race in general seems to have, i.e., the accumulation of wealth.” Magie’s clear implication is that the capitalist insistence on wealth being the only measure of value is unacceptable.Īt the same time, she sure did make the struggle for financial dominance look fun. “It’s a practical demonstration of the present system of land-grabbing with all its usual outcomes and consequences,” Magie remarked. In fact, it’s purposely designed to be incredibly unfair and frustrating-much like capitalism itself, according to Magie. Yes, you read that right: Monopoly isn’t a celebration of American entrepreneurialism. Her deliberate goal was to highlight the evils of capitalism. The truth is that a left-wing feminist named Lizzie Magie invented the game in 1903 under the title The Landlord’s Game. In fact, by that point, Monopoly had been entertaining players for three decades. Here’s the company line (and we mean that literally Hasbro, which now owns the rights to the game, prints this story on the first page of the rulebook): Monopoly was born when Charles Darrow presented the game, fully formed, to Parker Brothers executives in 1934. Before we get into the rules, it’s helpful to understand that… The man who invented Monopoly did not invent Monopoly. We decided to look a little more closely at the history of Monopoly, and along the way, we discovered that we’ve been playing the game incorrectly-and chances are, you have, too. Monopoly is the worst the game never ends, no one knows the (actual) rules, no pathway to victory without cheating & scandal … /oanJvcPUbc That’s partly due to the inevitable arguments about the rules. While most households have at least one version of the ubiquitous board game, we’ll be the first to admit that Monopoly isn’t very fun (at least, not if you’re losing).Īccording to the instruction booklet, an average Monopoly game takes from one to four hours, but we can remember some games dragging on for days. ![]() The Law Offices of Nicholas Rose, PLLC offers free consultations. If you have been injured through someone else’s negligence, an experienced New York construction accident attorney can help you. Additionally, the plaintiff was able to show that the defendant already had the information and therefore suffered no surprise or prejudice. Plaintiff’s counsel received the information late, through no fault of the plaintiff or counsel. The facts here supported allowing the plaintiff leave to supplement or amend the bill of particulars. The court granted the plaintiff’s motion. ![]() Additionally, the court noted the fact that plaintiff’s counsel did not receive the records indicating those conditions until August 2015. The court therefore found that the defense could not argue surprise because its own doctor had informed it of the diagnoses. ![]() The plaintiff pointed out that the report by the defendant’s doctor from May 2014 referenced the diagnoses included in the supplemental bill of particulars. The court found that the defendant had failed to show undue prejudice and that the supplement merely elaborated on the extent of the injuries without setting out new injuries or a new theory of liability. The defendant argued that the plaintiff had waited until close to the time of trial before supplementing or amending the bill of particulars, but the court granted the plaintiff’s motion. The court cited previous case law that a bill of particulars can be supplemented with continuing damages only if the continuing damages are “an anticipated sequelae” of the injuries alleged in the original. Leave to amend should be freely granted absent prejudice to the other party. The court noted that the standard used in determining motions for leave to amend a bill of particulars is similar to that in CPLR 3025. The plaintiff then moved for leave to serve the amended bill of particulars. The response stated that it had been improperly served and was being “rejected and returned.” The response stated that the injuries described in the supplemental/amended bill of particulars had not been alleged previously. The defendant’s attorney responded that the plaintiff could not amend the bill of particulars without leave of the court. Additionally, it stated that the plaintiff remained totally disabled from employment and continued to have lost earnings, as alleged previously. This document stated that the plaintiff had left common peroneal neuropathy and left tarsal tunnel syndrome as a result of the injuries he had already alleged. In August 2015, he served a supplemental/amended bill of particulars. The report was dated June 5, 2013, but plaintiff’s counsel said he had not received it before. Plaintiff’s counsel stated that there was a report in the records that indicated left common peroneal neuropathy and left tarsal tunnel syndrome related to the injury of the plaintiff’s left leg. The plaintiff’s attorney requested and received updated medical records from the plaintiff’s treating physician around August 2015. The plaintiff was examined by the defendant’s doctor on May 2, 2013. His verified bill of particulars, served on July 12, 2012, indicated his injuries were fractures in his left leg and ankle. The plaintiff filed suit after allegedly being injured while working in construction. The Supreme Court of the State of New York, County of New York, recently considered whether a plaintiff could amend his bill of particulars in Lopez v. Thereafter, a party may amend or supplement a pleading by stipulation or leave of the court, which is to be “freely given upon such terms as may be just….” Defendants will often object to a plaintiff amending his or her pleadings, arguing that they would be prejudiced by the amendment. Under Rule 3025 of the New York Civil Practice Law and Rules, a party may amend a pleading without the court’s leave within 20 days after the pleading is served or within 20 days after service of the response. While ideally all pleadings and other court documents would contain full and accurate information, sometimes a plaintiff needs to amend when new information comes to light. ![]() The first example is a expert-crafted, guided template available in Aha! software. You can also show dependencies between tasks. You can track the status of each task by changing the color of the bar. This Gantt chart shows the key activities that make up the overall release plan, including the features that will be delivered. ![]() Gantt charts show all the phases of work required from all teams involved to meet a release deadline - serving as a more detailed companion to the product roadmap, which typically displays the general timeline for multiple releases. In fact, release planning is one of the most common use cases for Gantt charts in product development. Release planning Gantt charts are great tools for product managers. If you are interested in exploring what you can do with dynamic, cloud-based Gantt charts, check out this Aha! tutorial: Static slides and spreadsheets are simple to create but can become cumbersome to update over time. Jump ahead to the different templates here: Save and share your Gantt chart with your product development team and other stakeholders. Adjust their lengths and positions to match the start and end dates you have set with your team.Ĭonsider changing the colors of certain elements to emulate your company's branding or show status, categories, and other indicators. ![]() You will find options in the Gantt chart templates below for weekly, monthly, and quarterly schedules.īegin populating the rest of the template components with relevant information - like assignees, milestones, and products.Įach horizontal bar represents a phase of work. Update the template to match your time frame. (e.g., "Fredwin Cycling Portfolio Gantt Chart - Q1 2023) Reference the current timeframe so you can keep track of different versions you may create. But in general, here is how to get started:īrowse the templates below and select one based on the type of plans you want to document - strategy, portfolio, product, release, or project management. The process for filling out these Gantt chart templates varies somewhat by use case. How to use Gantt chart templates in Excel or PowerPointĮxcel and PowerPoint templates offer a way to make Gantt charts with tools you already use. Once you download your template, you can begin customizing your Gantt chart. There are 10 templates for strategic planning, portfolio planning, product management, release planning, and project management. To create a Gantt chart in Excel or PowerPoint, download any of the chart templates below. Gantt chart Excel templates or cloud-based Gantt charts make it easy to get started. ![]() on New Year’s Day.įor more information about DART routes and schedules call (214) 979-1111. Call (214) 515-7272 and choose Option 3 to schedule.ĭART’s Customer Information Center will be open from 6 a.m. Next day scheduling will be accepted New Year’s Day using DART’s voicemail booking system and automated booking feature, X-Press Booking. 29 customers needing service New Year’s Day can begin calling Tuesday Dec. Customers needing service for New Year’s Eve should begin calling Monday Dec. Regular subscription trips will be canceled on New Year’s Day and the Paratransit Scheduling Center will be closed. Paratransit will operate on Sunday schedule. DART administrative offices will be closed and TRE will not provide service. Green Line to Buckner departs Victory at 2:02 a.m.ĭART buses and light rail operate on a Sunday schedule. Green Line to North Carrollton/Frankford departs Victory at 2:29 a.m. Orange Line to DFW Airport departs Victory at 2:05 a.m. Orange Line to Parker Road departs Victory at 1:57 a.m. Red Line to Westmoreland departs West End at 2:18 a.m.īlue Line to Downtown Rowlett departs West End at 2:17 a.m.īlue Line to Ledbetter departs West End at 2:16 a.m. Red Line to Parker Road departs West End at 2:14 a.m. The last scheduled DART trains from the Big D NYE event are: Two-car trains will operate on the Red, Orange and Blue lines with three-car trains on the Green Line. Trains will then run every 30 minutes until 2 a.m. 25: DART administrative offices and Paratransit Reservations/Scheduling Center will be closed buses and light rail will be on a Sunday schedule, and TRE will have no service.ĭART buses and the TRE will be on a regular weekday schedule, but an extra westbound TRE train will leave 20 minutes after the Big D NYE fireworks at Victory Station.ĭART Rail has a regular weekday schedule until midnight. DART’s Customer Information Center will be open from 6 a.m. 24 and 26: DART, TRE and Paratransit will follow regular weekday schedules. Weekend travel to UNT DALLAS STATION starts at Downtown Rowlett Station and 8th & Corinth Station.Dallas Area Rapid Transit (DART) and the Trinity Railway Express (TRE) are operating special Christmas and New Year’s holiday schedules and also adding service for the Big D NYE event.ĭec. Weekend travel to CEDARS STATION starts at Downtown Rowlett Station. Weekend travel to DOWNTOWN ROWLETT STATION starts at UNT DALLAS STATION and CEDARS STATION. Weekend travel to FAIR PARK STATION starts at UNT DALLAS STATION. Weekend travel to 8TH & CORINTH STATION starts at UNT DALLAS STATION. Weekends includes travel to 8TH & CORINTH STATION, FAIR PARK STATION, DOWNTOWN ROWLETT STATION, CEDARS STATION, and UNT DALLAS STATION. ![]() Due to the expected poor conditions, DART recommends passengers avoid travel wherever possible. ![]() ![]() DART will operate 14 shuttle bus routes between rail stations. DART bus routes will operate on a Sunday schedule. Weekday travel to UNT DALLAS STATION starts at Downtown Rowlett Station and 8th & Corinth Station. It will suspend DART rail operations from Thursday, Feb. Weekday travel to CEDARS STATION starts at Downtown Rowlett Station. Weekday travel to DOWNTOWN ROWLETT STATION starts at CEDARS STATION and UNT DALLAS STATION. Weekday travel to FAIR PARK STATION starts at UNT DALLAS STATION. Weekday travel to 8TH & CORINTH STATION starts at UNT DALLAS STATION. Weekdays includes travel to 8TH & CORINTH STATION, FAIR PARK STATION, DOWNTOWN ROWLETT STATION, CEDARS STATION, and UNT DALLAS STATION. ![]() So, to help us think about that I've copied and pasted this on my scratch pad and we can draw through it and the first thing that we might wanna think about is if you rotate, I've talked about this So, this would be 270 degrees in the counter-clockwise direction. ![]() The direction of rotation by a positive angle is counter-clockwise. We have this little interactive graph tool where we can draw points or if we wanna put them in the trash we can put them there. Triangle SAM, S-A-M, and this is one over here, S-A-M, is rotated 270 degrees, about the point four comma negative two. But it's easy to calibrate it if you want to specify another point, around which you want to rotate - just make that point the new origin! Figure out the other points' coordinates with respect to your new origin, do the transformations, and then translate everything back to coordinates with respect to the old origin. Now, since (a, b) are coordinates with respect to the origin, this only works if we rotate around that point. Rotating (a, b) 360° would result in the same (a, b), of course. Because the axes of the Cartesian plane are themselves at right angles, the coordinates of the image points are easily predictable: with a bit of experimentation, you could easily 'prove' to yourself that rotating (a, b) 90° would result in (-b, a) rotation of 180° gives us (-a, -b) and one of 270° would bring us to (b, -a). Here's my idea about doing this in a bit more 'mathematical' way: every rotation I've seen until now (in the '.about arbitrary point' exercise) has been of a multiple of 90°. |
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