How to Rotate a Shape
What Does it Mean to Rotate a Shape?
Rotating a shape means to change its direction by turning it. The size and shape do not change during rotation. A shape is often rotated about a specific point called the centre of rotation. We also need to know what angle and direction the shape is rotated in.
For example, this triangle has been rotated 90° counter-clockwise about the point.
You can tell that a shape has been rotated because it is not facing the same direction as it was originally. The original shape is called the object and the rotated shape is called the image.
When rotated, a shape remains the same distance away from the centre of rotation. It is just in a different direction.
Rules for Rotating a Shape About the Origin
The rules for rotating shapes using coordinates are:
|Clockwise rotation angle||Counter-clockwise rotation angle||Rule|
|90°||270°||(x, y) → (y, -x)|
|180°||180°||(x, y) → (-x, -y)|
|270°||90°||(x, y) → (-y, x)|
How to Rotate a Shape by 90 Degrees
To rotate shape 90° clockwise about the origin, all original coordinates (x, y) becomes (y, -x). To rotate a shape 90° counter-clockwise about the origin, the coordinates (x, y) become (-y, x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.
For example, use the rule (x, y) to (y, -x) to rotate the shape 90° clockwise.
To use this rule, simply switch the (3, 1) to (1, 3) and then make the 3 negative to get (1, -3).
How to Rotate a Shape by 180 Degrees
To rotate a shape by 180° clockwise or counter-clockwise, the rule is to replace the (x, y) coordinates with (-x, -y). For example, a coordinate at (3, 1) will move to (-3, -1) after a 180° rotation.
Simply multiply each coordinate by -1 to rotate a shape 180°.
If a coordinate is negative, it will become positive after a 180° rotation. For example, the coordinate (-1, -4), will move to (1, 4) after a 180° rotation.
How to Rotate a Shape by 270 Degrees
To rotate shape 270° clockwise about the origin, all original coordinates (x, y) becomes (-y, x). To rotate a shape 270° counter-clockwise about the origin, the coordinates (x, y) become (y, -x). Simply switch the x and y coordinates and multiply the coordinate with the negative sign by -1.
For example, to rotate the point (3, 1) 270° clockwise, it becomes (-1, 3). Simply switch the x and y coordinates to get (1, 3) and then multiply the 1 by -1 to get (-1, 3).
How to Rotate a Shape Using Tracing Paper
To rotate a shape using tracing paper:
- Place the tracing paper over the shape and draw around the shape.
- Draw an arrow from the centre of rotation pointing upwards.
- Keep the pen over the centre of rotation and rotate the tracing paper.
- Stop when the arrow is facing either right (for 90° CW / 270° CCW turn), down (for 180° turn) or left (for 270° CW / 90° CCW turn).
- Draw the shape in this new position below the tracing paper.
The easiest way to rotate a shape is to use tracing paper.
For example, use tracing paper to rotate the shape 90° clockwise about the point.
After a 90° clockwise rotation, the upwards arrow is facing right. Use the grid lines of the paper to help line up the arrow correctly, ensuring that it is completely horizontal.
Then draw in the shape below.
Here is another example. Rotate the shape 270° clockwise about the point using tracing paper.
Every lot of 90° is equivalent to a quarter turn. 270° is 3 lots of 90° and so, 270° is equivalent to three-quarters of a turn.
The upward-facing arrow will be facing to the left after a 270° clockwise rotation.
We draw the shape in below.
How to Rotate a Shape Without Using Tracing Paper
To rotate a shape without tracing paper, draw horizontal and vertical arrows from the centre of rotation to each corner of the shape. The new corner can be found by rotating each of these arrows according to the following rules:
|Original Direction||90° CW / 270° CCW||180° CW / 180 ° CCW||270° CW / 90 ° CCW|
For example, rotate the shape 90° clockwise without using tracing paper.
The first step is to draw horizontal and vertical arrows connecting the centre of rotation to a corner on the shape.
The corner selected below is one square right and one square up from the centre of rotation.
Using the rules above, a right-facing arrow will be facing down following a 90° clockwise rotation. An upwards-facing arrow will be facing right after a 90° clockwise rotation.
Instead of the corner being one right and one up, it will now be one down and one right.
Using the new position of this corner, the rest of the shape can be drawn in. The same process can be repeated for all corners to check the result.
Here is another example. Rotate the shape 270° clockwise without using tracing paper.
Draw horizontal and vertical arrows from the centre of rotation to each corner.
After a 270° clockwise rotation all upwards-facing arrows will be facing left and all left-facing arrows will be facing down.
Once all corners are drawn in their new positions, the rotated shape can be drawn by connecting these together.
3 Ways to Rotate a Shape
These 3 methods to rotate a shape were super helpful for my students!
Rotating a shape can be a difficult concept for both students and teachers. After seeing my students struggle with this topic, I came up with a few strategies to make rotations easier.
These 3 strategies work with different levels of learners. I encourage you to try all 3!
This article contains affiliate links to products. I may receive a commission for purchases made through these links.
1. Use Patty Paper
Patty paper or wax paper is useful for so many math concepts! I always had plenty of patty paper in my classroom. This is the patty paper I use from Amazon. I love using patty paper to rotate a shape because it helps visual learners “see” the rotation before actual graphing it.
For this method, you will place the patty paper over the graph and trace the shape. You will also plot the origin (0,0) on your patty paper.
Next, rotate the patty paper. For this example, I wanted to rotate 90 degrees clockwise. So, I turned the patty paper one quarter turn to the right. Make sure you keep the origin on the patty paper lined up with the origin on your graph.
Next, write down the coordinates of your new shape.
Finally, remove the patty paper and graph your image.
2. Use Coordinate Rules
Coordinate rules are a great tool for transformations. There are three coordinate rules for rotating about the origin.
Using the coordinate rules to rotate a shape are great if your students aren’t allowed to use patty paper on the test. Check out the rules below.
The first step for using coordinate rules to rotate a shape is to write the coordinate rule on your paper. For this example, I wrote the coordinate rule for 180 degrees.
Next, write the coordinates of your pre-image.
Then, use the coordinate rule to get the coordinates for your image.
For 180 degrees, the rule is (-x, -y). This means to change the signs of both the x value and the y value.
BE CAREFUL: The negative DOES NOT mean that the number must be negative. It means change the sign. So, if my pre-image coordinates are negative, they will change to positive.
Finally, graph the coordinates for your image.
3. Use the Quadrants
This method is a variation of method #2. It’s my favorite method! It uses the four quadrants of the graph to rotate the shape.
First, write down the coordinate rule and the coordinates of your pre-image.
Next, determine what quadrant your image should be after the rotation.
In this example, the pre-image is in the second quadrant. If I rotate 90 degrees clockwise, the shape will be in the first quadrant. Rotating 180 degrees, will put the shape in the fourth quadrant. If I rotate 270 degrees, the shape will be in the third quadrant.
So, all points should be in the third quadrant.
The quadrant tells you what the signs should be for all of your coordinates. In the third quadrant, the signs are (- , -).
All of my coordinates for my image will have those signs.
Finally, look back at the coordinate rule. For 270 degrees (and for 90 degrees), the rule tells me to switch the x and y values.
Since I already have my signs, I just need to switch the x and y values. So, since point W is (-4, 5), then W’ will be (-5, -4).
Do that for all the coordinates. Then graph your image.
I hope you liked these 3 strategies for rotating a shape!
Would any of these methods help you with rotations? Let me know below!
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About the shape of a rotating liquid / Sudo Null IT News
I was thinking this morning as I stirred two sugar cubes in a cup of freshly brewed tea. I thought about the shape of the liquid that it takes when it rotates.
Of course, everyone imagines what will happen if you start stirring sugar in a cup of tea very quickly. I wanted to consider this banal and familiar process in more detail and try to tell you some interesting things from the physics of the phenomena that surround us in everyday life.
Let’s imagine that we have some cylindrical container containing some liquid. A liquid can be made to rotate in at least two obvious ways: to stir it with some object or to start rotating a cylindrical container, which, due to the frictional forces between the liquid and the surface of the vessel, will lead to the rotation of the liquid entrained by the rotating vessels containing it.
Let’s take a look at the second option. So, we have a vessel rotating with a constant cyclic frequency, in which, under dynamic equilibrium with a constant cyclic frequency, the liquid rotates in the same direction.
Let’s cut out from the entire liquid an elementary infinitesimal volume near the surface and consider what forces act on it. Due to the symmetry of the problem, we will focus on cylindrical coordinates, which will significantly simplify the calculations.
Qualitative calculation of the surface shape
Let’s write Newton’s second law for an elementary piece of liquid volume:
For example, after stirring a spoon of sugar in a cup of freshly brewed tea, the liquid rotates around the axis of symmetry, hence our elementary piece of volume has centripetal acceleration. Therefore, we project our Newton’s law onto an axis coinciding with the radius vector from the elementary volume to the axis of symmetry. We will not take into account viscosity and surface tension. The force that imparts centripetal acceleration (on the right side of our law of motion) will arise due to the pressure difference of the fluid columns, which can be seen in the enlarged part of the first figure.
Thus, we get the following expression:
, where , and the same force is defined as , where the area of the effective section denotes the area of our elementary volume, which is affected by the pressure difference of the liquid columns.
We get the force
The mass of our volume element is determined by the formula familiar to everyone, and the volume itself will be equal to (elementary volume in cylindrical coordinates).
As a result, Newton’s 2nd law for our little problem is written in the following expression:
After small reductions and transformations, we get:
Now we integrate both parts of the expression using indefinite integrals:
Detailed calculation of the surface shape
Now we have got a quite clear dependence for the shape of the surface and we can say with confidence that it is a paraboloid. But we do not know the constant value . Let’s define it to fully understand the physics of the process.
Since the volume of the liquid does not change (we assume that we did not spill a drop while we were stirring our tea ツ), we will write down the volumes before and during rotation with a constant cyclic frequency.
, where is the height of the fluid in the cylindrical surface at rest (no rotation).
These volumes are equal, therefore:
From here, a previously unknown constant is expressed:
And the final equation for the shape of the surface of a rotating liquid is:
or by transforming
We would like to draw attention to the fact that the shape of the surface depends on the frequency of rotation, acceleration of free fall, the geometrical parameters of the vessel, the initial volume of the liquid, but does not depend on the density of the liquid. This expression seemed to me quite interesting, since it can be used to easily model the approximate location of a liquid inside a cylindrical vessel rotating around its axis of symmetry. To do this, you can use MathCAD and build several graphs.
Graphical representation of calculation results
Let’s take quite real parameters of the system, commensurate with the size of a cup or glass.
Cylindrical surface radius:
Liquid height in a cylindrical surface without rotation:
Free fall acceleration:
Cyclic frequency of rotation of a cylindrical surface:
(All values of these quantities are given in the C system)
Next, we will rewrite our function to display it in MathCAD.
For 2D display of the section:
For 3D display of the surface:
As a changing parameter, we will change the cyclic rotation frequency . The results can be seen in the pictures below:
At cyclic frequency
At cyclic frequency
At cyclic frequency
At cyclic frequency
At cyclic frequency
At cyclic frequency
It can be seen that if the cyclic frequency exceeds the value , then we will see the bottom of a rotating cylindrical vessel, and, starting from this frequency, the liquid will smoothly «pass» to the walls of the vessel, exposing the bottom more and more. Obviously, at very high frequencies, the entire liquid will spread over the walls of the vessel. Now we know all the parameters of such a liquid. Knowing its equation, it will not be difficult to calculate the thickness of the liquid layer on the vessel wall at a certain height at a certain frequency.
upd. Separately, I would like to emphasize those contradictory assumptions that were made when considering the problem:
1. It was believed that the liquid rotates due to the rotation of the vessel that contains it. This can be only when internal friction, viscosity and surface tension are taken into account.
2. But when deriving the surface shape, these phenomena are not taken into account in order to simplify the solution and show only qualitative simulation results. Those. the solution slightly contradicts the model described initially. Accounting for all phenomena, including the nonlinearity of the process at high frequencies, would complicate the task so much that it could hardly be solved analytically and show an approximate and understandable model for a person who is not related to mathematics / physics.
3. The goal was to show only a very approximate and simplest solution, which includes a number of assumptions.
How to choose a toothbrush
- Manual toothbrushes
- How to choose a manual toothbrush
- Bristle stiffness
- Electric toothbrushes
- What are electric toothbrushes? brushes
- Popular brands of electric toothbrushes
- How to choose an electric toothbrush
- Ultrasonic toothbrushes
- How to choose an ultrasonic toothbrush
- How to choose an electric toothbrush: video
- Myths about electric toothbrushes: video
Choosing the right toothbrush affects the health of the entire oral cavity, and it is no less important than the choice of toothpaste. A toothbrush is the most important thing in caring for your teeth and gums. Only a brush can remove food debris and soft plaque from the surface of the teeth, prevent the formation of tartar, caries and other oral diseases.
Today, manufacturers offer a lot of brush options that differ not only in materials, but also in the principle of operation, which makes the choice not the easiest. We tell you how to choose a toothbrush, what they are and what myths exist about them.
The most popular and familiar. The handle of a classic manual brush is usually made of plastic, and the bristles are made of synthetic fibers. The packaging always indicates the degree of hardness.
There are also special manual brushes for orthodontic constructions and mono-beam brushes for cleaning braces, bridges, gingival sulcus, back teeth and fissures.
How to choose a manual brush
Manufacturers of manual brushes are constantly changing design — there are rubberized areas, plastic and rubber inserts on the head of the brush itself. But all of them do not affect the quality of teeth cleaning — it depends on the correct technique, the quality of the brush and its bristles. All these inserts will not make your teeth cleaner, but will only damage the gums and mucous membranes.
If you have settled on classic toothbrushes, pay attention to the following points when choosing.
- The bristles must be synthetic.
- The brush head should not have any plastic or rubber inserts, only bristles.
- The length of the brush head should cover 2-3 teeth. The smaller it is, the more convenient it will be to brush your teeth and the more effective the cleaning will be.
If you have braces, choose a special orthodontic brush. The design of its bristles provides a V-shaped notch, which helps to thoroughly clean not only the teeth, but also the bracket system. Monobeam brushes are indispensable for cleaning braces.
Stiffness of the bristles
The bristles are the most expensive part of a toothbrush, on which the thoroughness and effectiveness of brushing depends. In the production of modern brushes, artificial bristles are used, which not only last longer, but also do not become a habitat for bacteria and microorganisms.
According to the stiffness of the bristles, the brushes are:
- soft and ultra soft;
- medium hard;
Do not choose a hard brush yourself — it may not fit and damage the enamel and gums. These brushes are suitable for the care of some removable dentures.
The soft and ultra-soft brushes with a large number of bristles are suitable for most people. If you like coffee and smoke, choose a brush with medium hardness. The main thing — remember that enamel is easier to damage than it seems.
Electric toothbrushes make several thousand strokes per minute, cleaning teeth in a way that a manual toothbrush never can.
Most electric toothbrushes are battery powered, but battery-powered models are also available.
Electric brushes have a number of advantages over manual brushes.
- They are much easier to use — no need to learn complex techniques for proper cleaning.
- They have modes of different intensity and for solving various tasks: for daily brushing, for sensitive teeth and gums, for braces and dentures, for the tongue.
- One brush with different attachments can be used by several people.
Types of electric toothbrushes
In any electric brush, the bristles are driven by a built-in motor. The frequency of their fluctuations, depending on the brand, is from 10 to 30 thousand movements per minute. It makes no sense to somehow classify such brushes, because there are so many criteria that you can go into the wilds and get confused.
On the Internet, it is customary to single out sonic brushes as a separate type. This concept itself appeared thanks to the word sonic in the name on the boxes of brushes of different brands. Due to the special frequency and amplitude of the movement of the bristles, the mixture of saliva and paste begins to circulate during brushing, resulting in a dynamic flow of liquid in the mouth, which cleans hard-to-reach places.
Therefore, everyone got used to the word sonic in relation to electric brushes, and then new manufacturers began to use it. In fact, sonic toothbrushes are electric toothbrushes, in which the bristles move at a speed sufficient to achieve the turbodynamic effect in the oral cavity. Some brushes do not reach it — they do not have enough speed. For example, B.Well models do not accelerate to the required speed and do not create a flow — they are electric, but not sound.
Philips patented Sonicare technology: the generator built into the brush generates special sound waves that accelerate the bristles up to 31,000 strokes per minute. It’s a unique technology, but that doesn’t mean other toothbrushes can’t be called sonic. They can if they create a dynamic flow of fluid in the mouth.
Popular brands of electric toothbrushes
Today, the electric toothbrush market has its own leaders — Philips, Oral-B, Xiaomi Soocas, Revyline and CS-Medica. We will tell you about their features and help you understand which brand is right for you.
Oral-B uses two technologies: 2D and 3D. Brushes with 2D technology produce reciprocating movements and make up to 10,500 revolutions per minute. With 3D technology, the brush performs both reciprocating and rotational movements, and ripple . The frequency of movements depends on the model. Oral-B is a trusted brand recommended by dentists worldwide. These brushes are also called indestructible.
Philips brushes have a different way of working — patented Sonicare technology. The bristles perform sweeping movements which are combined with pulsation , which creates an additional irrigation effect. All Philips models perform the same number of strokes per minute — 31,000. Philips is renowned for its high technology and eye-catching design.
Soocas brushes are loved for their technology and cool minimalist design. CS Medica — brushes with the usual shape of the head and quite democratic in price. Revyline is a young but promising brand whose brushes are the best value for money.
How to choose an electric brush
When choosing an electric brush, pay attention to the following parameters.
Can be round or oblong and is associated with bristle movement technology. The shape of the head dictates the cleaning technique itself. Therefore, it is important to understand which technique is more comfortable for you. With an oblong nozzle you cover two teeth and two gaps, with a round one — completely one tooth.
Brushes with a round head (eg Oral-B) have bristles that rotate back and forth. In brushes with an oblong head (for example, Philips), the movements of the bristles are sweeping. However, both technologies are effective.
We generally recommend Philips toothbrushes for people with sensitive enamel because the bristles are soft and gentle on the teeth. Those who want to get rid of stubborn plaque and love intensive cleaning should pay attention to Oral-B.
Number of operating modes
The more expensive and modern the model, the more modes it has. In inexpensive brushes — 1-2 modes. Oral-B Genius 20000 has 6 of them.
But it’s not the number of modes that matters, but the ability to adjust them. This is true for people with sensitive enamel and gums, and also if several family members use the brush.
Design and ergonomics
We hold the brush in our hand twice every day, so it should fit comfortably in the hand and please. For ease of cleaning, the brush bodies are covered with non-slip, textured materials. For example, Oral-B adds rubberized inserts to brush handles. Philips makes stone-look pens.
Appearance is also an important point when choosing.
- If we talk about sophistication and luxury, this is for Philips. In addition to classic white and elegant black, the range includes frosty pink, mint, lavender, turquoise, blueberry, and steel colors.
- Oral-B has a simpler design, but it also has its stars. For example, the latest Oral-B Genius 20000 in black and rose gold.
- The Soocas X3 in Black Gold also looks impressive. The entire range of this brand is stylish and technological.
- Of the democratic options, Revyline brushes have a good performance. Model RL 010 in pink and black looks very stylish.
Battery capacity and type
An important point is how long the brush works without recharging. This is especially important if the brush will be used by several people: this way it will sit down much faster. In this article, we are only talking about battery brushes, because they are in demand more than battery ones.
Batteries come in Lithium Ion and NiMH . The first can be regularly recharged a little bit, they hold a charge longer and do not require replacement for about 10 years. The latter charge faster, but last less.
The most affordable model of Oral-B — Vitality — is designed for autonomous operation for 12 days . Oral-B Pro and Smart series brushes last on average about 20 days without recharging.
Philips Sonicare’s longest lasting brush — 3 series Easy Clean — works about a month without recharging. On average, Philips batteries retain the charge required for cleaning for about 3 weeks, that is — 20-25 days .
Soocas and Revyline brushes break real records for autonomy — they clean without recharging up to 2 months .
These data are not from the manufacturers’ official websites. We received them as a result of our own independent test , as we always test everything that we sell.
Additional selection criteria
We have dealt with the main selection criteria and move on to optional, but useful, which should also be taken into account.
Dentists recommend brushing your teeth for at least 2 minutes, and this is where the timer comes in. For some brushes, it is divided into intervals of 30 seconds — that is how much time you need to spend on a quarter of the dentition (or, as hygienists call it, a quadrant). This is not a very important criterion, because the timers in all electric brushes are plus or minus the same.
A very important function is the pressure sensor. While brushing, it is easy to think about something and put too much pressure on the enamel and gums. To protect them, manufacturers have made pressure sensors that tell them when it’s time to slow down. Well this option is done by Oral-B. If the pressure is too high, the red light will turn on. At Philips, depending on the model, the brush gives a sound or light signal.
Bristle wear indicator
Brush heads need to be changed, just like the manual brush every 3 months . Philips BrushSync technology reminds you when it’s time to change your brush head. The indicator on the brush will light up and a short beep will sound.
The BrushSync sensor must be in both the head and the brush for the technology to work. More affordable Philips models do not have such sensors, so indicator bristles will help determine the time of replacement — they discolor when it is time to change the nozzle.
Oral-B does not have a BrushSync equivalent, but has special blue bristles. When they turn white, it means it’s time to change the nozzle.
Another great feature from Philips is Easy Start. It gradually increases the brushing power over the first 14 treatments, which helps you get used to the electric toothbrush.
Some brushes have their own mobile app. If you draw an analogy, then this is like a personal fitness trainer. Only in this case it is a brushing trainer.
- The Philips app tracks your brushing progress and makes personalized recommendations.
- With Oral-B, the toothbrush recognizes brushing style, monitors its quality and also makes recommendations.
- Xiaomi Soocas brushes do not work with the Mi Home app, but they have their own Soocas app.
Some customers think that ultrasonic toothbrushes are better than sonic toothbrushes because they have a higher vibration frequency. To dot the i’s, you need to clarify — these are two completely different technologies.
Electric toothbrush boxes are often labeled with the number of bristle oscillations or head rotations. Ultrasonic brushes work due to the piezoelectric element, which creates an ultrasonic wave with a frequency of 1600 kHz. These are also vibrations, but vibrations of the wave itself, and not of the bristles.
The bristles themselves can remain motionless — cleaning is carried out by ultrasonic action. The human ear cannot pick up sound at these frequencies. Therefore, when cleaning, many people have the feeling that some ultrasonic brushes do not work at all.