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Rigid transformations intro

Transformations in math involve changing a shape's position or which way the shape points. There are three main types: translations (moving the shape), rotations (turning the shape), and reflections (flipping the shape like a mirror image). Rigid transformations keep the shape's size and angles the same. The image is the shape in its new position and direction.

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  • piceratops ultimate style avatar for user MFZ
    Is a translation and a transformation the same thing?
    (79 votes)
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  • spunky sam blue style avatar for user celloben
    what other types of transformations are there besides rigid transformations?
    (22 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Nice question!

      A common type of non-rigid transformation is a dilation. A dilation is a similarity transformation that changes the size but not the shape of a figure. Dilations are not rigid transformations because, while they preserve angles, they do not preserve lengths.

      Have a blessed, wonderful day!
      (44 votes)
  • duskpin ultimate style avatar for user 💜PURPLE4VICTORY💜
    Hi! Someone down in the thread asked about other ways of translating, rotating, and reflecting points, and I answered their question. Because I think that this information is valuable for the KA community, I also decided to post my answer here. Sorry if this isn't a question, and I hope it helps! :)

    There is another way--*algebraic representation
    . For the sake of this video, Sal used a graph to translate, rotate, and reflect the quadrilateral.

    Translation: Right/Up -> (x + n, y + n) & Left/Down -> (x - n, y - n)
    You can translate points and shapes in any direction using this key by adding or subtracting the distance moved.

    Example: Pretend that there is a point like (1, 2) and you want to move the point up 1 and to the right 2. In this scenario, you would add 1 and 2 to each of the point's coordinates making it look something like this: (1 + 2, 2 + 1). You would eventually end up with the translated point as (3, 3). If you wanted to translate an entire shape without using a graph, you would do this method with all of the necessary points of the shape.

    Rotation: 90 degrees clockwise -> (y, -x), 180 degrees clockwise -> (*-x, -y*), 270 degrees clockwise -> (*-y, x*)
    If you do not want to use a graph, using the key above can help you rotate points and shapes in a clockwise direction.

    Example: Say there is a point (2, 5) and you wanted to rotate the point 90 degrees clockwise. If you follow the key, the point would become (5, -2).

    Reflection: Over the X-Axis -> (x, -y) & Over the Y-Axis -> (*-x, y*)
    You can also use this key when reflecting points or shapes across the x-axis or y-axis.

    Example: If you wanted to reflect (5, 3) over the y-axis, you would end up with (-5, 3) based on the key.

    Regardless of whether you find algebraic representation easier or more efficient than using a graph, I hope that this helps you and gives you a clearer understanding of translating, rotating, and reflecting. :)
    (32 votes)
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  • duskpin tree style avatar for user divyashshah05
    Is Dilation a Rigid Transformation?
    I'm not sure about it.
    (16 votes)
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  • leaf green style avatar for user daniel kerbel
    At 0.48 seconds, Sal said that there are an infinite number of points along the shape. I am just checking my understanding; I get that there a LOT of points but surely the number is finite as it is along a fixed 2D shape with lines connecting or have I not understood it?
    (17 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      A side of a polygon is a type of line segment. Any line segment has infinitely many points, though its length is finite. Note that for any two distinct points P and Q on a line segment, no matter how close they are together, there are points (besides P and Q) on that line segment that are between P and Q.
      (21 votes)
  • winston default style avatar for user Maulika Mehra
    What are the different types of translations?
    (15 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      A translation (or "slide") is one type of transformation.

      In a translation, each point in a figure moves the same distance in the same direction.

      If each point in a square moves 5 units to the right and 8 units down, then that is a translation!

      Another example:
      If each point in a triangle moves 3 units to the left, and there is no up or down movement, then that is also a translation!

      Hope this helps!
      (21 votes)
  • starky sapling style avatar for user Ishan
    Can someone explain rotations. I think I got Translations and Reflections, but not rotations I have always been stuck on it. Thank you!
    (11 votes)
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    • hopper cool style avatar for user CalvinatorIA
      There are 3 main types of rotations:
      1.) 90∘ clockwise - To move a point or shape 90∘ clockwise, simply use this equation: (x, y) → (y, −x).
      2.) 90∘ counterclockwise - To move a point or shape 90∘ counterclockwise, simply use this equation: (x, y) → (−y, x).
      3.) 180∘ rotation - To move a point or shape 180∘, simply use this equation: (x, y) → (−x, −y).
      If a question asks for a 270∘ clockwise rotation, simply change it to a 90∘ counterclockwise, and vice versa. I hope this helped! Thank you for asking! :)
      (22 votes)
  • aqualine seed style avatar for user desiree chong
    will this be taught in geometry?
    (10 votes)
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  • primosaur seed style avatar for user saleomight riddle
    How do you know how many degrees to turn the shape for rotation?

    deeply greatfull
    (11 votes)
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  • spunky sam blue style avatar for user celloben
    what kind of transformation is a dilation?
    (7 votes)
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Video transcript

- [Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context? Well, it could mean that you're taking something mathematical and you're changing it into something else mathematical, that's exactly what it is. It's talking about taking a set of coordinates or a set of points, and then changing them into a different set of coordinates or a different set of points. For example, this right over here, this is a quadrilateral we've plotted it on the coordinate plane. This is a set of points, not just the four points that represent the vertices of the quadrilateral, but all the points along the sides too. There's a bunch of points along this. You could argue there's an infinite, or there are an infinite number of points along this quadrilateral. This right over here, the point X equals 0, y equals negative four, this is a point on the quadrilateral. Now, we can apply a transformation to this, and the first one I'm going to show you is a translation, which just means moving all the points in the same direction, and the same amount in that same direction, and I'm using the Khan Academy translation widget to do it. Let's translate, let's translate this, and I can do it by grabbing onto one of the vertices, and notice I've now shifted it to the right by two. Every point here, not just the orange points has shifted to the right by two. This one has shifted to the right by two, this point right over here has shifted to the right by two, every point has shifted in the same direction by the same amount, that's what a translation is. Now, I've shifted, let's see if I put it here every point has shifted to the right one and up one, they've all shifted by the same amount in the same directions. That is a translation, but you could imagine a translation is not the only kind of transformation. In fact, there is an unlimited variation, there's an unlimited number different transformations. So, for example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it, and I rotate it I would rotate it around the point. So for example, I could rotate it around the point D, so this is what I started with, if I, let me see if I can do this, I could rotate it like, actually let me see. So if I start like this I could rotate it 90 degrees, I could rotate 90 degrees, so I could rotate it, I could rotate it like, that looks pretty close to a 90-degree rotation. So, every point that was on the original or in the original set of points I've now shifted it relative to that point that I'm rotating around. I've now rotated it 90 degrees, so this point has now mapped to this point over here. This point has now mapped to this point over here, and I'm just picking the vertices because those are a little bit easier to think about. This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation that one actually has not shifted, and just 'til you get some terminology, the set of points after you apply the transformation this is called the image of the transformation. So, I had quadrilateral BCDE, I applied a 90-degree counterclockwise rotation around the point D, and so this new set of points this is the image of our original quadrilateral after the transformation. I don't have to just, let me undo this, I don't have to rotate around just one of the points that are on the original set that are on our quadrilateral, I could rotate around, I could rotate around the origin. I could do something like that. Notice it's a different rotation now. It's a different rotation. I could rotate around any point. Now let's look at another transformation, and that would be the notion of a reflection, and you know what reflection means in everyday life. You imagine the reflection of an image in a mirror or on the water, and that's exactly what we're going to do over here. If we reflect, we reflect across a line, so let me do that. This, what is this one, two, three, four, five, this not-irregular pentagon, let's reflect it. To reflect it, let me actually, let me actually make a line like this. I could reflect it across a whole series of lines. Woops, let me see if I can, so let's reflect it across this. Now, what does it mean to reflect across something? One way I imagine is if this was, we're going to get its mirror image, and you imagine this as the line of symmetry that the image and the original shape they should be mirror images across this line we could see that. Let's do the reflection. There you go, and you see we have a mirror image. This is this far away from the line. This, its corresponding point in the image is on the other side of the line but the same distance. This point over here is this distance from the line, and this point over here is the same distance but on the other side. Now, all of the transformations that I've just showed you, the translation, the reflection, the rotation, these are called rigid transformations. Once again you could just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't stretch or scale up or scale down it kind of maintains its shape, and that's what rigid transformations are fundamentally about. If you want to think a little bit more mathematically, a rigid transformation is one in which lengths and angles are preserved. You can see in this transformation right over here the distance between this point and this point, between points T and R, and the difference between their corresponding image points, that distance is the same. The angle here, angle R, T, Y, the measure of this angle over here, if you look at the corresponding angle in the image it's going to be the same angle. The same thing is true if you're doing a translation. You could imagine these are acting like rigid objects. You can't stretch them, they're not flexible they're maintaining their shape. Now what would be examples of transformations that are not rigid transformations? Well you could imagine scaling things up and down. If I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved that would not be a rigid transformation. If I were to just stretch one side of it, or if I were to just pull this point while the other points stayed where they are I'd be distorting it or stretching it that would not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting. When you use an art program, or actually you use a lot of computer graphics, or you play a video game, most of what the video game is doing is actually doing transformations. Sometimes in two dimensions, sometimes in three dimensions, and once you get into more advanced math, especially things like linear algebra, there's a whole field that's really focused around transformations. In fact, some of the computers with really good graphics processors, a graphics processor is just a piece of hardware that is really good at performing mathematical transformations, so that you can immerse yourself in a 3D reality or whatever else. This is really really interesting stuff.