Something You Didn't Learn About Reflections from School!

what are different ways we can transform objects without changing their size or shape can we grow or Shrink them can we stretch them can we cut them and bend them into other shapes after some thought you might come to the conclusion that we can only move them in physics we have a similar question therefore what are different types of ways we can move [Applause] objects what do we mean by a transformation we simply mean a map or a function like f ofx y a function Maps points to points as seen here in stretching the circle into an ellipse you might notice there are two parts to a transformation one is the final result which would be the image of the original space otherwise known as The Domain two is the animation in getting from the original space to the final space over some period of time what is an animation mathematically is a sequence of maps where one parameter is usually in the interval from 0 to one Alan howcher book algebraic typology alludes to this idea when he talks about deformation retracts in in the plane we can move objects side to side or up and down this motion is known as a translation and you can see it does not change the size or shape if we input a coordinate system we can construct an explicit expression for the function the coordinates XY is any point on the shape another distinct motion is rotation which is very important in physics pick any point in the plane and make the object spin About It by some angle the point could be at the center it could be somewhere in the interior it could be on a Vertex or it can be exterior to the shape if we introduce a coordinate system we can write down an explicit formula for rotation about any point AB in angle Theta this formula can be found in the book learning isn't linear now we might think what about flipping an object to change its orientation the usual way this transformation is constructed in Geometry is to draw lines from the vertices perpendicular to something called the line of reflection and then the mirror image appears on the other side notice that the orientation or order of the vertices changes from counterclockwise to clockwise this is a distinct motion as we cannot translate and rotate the mirror image back onto the original this is the 2D perspective however and aren't congruence mappings AKA rigid motion transformations supposed to be continuous it looks like a magic trick is performed where disappears and reappears what is really going on here what really is a reflection again we witnessed this disappearing act maybe the answer can be found by jumping up a dimension now we see the motion is a continuous flip over or about a line actually it looks a lot like 180° rotation about an axis we see that if this line or axis intersects the shape in a point point we get a rotation as before now what if we have a 3D object like a cube we can also rotate it or reflect it but they look exactly the same rotating or reflecting about the z-axis is equ equivalent to translating to the left and then rotating or reflecting 180° about the Cube's Center axis this idea that a reflection is actually a 180° rotation about an axis in three space space is quite literally coded into the program is a nice activity to use Desmos geometry to create two Reflections for any given series of translations and [Music] rotations let's take a closer look at why it seems any composition of translations and notations can be written as a composition of only two Reflections if we only look at the equation of a straight line in a static way we won't see it taking a dynamic perspective however we see that each line of reflection has rotation and translation built into it as the change in slope and change in y intercept respectively [Music] n [Music] [Music] where is the pivot point is it the x intercept or the Y intercept it is the y [Music] [Applause] [Music] intercept and here's a nice little tidbit we can extract an elliptical rotation Matrix from a point on the line using the slope m equal tangent Theta [Music] now let's summarize our results [Music] one last idea to think about looking back in the video it looks like we created something one might call a gyration what is a gation could this be a generalization of a rotation thanks for watching if you enjoyed watching this video please consider liking subscribing and sharing and also if you're learning from the free resources from this channel then you'll love my book learning isn't linear available on amazon.com it contains crisp illustrations to accompany non-standard problems with extremely detailed Solutions the link is in the description if you get the ebook version using the Kindle there are clickable links throughout the book to Desmos and geogra projects it's very easy to search within the book for specific topics as you can see chapters range from algebra to calculus whether you are a self-motivated student solving problems on your own or a teacher preparing for a math class this book is a valuable resource to add to your repertoire okay till next time and remember Pro prioritize long-term learning over short-term test taking [Music]

Channel: Learning Isn't Linear

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Id: eX97nKSVLXE

Length: 9min 33sec (573 seconds)

Published: Fri Jul 05 2024