There are many pairs of points that describe the same vector (and the same Then the invariant line g passesįind the glide vector: Observe that T(P) = M(R(P)) = M(Q) = This time put A in the center of a square BCDE, with m = CD.įind the invariant line: Let P be the midpoint of BC and let Q be the midpoint of CD. Use the image of special points to find the defining data. Squares shown, and the distance between the two intersection points, and theirĪgain we use the knowledge that T is a glide reflection and Intersections of the perpendiculars with line MN are the centers of two of the Start with A and its image A' and construct lines through A and A' Note : The glide vector can be found from the Midpoint AA'' and then reflecting in m to get N. (It is also possible to see that T(M) = N by first finding R(M) = Maps AA' to A'A''' the midpoint of AA' is mapped to the midpoint N of A'A''', Through M that makes a 45-degree angle with m, as shown.įind the glide vector: Since M is on the invariant line, the Invariant line g = line MN, which is parallel to line AA'''. M and the midpoint N of A'A''' are both on the invariant line, so the It is easy to see that AA'A''' is anįind the invariant line: The midpoint M of AA' is on m. T(A) = M(R(A)) = M(A) = A', the reflection of A in m. Use special choices that are easy to compute. This can be done by finding T(P) for any points P but it is easier to Not concurrent, so we know that T must be a glide reflection. We take the product with M we get a product of reflections in 3 lines that are Since R is a rotation, it is a product of the reflections in In order to see more clearly where points are located, aĬoordinate system of squares has been added to the figure with A on the y-axis Line reflections as in the proof of this theorem. The third way uses factorization into the product of three Trace the images of special points to find T exactly. Knowledge of T from a general theorem about triple line reflections and then Here this question is answered in three different ways. Patterns that have both 90-degree rotational symmetries and line symmetries. Isometry is T and what is it exactly? This product T is an important one, since it occurs in wallpaper Let R = A90, the rotation by 90 degrees, with center a pointĪ not on m. What is the product of a rotation and a line A few years ago, I wrote a script that did that for one specific case, but unfortunately, I don't have access to that script anymore.Three Ways to Analyze the Composition of a 90-degree Rotation and a Line If you know what the source rotation was, and what the target rotation needs to be, you can take the coordinates and rotation flags from the saved comments file, and transform them so that they match the new file, and write out a modified comments file. The same thing happens when you export the comments from one document and import into a document with a different page rotation. in MS Word, that page is very likely not rotated, and therefore does not have the rotate property (or it is set to 0 deg), This replaces your page image, but all of a sudden, the orientation (and position) of your annotations is wrong. However, if you now replace one of these pages with a page that you created e.g. Now the problem is that when you add annotations to such a page, they will be rotated as well, so that they match the page (and look correct when you display them in Acrobat). Acrobat does not actually rotate the page contents, it just saves a rotate property for the page(s), and Acrobat knows that before the page gets displayed, it needs to be rotated 90 deg ccw. You don't want to have to tilt your head every time you want to read this document, so you rotate them in Acrobat by 90 degrees counter clock wise (ccw), and the content finally looks correct. Let's say you scan a few letter sized pages, but the way your scanner rotates the pages, it looks like they are all rotated by 90 degree clock-wise. #90 CLOCKWISE ROTATION PDF#I can explain why this happens, but unfortunately don't have a good solution to avoid the problem:Ī PDF file can have a rotation flag that indicates if Acrobat (or any well behaved PDF viewer and printer) should rotate the page before it gets displayed.
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