![]() Then I went back to school to set up the lighting for a dance show. This week I was trying to get some manhole covers up as the architect for our building project needs to know what is underneath. I may log in again at some time between GMT 1600 and GMT 1900 and again maybe late at GMT 2300.īut none of this is guaranteed as I'm often out doing other things. But if I'm leaving my computer to do something else then I log off, otherwise you'd be posting away, wondering why I'm ignoring you when, actually, I'm not even in the room.Īt lunch (around GMT 1300) I log back in again for a while. When I'm having breakfast (around GMT 8.00) I log on and see what's happening. This brings the shape to the same position as T(RR). If you apply the glide reflection to this you have to reflect in x = 6.5 and go up 4. It is a glide reflection specifically, reflect in the line x = 6.5, and 'glide' up parallel to this line 4 units. T(RR) = T since the reflections 'cancel out'. ![]() If F is translated by T and then rotated by R2 the resulting shape must be congruent to F but rotated so R2T is another rotation of 180 around the point shown. ![]() R1 and R2 are rotations of 180 about the highlit (highlighted?) points. Thus, for example, QP must also represent a transformation of points in the plane. When a point is transformed by one transformation and the result transformed again by another, the result will, of course, be another point and since each transformation is bijective, the combination will be too. I shall adopt the 'reverse order' convention for the mappings namely P (on A) followed by Q will be written QP(A). P, Q and R are transformations such that all points are transformed and no two points are transformed to the same point.Ī, B, C and D are points in the plane such that Comments welcome, but, if I'm wrong, break it to me gently please.Įach circle represents the points of a cartesian plane. The proof came to me fairly quickly but seemed too easy, so I've been trying to find a loophole ever since.Īs I haven't found one, I thought I'd post it. I think what follows is a proof that all bijective transformations of the plane may be combined associatively. Such a transformation would have no inverse. Rotations, reflections, translations, glide reflections, shears, stretches and enlargements all appear to be ok.īut it is possible to define a transformation that maps the whole plane onto, for example, a single line. Transformations that are not bijective will not have inverses. But I'm unsure about associativity.ĮDIT: I think associativity does hold. Identity, inverse and closure are all ok. You may be wondering: "Is the set of all possible transformations, a group?" Similarly the set of rotations about a single point.īut the set of reflections is not a group because two reflections do not make another reflection. (iv) To combine three translations you would just use the rules of arithmetic so combining is associative. (iii) A combination of two translations is also a translation So the set is closed. (ii) Each translation has an inverse translation. (i) There is an identity, the 'stay where you are' translation. ![]() ![]() Does a set of transformations form a group under combination of transforms?Įg. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |