Can non-zero translations be composed of two reflections in perpendicular lines?

  • Thread starter zcdfhn
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In summary, it can be proven that every non-zero translation Tb is a composition of two reflections in lines perpendicular to the direction of vector b. This can be done by using the formula for a reflection and showing that successive reflections in two perpendicular lines will result in Tb.
  • #1
zcdfhn
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Prove that every non-zero translation Tb is a composition of two reflections in lines which are perpendicular to the direction of vector b.

Note: Tb(z) = z+b where z,b[tex]\in[/tex]C

My guess at how to start this is to assume b = rei[tex]\theta[/tex] where r = |b| and [tex]\theta[/tex]=arg b, so then the direction of b is b/|b| = ei[tex]\theta[/tex]. Therefore the unit vector with the direction orthogonal to b would be c = ei([tex]\theta[/tex] + [tex]\pi[/tex]/2). From there, I am shaky about what to do. I attempted to create two reflection f,g that reflection over lines with the same direction as c and I attempted to do g[tex]\circ[/tex]f and I should end up with Tb but my work gets more and more complicated.

Thanks for your help.
 
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  • #2
What kind of formula do you know for a reflection? The reflections you are talking about not just reflections through the origin. You'll never get a translation out of that.
 
  • #3
I know M(z) = a(z-bar)+b, where |a| = 1 and a [tex]\neq[/tex] 1, a(b-bar) + b = 0

and i also know M(z) = z0 + ei2[tex]\eta[/tex](zbar - z0bar)
 
  • #4
You seem to be confusing "reflection" with "rotation". [itex]re^{i\theta}[/itex] will give a rotation by angle [itex]\theta[/itex] together with an expansion (or contraction) by r, not a reflection. Given any point p, draw the straight line between p and Tb(p) and draw lines L1 and L2 perpendicular to that line 1/3 and 3/4 of the way between p and Tb(p). If R1 is reflection in L1, R1(p) will be the point, p2, exactly half way between p and Tb[/sup](p), and reflection of p2 in L2 will be Tb will be Tb(p) itself. Show that successive reflections of any point x in L1 and then L2 will give Tb[/sup](x).
 

1. What are isometries?

Isometries are transformations in geometry that preserve the size and shape of an object. They include translations, rotations, reflections, and glide reflections.

2. How are translations different from other isometries?

Translations involve moving an object without changing its orientation or shape. This is done by sliding the object in a specific direction, distance, and magnitude.

3. What are some real-life examples of translations?

Some common examples of translations include sliding a book across a table, pushing a chair in a straight line, or moving a car from one parking spot to another.

4. How can translations be represented mathematically?

In geometry, translations can be represented by a vector, which specifies the direction and magnitude of the translation. In algebra, translations can be expressed as linear functions, where the input is shifted by a constant value.

5. Do translations preserve all properties of objects?

Translations preserve some properties of objects, such as size and shape, but not others. For example, translations do not preserve angles or distance between points. These properties are preserved by other isometries, such as rotations and reflections.

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