This my second answer to this question. I felt like writing a new one Instead of editing the old one because this a completely new approach and is way more simpler.
It all comes from how we define left and right and that mirror changes the direction in whic the object is facing when by inverting the Z-axis. We assume that the ponits on vertical axis, since it seperates the left and right, remain unchanged after mirroring, we can draw a similar axis between any two points and say that the mirror has flipped along that axis.
Mirror does nothing apart from inverting the Z-axis, as <PERSON> said. Each point on your body that is facing the mirror will remain with the same X and Y coordinates(or to say in simple language, image of each point is right in front of the actual point). Mirror doesn't differentiate any points as in such. 586
The asymmetry arises from our definition of left and right, here's how we try to find out which part of an object is the right and which is left.
Step 1: I start by drawing a hypothetical line between what I assume are the top and bottom of the object.
Step 2: Then, I orient myself(hypothetically) along that axis and face in the same direction as that(I assume) the object is facing.
Step 3: Now, whichever half of the object contains the right half of the body, I call it the right side of the object, and whatever is 'left' (some clever pun by the author), I call it as the left side of the object.
When I see the reflection of the object in the mirror all the points (including top, bottom, left and right most points) are exactly in front of each other but the direction the object is facing has changed by 180°, Owing to the Z-axis inversion by mirror as <PERSON> said. This is where it all happened!!! Now that the image is 'facing' the opposite direction to the actual object, I’ll have to position myself facing opposite direction to what I did earlier in step 2, This means in step 3, I’ll find what earlier was to my right will be on the left and vice-versa and hence the right and left of the image appear to be swapped (Recall the PET classes when we as kids, used to confuse left and right beacuse our instructor in front is facing you and his 'left' is your 'right'). When we drew the axis we assumed that the points along the axis remained even in the image and everthing else is swapped along it. But in truth all points remained at the same place(Only Z inversion happened), Infact If I draw this line(which seperates right and left of a body), between any other two points and call the two halfs obtained as left and right, It'd appear that all the points on that axis remained at the same place and the inversion happened along the new axis.
If it is still confusing you, let’s see how we can swap our top and bottom as opposed to left and right just by redefining ‘left’ and ‘right’.
I draw a horizontal axis between the right most part of the object and left most and repeat step 2 with my head at the right most part of the object, this means what was the 'up'per half of the object earlier will now be the 'left' half and what was the 'bottom' half will be the 'right' half. On looking at the the mirror image and and repeating step 2 and 3 assuming that the ponits along the the new axis remained same, It would tell me that the new left and right (the former 'up' and 'down') have been swapped. 586