Reflections
In order to understand the dynamics of a bounce-pass, one must understand the basics of reflections. Examine the figure below. Think of the point A at the left as the original.
It is called the preimage. The reflection image A' at the right can be
drawn by folding over the line m and then tracing. Line m is called
the reflecting line or line of reflection.
The apostrophe (') indicates corresponding points; A' (read "A prime")
corresponds to A. Line m is the perpendicular bisector of the segment
connecting the corresponding points A and A'.
Properties of Reflections
* For a point P not on line m, the reflection image of P over line
m is the point Q if and only if m is the perpendicular bisector
of segment PQ.
* For a point P on m, the reflection image of P over line m
is P itself.
You can use abbreviations instead of writing statements like "P is the
reflection image of R over line m" and "A' is the reflection of
A." Use the lower case letter r, usually in italics to refer to
a reflection. When discussing reflections in general, or when the reflecting
line is obvious, you can write "r(A) = A'" for "The reflection image
of A is A'." It is read "r of A equals A'." When you want
to emphasize the reflecting line m, write r m(Q)
= P for "The reflection image of Q over line m is P." It is read
"r of Q over line m equals P."
Reflecting images are often needed for points in the coordinate plane.
In order to reflect (4,5) over the x-axis (y=0), you must multiply the
y value by -1. That means that r y=0(4,5)=(4,-5). Another
example, drawn on the graph below is r x=5(1,2)=(9,2).
Take a look at the picture above: when A is reflected over l, the y value of A' is the same as the y value of A. The x value of A' is the [perpendicular distance between A and l]+[the x value of l]. Substituting the actual values, we arrive at [4]+[5]. Thus, the x value of A' is 9. The y value stays the same, 2. So, the ordered pair location of A' is (9,2).
Here is a set of simple rules to follow when reflecting in the coordinate plane:
When reflecting over the line y=x: if A=(m,n), then A'=(n,m)
When reflecting over the x-axis (the line y=0): if A=(m,n), then A'=(m,-n)
When reflecting over the y-axis (the line x=0): if A=(m,n), then A'=(-m,n)
When reflecting over the the line x=a: if A=(m,n), then A'=([a-m]+a,n)
When reflecting over the the line y=a: if A=(m,n), then A'=(m,[a-n]+a)
A note on notation: "r l º r m(A)" means "the reflection of point A first over line m, then over line l".
Great, let me try some problems myself!