Monday, September 21, 2015

Unizor - Geometry3D - Axis Symmetry

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Symmetry in 3-D - Symmetry around an Axis

In
this lecture we will discuss the symmetry about an axis (that is,
relative to a straight line) in three-dimensional space that assumes the
existence of an axis of symmetry.

Recall from the symmetry on a
plane that point A', symmetrical to point A relative to an axis of
symmetry (straight line) s, can be constructed by dropping a
perpendicular AP from point A onto axis s (P∈s, AP⊥s) and extending this
perpendicular beyond point P by the same length, thus obtaining point
A'.

In three-dimensional space symmetry about an axis requires analogous construction.
If
we are given point A and an axis of symmetry (straight line) s, then
point A', symmetrical to point A relatively to axis s, is located on a
continuation of perpendicular AP to axis s (P∈s, AP⊥s) beyond point P by
the same distance as the length of segment AP.

The construction
of a symmetrical point is, obviously, reversible. If we start from point
A', drop a perpendicular to an axis of symmetry A'P and extend by the
same length, we will be at point A. It follows from the uniqueness of a
perpendicular from a point onto a straight line. So, if point A' is
symmetrical to point A relatively to axis s, point A is symmetrical to
point A' relatively to the same axis of symmetry s.

The other
viewpoint on symmetry around an axis in three-dimensional space is
considering the symmetrical reflection relative to an axis as a result
of rotation in space around this axis by 180o.

Rotation of point A in space around an axis s by an angle φ can be constructed as follows:
(a) draw a plane β perpendicular to axis s through point A (s⊥β, A∈β);
(b)
draw a line on plane β connecting point O of intersection of plane β
and axis s (O = β∩s) to point A; since OA∈β and s⊥β, OA⊥s;
(c) within plane β rotate segment OA by angle φ around point O, so point A would take position A'.
In
particular, if φ=180o, point A' would be on continuation of segment OA
beyond point O by the same distance as the length of segment OA, which
exactly corresponds to the rules of construction of a point symmetrical
relative to an axis.

We can say now that two points A and A',
symmetrical relatively to axis s, are centrally symmetrical within plane
β drawn perpendicularly to axis s trough point A. The center of
symmetry within plane β is a point of its intersection with axis s.

The
symmetry can be viewed also as an operator on points of
three-dimensional space. Given an axis of symmetry, this operator
transforms each point into its image constructed by the rules above. We
will sometimes relate to symmetry as a transformation assuming exactly
this type of operation.

There are a few simple theorems we'd like to present about symmetry about an axis in three-dimensional space.

Theorem 1
Symmetry about an axis preserves the distance between points.

Proof
Consider two points A and B and axis of symmetry s, so each point has its symmetrical counterpart - A' and B' correspondingly.
We have to prove that AB=A'B'.
Draw
plane γ trough point A perpendicular to axis s, intersecting it at
point P, and plane δ through point B also perpendicular to our axis s,
intersecting it at point Q:
A∈γ, γ⊥s (⇒ A'∈γ)
B∈δ, δ⊥s (⇒ B'∈δ)
Let C and C' be projections onto plane γ of points B and B' correspondingly.
Obviously, BB'C'C is a rectangle.
First,
let's prove that projections AC and A'C' of segments AB and A'B' onto
plane γ are congruent. Then it will easily follow the congruence of
segment AB and A'B'.
But congruence between AC and A'C' is obvious
since points C and C' are centrally symmetrical relatively to the same
center of symmetry P as points A and A' (a trivial statement of the
plane geometry).
Since AC=A'C', right triangles ΔABC and ΔA'B'C' are
congruent - their catheti AC and A'C' are congruent and their catheti BC
and B'C' are both equal to a distance between two parallel planes γ and
δ.
Therefore, hypotenuses AB and A'B' are congruent. The length of a
segment AB is preserved by symmetry relative to an axis. Since the
lengths of segments are preserved, angles between intersecting lines are
preserved as well, as it's sufficient to include any angle into a
triangle and symmetry around an axis will preserve the length of its
sides and, therefore, all its angles.
End of proof.

Consequently,
symmetry around an axis is an invariant transformation, that is the
transformation that preserves lengths of segments and values of angles.

Theorem 2
An object symmetrical to a straight line relatively to some axis is a straight line.

Theorem 3
An object symmetrical to a plane relatively to some axis is a plane.