Fixed points of isometry groups in Euclidean space

Unizor - Geometry3D - Axis Symmetry

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• published: 21 Sep 2015
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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.