Tuesday, June 2, 2015

Unizor - Geometry3D - Lines and Planes - Angles between Planes

Unizor - Creative Minds through Art of Mathematics - Math4Teens

If we draw a straight line on a plane, it divides the plane into two half-planes. We will consider the points on that border line as being a part of either half-plane and will call it an edge of a half-plane.

Consider two intersecting planes σ and τ with line d being their intersection. This line d divides each plane in two half-planes:
σ = σ1∪σ2 and τ = τ1∪τ2
σ∩τ = d;
σ1∩σ2 = d and τ1∩τ2 = d;

Definition: an object formed by two half-planes from two different planes that share the common edge is called a dihedral angle.
Thus, in our case, σ1∪τ1 constitutes a dihedral angle.
Other three dihedral angles formed by two intersecting planes σ and τ are:

Line d is an edge of a dihedral angle, half-planes forming the angle are called its faces.

A notation for a dihedral angle that includes the names of its two faces and an edge in between similar to this: ∠σ1dτ1 can be used (but rarely).

Consider a dihedral angle ∠σ1dτ1 and a plane γ intersecting its edge d and faces σ1 and τ1 perpendicularly to the edge d.
This plane γ intersects half-plane σ1 by ray s and intersects half-plane τ1 by ray t. These two rays, s and t have a common vertex at point A of intersection of plane γ with the edge of our dihedral angle d.
Obviously, since s∈γ, t∈ γ and d⊥γ, we can state that d⊥s and d⊥t.
Angle formed by two rays s and t with a common vertex A is called a linear angle of a dihedral angle ∠σ1dτ1.
If, instead of plane γ, we choose any other plane δ perpendicular to edge d, the other two rays formed by the intersection of plane δ with faces of our dihedral angle will form another linear angle congruent to the one produced by plane γ because corresponding rays will be parallel. Quick proof is based on two theorems we have already covered: two planes perpendicular to the same line are parallel to each other and, if two parallel planes are intersected by a third one, the lines of intersection are parallel.

So, as we see, any dihedral angle determines its corresponding linear angle. To obtain this linear angle we just have to construct a plane perpendicularly to the edge.
The reciprocal statement is true as well. If linear angle is given, we can construct one and only one dihedral angle by performing the following:
1. Construct one and only one perpendicular to a plane where two forming rays of our linear angle are lying through its vertex. This will be the edge of a dihedral angle.
2. Construct one and only one first face of a dihedral angle through one ray of a linear angle and an edge constructed in the previous step.
3. Construct one and only one second face of a dihedral angle through another ray of a linear angle and the same edge.

Both procedures, constructing a linear angle from a given dihedral angle and, opposite, constructing a dihedral angle from a given linear angle, involved steps where one and only one element can be constructed. This leads us to a statement that we can considered as proven by this uniqueness of construction elements: congruent dihedral angles have congruent corresponding linear angles and, vice versa, if corresponding linear angles of two dihedral angles are congruent, dihedral angles are congruent as well.

As an immediate consequence of the above correspondence between dihedral and linear angles, we can establish a measure (degrees or radians) of dihedral angles as a measure of corresponding linear angles. We can talk about two half-planes with common edge forming an acute or obtuse angle or being perpendicular to each other based on corresponding properties of their linear angles.

In plane geometry two rays a and b with common vertex form, in theory, two angles, sum of which equals to 3600. Moreover, if we take into consideration the direction (clockwise or counterclockwise), each of these two angles can be positive or negative, thus making four different numeric values.
Traditionally, dihedral angles are considered in a simplified manner. Indeed, there are two different dihedral angles formed by two half-planes with common edge, but we usually consider only the smaller one that correspond the smaller linear angle and always measure it in positive units. Thus, we will only be dealing with dihedral angles measured from 0 to 180 degrees (or from 0 to π radians).

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