Monday, June 22, 2015

Unizor - Geometry3D - Lines and Planes - Angles between Lines in 3D

Unizor - Creative Minds through Art of Mathematics - Math4Teens

The purpose of this lecture is to introduce a concept of an angle between any two (even non-intersecting) lines in three-dimensional space.

First of all, we will consider angles between rays (half-lines) to define angles. Two rays in three-dimensional space with a common point of origin form an angle that we can consider from a two-dimensional viewpoint by constructing (one and only) plane that contains these two half-lines.

Whatever the measure of an angle between two rays with a common origin in two-dimensional sense on the plane that contains them is - that, by definition, would be the measure of an angle they form in three-dimensional sense.

There is one nuance in this case. On the plane there are four angles between two rays with a common origin - one smaller, another that completes a full circle, and both with positive (counter-clockwise) or negative (clockwise) value in degrees or radians. Traditionally, to simplify the issue in the three-dimensional space, we consider only the smaller angle between two rays with a common origin and always measure it in positive units, so its value is always between zero and 1800 or π radians (inclusive on both sides).

Now we are ready to define the angle between two rays that do not share a common origin.

Consider we have two points in three dimensional space, A and B, and two rays, a and b, correspondingly originated from these points in some directions.
Let's choose any point P in space (in can even coincide with A or B) and construct two rays originating in this point P: a' ∥ a and b' ∥ b.
We assume that rays a' and b' are constructed not only parallel to, correspondingly, a and b, but also similarly directed, so, if we make a parallel shift from point P to point A, rays a' and a will coincide and, analogously, with shifting point P to point B in regards to rays b' and b.

A three-dimensional angle between rays a' and b' was defined above.
By definition, this is the angle between rays a and b.

Thus, we have defined the angle between two rays in three-dimensional space that share or do not share a common origin. Let's discuss the correctness of this definition.

Our first task was to construct two rays, a' and b' originating from any point P and correspondingly parallel to rays a and b.
This construction has been discussed in the previous lectures and is based on the fact that we can always construct one and only one plane that contains a straight line (a or b) and a point P, after which we can construct one and only one line in that plane through point P parallel to a line (a or b).
So, construction is always possible and unique.

Our second task is to prove that, if we choose a different point Q in space as a new origin and construct rays a" and b" correspondingly parallel to rays a and b, we will end up with an angle between new rays a" and b" congruent to the one between rays a' and b'.
Here is the proof:
Ray a' is parallel to ray a, ray a" is parallel to ray a. Therefore, rays a' and a" are parallel to each other, as has been proven before. Similarly, b' ∥ b". Therefore, angles formed by pairs of rays, one - by a' and b', another - by a" and b", have correspondingly parallel sides. As has been proven before (see topic "Plane ∥ Plane" - Theorem 4), these angles are congruent.
Therefore, the value of the angle between two rays in three-dimensional space is independent of the position of a new point of origin P that we used to define this angle, which proves that our definition makes sense.

Switching from rays to full lines presents no additional problems. There are four ways we can construct pairs of rays from two straight line. We define the smaller angle among these four combinations as the angle between the lines.

Now we can talk about perpendicularity of two lines, even if they do not intersect in three-dimensional space (skew lines).

Theorem 1
Perpendicular to a plane is perpendicular to any line on that plane.

Theorem 2
Perpendicular to a plane is perpendicular to any line parallel to that plane.

Theorem 3
A plane, that is perpendicular to one of two skew lines perpendicular to each other, is parallel to another line.

No comments: