Thursday, April 23, 2015
Unizor - Geometry3D - Elements - Points, Lines, Planes
Geometrical points, lines and planes are abstractions, as most other objects of mathematics, like numbers, functions etc.
It is appropriate to present definitions of these concepts given by Euclid around 300BC:
"A point is that which has no part."
"A line is breadthless length."
"A straight line is a line which lies evenly with the points on itself."
"A surface is that which has length and breadth only."
"A plane surface is a surface which lies evenly with the straight lines on itself."
As definitions, from a modern viewpoint, these do not serve the purpose because they use other concepts like "part", "length" etc., which are not defined. So, consider them not as definitions, but rather explanations of mathematical concepts of a point, a line and a plane.
The contemporary set of axioms of geometry that includes those related to solid geometry in three-dimensional space were developed by David Hilbert in the beginning of the 20th century. They are presented on the Web site that hosts this and all other lectures - UNIZOR.COM - in notes for this lecture.
We will use the following short list of main propositions related specifically to planes, that we take as axioms, and that are based on the more formal list of axioms above:
A1. If two points of a straight line belong to a plane, every point of this line belongs to this plane.
A2. If two planes have a common point, they intersect along a straight line passing through this point.
A3. For any three points not lying on the same straight line there is one and only one plane that contains them.
From the above axioms we can derive the following simple theorems:
T1: For any straight line a and a point X outside of this line there is one and only one plane α that contains both of them, line a and point X.
We have to prove two things: the existence of a plane that contains a given straight line and a point outside it and the uniqueness of this plane, that is that there is only one such plane.
Choose any two points A and B on a given line a. Using axiom A3 above, there is one and only one plane α that contains these two points and the one outside of a line - point X.
Now, using axiom A1, we can say that an entire straight line a belongs to this plane.
So, plane α satisfies the condition of existence. But is it the only one?
Assume that with two different points on line a, C and D and the same point X we construct a different plane β that contains them. Since points C and D lie on line a, by axiom A1 an entire line a belongs to plane β as well as to plane α. Hence, three points X, C and D not lying on the same line belong to both plane α and plane β. Based on axiom A3, there is one and only one plane that contains these three points. Therefore, planes α and β are identical.
T2: For any pair of intersecting straight lines a and b there is one and only one plane α that contains them both.
Let X be a point of intersection of lines a and b. Choose a point A on line a and a point B on line b.
Using axiom A3, construct a plane α that contains three points X, A and B.
This plane α contains an entire line a because it contains two points on it - X and A (axiom A1). Plane α also contains an entire line b because it contains two points on it - X and B. Therefore, the condition of existence of a plane α
is satisfied. But is it unique?
Yes, because any other plane β that contains both lines a and b would inevitably be identical to α since both planes must contain the three points mentioned above - X, A and B.
T3: For any pair of parallel lines there is one and only one plane that contains them both.
The existence of such a plane follows from the definition of the parallel lines. The uniqueness of this plane follows from axiom A3, if we choose any three points - one on one line and two on another.
T4: For any given straight line in space there are infinite number of planes that contain it.
Choose any point outside the line and construct a plane that contains both this point and an original line. Now you can choose any point outside this plane and construct another plane that contains this new point and an original line. Choose yet another point outside both previously constructed planes and construct a third plane that contains this new point and an original line. This process can continue infinitely.
All these planes are different and all contain an original straight line. They can be considered and a set of positions of one plane that rotates around a straight line - an axis or rotation.