One of the hardest concepts to assimilate in the first classes of projective geometry is the improper point. A improper point is a point at infinity and can be translated or interpreted as direction.
While metric geometry two lines intersect or are parallel, in projective geometry always intersect at a point proper or improper, what does not change in any way the operation with this geometric-mathematical model.
My students have wanted to highlight this aspect in their jobs and, experience in educational innovation in which the subject developed in blogs, offered this curious article. The group “Projecting-ando” he displayed his name:
Parallel lines intersect at infinity, Myth or Reality?
We've always heard that two parallel lines are those that extend much never get to cut, but we also know the concept that two parallel lines intersect at infinity. Which of these two statements is true? Then try to answer the dilemma in which we.
Parallel Lines? ?
Euclid it was a Greek mathematician and geometer, who lived around 300 A.C. It is known as “Father of Geometry” and was the creator of the geometry that bears his own name.
The Euclidian geometry is one that studies the properties of flat and three-dimensional space. The presentation of this is done through a system of axioms that, from a number of assumptions that are assumed to be true and through logic operations, generates new assumptions whose truth value is also positive. Euclid's five postulates raised in your system:
- Given two points you can draw one and only one straight line joining.
- Any segment can be extended continuously in either direction.
- You can draw a circle with center at any point and any radius.
- All right angles are equal.
- If a line, by cutting two, form angles smaller than a right angle, these two lines intersect indefinitely longer side at angles that are less than two right.
The latter assumption, which is known as the parallel postulate, fue reformulated as:
5. For un punto una outside the line, se puede una trazar only parallel to it given straight.
Euclid asumió sus postulates that all the self-evident axioms eran y by both Acts that the requerían demostración. Sin embargo, el fifth postulate that resultó bien si es compatible con otro los cuatro, es cierto so Independient. Namely, both el fifth postulate it as negation del fifth postulate, son compatibles con los otros cuatro postulates. Las geometries where no es el fifth postulate is valid llaman in-Euclidean geometries.
En el Renaissance las nuevas needs for Representación del arte y la technique empujan the ciertos humanists studio propiedades geometrical. Al discover her perspective y la sección, create the need to lay the formal foundation on which build new forms of geometry that this implies: the Geometría proyectiva, principles which appear in the seventeenth century:
- Two points define a line.
- Every pair of lines intersect at a point (when two lines are parallel we say that intersect at a point of infinity known as improper point).
Through these two principles we can get the answer to our question. The difference is found in the fifth postulate of Euclid (of the parallel); says: "Through a point outside a line, you can draw a single parallel to the given line ". This axiom, in projective we have seen that there, so that there are no “parallels”; all lines are secant, namely, intersect at a point. Therefore, point is the concept of improper (the subscript infinity; because they do not represent a particular place as the other points); which would determine the “address” the line. All straight-euclideanamente- would “parallels”, projectively intersect at the same point improper turn and improper all points determine a line plane improper, unique in that plane.
Although we just stated, in conclusion the answer to our question of whether parallel lines intersect at infinity is the following: PARALLEL LINES FROM THE POINT OF VIEW OF GEOMETRY Projective ARE CUT IN THE INFINITE, But based on Euclidean geometry RECTAN NOT REACH THE NEVER CUT.
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