Learning Path Metric Geometry

In addressing the study of a science we can follow different paths that lead to learning. Chaining concepts linked to each other allow us to generate a mental representation of abstract patterns, facilitating their assimilation and subsequent application in problem solving.
In these pages two images that summarize a possible strategy or sequence of progressive incorporation of the basics of this branch of science in the education of our students are proposed.

Diédrico System: Straight lines in a plane parallel to the projection

Under the so-called category “notable lines” plane are those that are parallel to the planes of projection diedricos. These lines are very useful in the operation that we will develop in this system of representation.

Diédrico System: Theorem of the three perpendicular

One of the most important theorems of descriptive geometry is the so-called “Theorem of the three perpendicular”, It establishes a relation between two lines perpendicular when one of them is parallel to a plane of projection.

Diédrico System: Projection of points in the plane

Can you get from a projection of a belonging to a flat point another projection on the plane dihedral to the full? For example, If give us the horizontal projection and vertical of a plane and a point in the latter as determinaríamos the projection on the horizontal plane?

Diédrico System: Projection of the plane

A plane is determined by three unaligned points, so adding a new point to a straight line projections can define it. In this case we will give at least two related dimensions on each plane of projection in order to become independent projections of these plans support of representation. We will learn to represent maps and items belonging to them.

Geometría proyectiva: Conjugate polar diameters

We have seen the definition of polar conjugate diameters, given to analyze the concept of Conjugate directions:

Conjugate polar diameters: They are polar two conjugated improper point.
Let's see how we can relate this concept with the triangle's autopolar seen in Involutions in second-order series.

Geometría proyectiva: Conjugate directions

The concepts of polarity we've seen to determine the polar of a point on a line, you have allowed us to obtain the autopolar triangle of a conical setting three different involuciuones with four points, They allow us to advance in the projective definition of its notable elements, diameters, Center and axis.

One of the basics is the of “Conjugate directions”

Geometría proyectiva: Tangent from a point to a conical

We have seen how to determine the points of intersection of a straight line with a Conic defined by five points. We will then see the dual problem.

This problem consists of determining the possible two straight tangent from a point to a Conic defined by five tangent.

Geometría proyectiva : Center of involution

We have seen how to determine the axis of an involution and, based on the concept of polar of a point with respect to two lines, possible Involutions which can be set from four points, with their respective shafts of involution, obtaining the autopolar triangle associated which are harmonious relations of the cuadrivertice full.

In this article we will continue to enhance these elements, in particular in the autopolar triangle vertices that will determine what is known as “Center of involution”.

Projective Geometry: Autopolares triangles in Involutions in second-order series

Connecting four points of a conical proyectivamente by Involutions we determine the axis of involution of these proyectividades.

Given the four points needed to define an involution, We can ask many different Involutions can establish between them.

Polar of a point with respect to two lines

The concept of polarity is linked to the harmonic separation.

This concept is Basic for the determination of the fundamental elements of conics, as its Center, conjugate diameters, axes ….

It will allow to establish new transformations which include homographies and correlations of great importance.