Graphic PIZiadas

Graphic PIZiadas

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Representation systems : Incidence (Intersections) [ Descriptive geometry ]

intersección recta y planoIncidence problems trying to identify common elements of two geometric figures; can be defined as special cases of belonging. Are independent of the metric system of representation and they can be solved by generalized models all.

Starting from the basic geometric elements straight and flat, We can apply the concepts of duality to analyze the possible problems that may occur.

  • Section a straight for plane is defined the point belonging to the two elements
  • Section a straight other straight is defined the point belonging to the two elements
  • Explode a plane other plane defining the line is owned by both elements
  • Explode a plane by straight is defined the point belonging to the two elements
In general, these problems will be solved by auxiliary planes conveniently choose as we will see later.
As general concepts apply in different cases for determining intersections, we can state that:
  • The intersection of two planes having a common address at both levels
  • Three planes intersect at a point
  • On sectioning a plane parallel planes parallel lines are determined together.
  • A straight, projection onto a given plane intersect on the plane of projection.

Intersection of line and plane

We will solve this problem in Diédrico system without reducing generality in the model resolution. Spatial concepts are identical, paths as well as derivatives.

The beam of a straight flat base (r) sectioned in a plane p according to a straight beam vertex point (I) intersection (r) and the plane p.

Intersección de recta y plano

Intersection of line and plane

To determine the intersection of a plane (a) and a straight (r) a plane is used (b) assistant to the line containing. Intersection (i) between the planes containing the point (I) Searches

The auxiliary plane is chosen so that it is projecting on the projection plane. This means it contains the projection direction and thus be represented as a straight line. In addition dihedral will be fulfilled that, when the projection direction normal to the plane, be perpendicular to the plane of projection.

Intersección de recta y plano

Intersection of line and plane

Assume the following example in which the intersection is sought to produce a line in the plane defined by two intersecting lines.

ejemplo Intersección de recta y plano

Example: Intersection of line and plane defined by two straight

  • Straight (r) and (s) pass through the point (P) and determine a plane (a).
  • The line (to) intersects the plane at the point (I) which is what we want to determine the dihedral projections.

El plano (b) contains the line (to) still projecting on vertical projection, and its intersection with the plane (a) determines the straight i, (a ∩ b), containing the point (I).

Resolucion Interseccion de recta y plano

Resolución : Intersection of line and plane

Junction plans

Consider first a spatial approach to the problem can allow us to reduce the problem to the previous case of intersections.

We can perform two approaches to this problem.

Intersección de dos planos

Analysis of the Intersection of two planes

  • First we use two auxiliary planes severed at the alpha and beta levels in two lines each. These lines in turn are cut into two point (I1 and I2) which belong to the searched intersection.
  • The second approach is to choose two lines of one of the plans and determine the points of intersection occur in the other plane, as seen in the example of intersection of line and plane.

In both cases the use of auxiliary planes is part of the methodology of resolution.