Piercing Points and Plane Intersections | Basic #2

Описание: 📐Edge-View Method

Assuming that the straight line is neither parallel to nor in the plane, it will intersect the plane at a point common to both the line and the plane. The limits of the line and plane as given may have to be extended to determine this "pierce point". Since an edge-view of the plane contains all points in the plane, the view showing the plane as an edge will also show the point where the line pierces the plane.

❗️Another method would involve showing the line as a point. This point of view of the line will contain the pierce point common to both the line and the plane.

📋Example 1: (Given Plane Appears as an Edge)

The plane ABCD and the line EF are given in both the plan and front elevation views. The plane appears as an edge in the plan view. The intersection of the line and the plane is determined by point M which is common to both. Using careful visualization, we notice that the FM portion of the line lies wholly in front of the plane and therefore will appear visible in the front view. The plan view shows that the EM Jportion of the line lies wholly behind the plane and, therefore, will be hidden in the front view.

📋Example 2: (Oblique Plane)

The plane ABC and the line EF are given in both the plan and front elevation views. Draw an auxiliary elevation view showing the plane as an edge. The pierce point M in this view is the point of intersection which can now be projected back to the plan and front elevation views. Careful visualization reveals the visible and hidden portions of the line.

❗️The pierce point could also have been determined by showing the plane as an edge in an inclined view projected from the front view.

📐Two-View Cutting Plane Method

The intersection of a line and an oblique plane can be determined by using a vertical cutting plane that contains the given line. The line of intersection of the cutting plane with the oblique plane and the given line must intersect or be parallel because they both lie in the vertical cutting plane. Since the cutting plane appears as an edge in the plan view, the relationship between the line of intersection and the given line is not apparent in the plan view. The related view, however, reveals this relationship. Should the two lines intersect in the related view, it is evident that the point of intersection is common to both the given plane and the given line and therefore determines the pierce point of the given line and given plane.

📋Example

The oblique plane ABC and the line EF are given in both the plan and front elevation views. A vertical cutting plane, coincidental with and containing the given line EF, appears as an edge in the plan view. The intersection of the given plane ABC and the vertical cutting plane containing EF is line 12. The lines EF and 12 both lie in the vertical cutting plane and intersect each other at point M in the front view. Since point M is on line 12, it is also on plane ABC because line 12 is on plane ABC. Therefore point M is the required point, being common to both the given line EF and the given plane ABC. It can now be projected to the related view. Use careful visualization to determine what portion of the line should be visible in each view.

❗️If line 12 had appeared parallel to EF in the front view, it would have indicated that the line EF was parallel to plane ABC and therefore it would have no point of intersection with the given plane.

#descriptivegeometry #point #intersection