Plane-line intersection

plane-line intersection
By Álvaro luna Bautista.
alvarolunab@yahoo.es

1.1. This article deals with the old plane-line intersection exercise that every student Will find in his descriptive geometry practices. Well, everyone knows that the plane-line intersection is actually a point (fig. 1). The process described in this article Will help us todo find that point: a) calculated with Blender modelling/editing tools, b) no scripts involved, c) with some geometric reasoning behind. The result therefore wont be mathematically exact (we would ned todo do some maths and coding todo achieve that) but quite an aceptable one, a very god approacching and backed by geometric logic.

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Figure 1. The famous plane-line intersection.
1.2. The main purpose of exercises like that is todo bypass current and unpredictable Blender booleanas tools. Once we are able todo find that plane-line intersection, we can apply this process todo a wide range of situations were we ned todo know which is the intersection between two objects of the scene. In the case below (fig.2) ive calculated the intersection between the prisma and the plane, using the principales described in the paragraphs below.

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Figure 2. An application of this practice.
1.3. The first exercise consists on finding the projection of an object (line) onto another (plane). Well find that by using consecutively side and front view in orthographic mode. Before starting, lets take a look on the Snap menú (shift+s keys). On that menú, with the cursor todo selection option we can place the cursor on a vértice previously selected. Then, if you put your pivot in cursor mode (period key), that vértice Will be the pivot for scaling and rotating operations. Lets go. We start from a single object in Edit Mode. That object consist in a plane and a line that intersect one each other(fig 3).

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Figure 3. A plane and a line that intersect each other.
1.4. In a Ortho side view (3 numkey), select an upper vértice of the plane as pivot. Then select the opposite vértice of that edge, duplicate it (shift+d key) and scale it (s key) until it coincides with the line. Do the same process with the lower vértices. (fig.4). Select the two vértices weve created and join them by an Edge (f key).

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Figure 4. Projecting the line onto the plane.
1.5. Then we change todo the orthographic front view (1 numkey). We select a vértice from the resulting Edge as pivot and scale the opposite one until it coincides with the line (fig.5 i 6). The closer your are todo the Edges, the more precise the result. And that it. That point Will be the plane-line intersection. You can chek the result of the exercise by rotating the view (Mb).

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figure 5. Front view of the resulting line.

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figure 6. Scaling the line in the front view.
1.6. we can develop further this kind of geometric reasoning todo solver more descriptive geometry exercises. In fact, if Blender can help us todo solver this simple principle, then it could help us todo solver any descriptive geometry exercise, Even the most dificult ones. The next challenge Will be the intersection between two planes (fig.7). Is that posible?
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figure 7. Two planes that intersect one each other.
1.7. We start from a single object in Edit Mode. That object consist in two planes that intersect one each other (fig. 7). Then, in a Ortho side view (3 numkey) we Project the Edges of a plane onto the other as described in paragraph 1.4 (fig. The left vértice of the upper Edge is the 1st, pivot for scaling vértices two times, one for the left Edge and other for the right one. The same for the 2nd, pivot. We join the vértices weve calculated by Edges.

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Figure 8. Projections of plane Edges onto the other.
1.8. As we se in the sequence below, when we change todo the front view, we Will probably ned todo change the vieport Shading into solid mode todo get a grasp of the situation. Then, scale your resultant lines according with what you se. When your lines have ben scaled, join then and that Will be the intersection between those planes. You can chek your result by rotating the view.

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1.9. That it. This time weve had a bit of fun with Blender by applying old principales of descriptive geometry, in fact a couple of hundred years old. A buch of interesting enlaces about descriptive geometry:
Written by Álvaro luna Bautista, draftsman.
alvarolunab@yahoo.es. http://blenderart.org/

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