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 to find that point: a) calculated with Blender modelling/editing tools; b) no scripts involved; c) with some geometric reasoning behind. The result therefore won't be mathematically exact (we would need to do some maths and coding to achieve that) but quite an aceptable one, a very good approacching and backed by geometric logic.
Figure 1. The famous plane-line intersection.
1.2. The main purpose of exercises like that is to bypass current and unpredictable Blender booleans tools. Once we are able to find that plane-line intersection, we can apply this process to a wide range of situations were we need to know which is the intersection between two objects of the scene. In the case below (Fig.2) I've calculated the intersection between the prisma and the plane, using the principles described in the paragraphs below.
Figure 2. An application of this practice.
1.3. The first exercise consists on finding the projection of an object (line) onto another (plane). We'll 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 menu, with the Cursor to Selection option we can place the cursor on a vertice previously selected. Then, if you put your pivot in cursor mode (Period Key), that vertice 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).
Figure 3. A plane and a line that intersect each other.
1.4. In a ortho side view (3 NumKey), select an upper vertice of the plane as pivot. Then select the opposite vertice 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 we've created and join them by an edge (F key).
Figure 4. Projecting the line onto the plane.
1.5. Then we change to the orthographic front view (1 NumKey). We select a vertice 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 to the edges, the more precise the result. And that's it. That point will be the plane-line intersection. You can chek the result of the exercise by rotating the view (MMB).
. Front view of the resulting line.
. Scaling the line in the front view.
We can develop further this kind of geometric reasoning to solve more Descriptive Geometry exercises. In fact, if Blender can help us to solve this simple principle, then It could help us to solve any Descriptive Geometry exercise, even the most difficult ones. The next challenge will be the intersection between two planes (Fig.7
). Is that possible?
. 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 vertice 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 we've calculated by edges.
Figure 8. Projections of plane edges onto the other.
1.8. As we see in the sequence below, when we change to the front view, we will probably need to change the vieport shading into solid mode to get a grasp of the situation. Then, scale your resultant lines according with what you see. When your lines have been scaled, join then and that will be the intersection between those planes. You can chek your result by rotating the view.
1.9. That's it. This time we've had a bit of fun with Blender by applying old principles of Descriptive Geometry, in fact a couple of hundred years old! A buch of interesting links about Descriptive Geometry:
Written by Alvaro Luna Bautista, draftsman.