For instance, consider the triangle in space (see Figure 2.4). Assume that a triangle $A,B, C$ is projected to two different mutually perpendicular projection planes. The vertices of the triangle are mapped to points $A',B',C'$ and $A'',B'',C''$ in the projection planes. Furthermore, assume that the plane that supports the triangle contains the line $l$ in which the two projection planes meet. Under this condition the images $ab'$ and $ab''$ of the line supporting the edge $AB$ will also intersect in the line $l$. The same holds for the images $ac'$ and $ac''$ and for $bc'$ and $bc''$. Now let us assume that we are trying to construct such a descriptive geometric drawing without reference to the spatial triangle. The fact that $ab'$ and $ab''$ meet in $l$ can be interpreted as the fact that the spatial line $AB$ meets $l$. Similarly, the fact that $ac'$ and $ac''$ meet in $l$ corresponds to the fact that the spatial line $AC$ meets $l$. However, this already implies that the plane that supports the triangle contains $l$. Hence, line $BC$ has to meet $l$ as well and therefore $bc'$ and $bc''$ also will meet in $l$. Thus the last coincidence in the theorem will occur automatically. In other words, in the drawing the last coincidence of lines occurs automatically
I am totally confused on whatever is going on in the above paragraph.
Here is what I understand so far:
Monge's method involves projecting a 3-d Object onto some planes and then reconstructing the object back in 3d from the projections. An analogy I found helpful for thinking about this is how we can construct a vector by projecting it's component with respect to a set of standard bases.
What I don't understand:
- What is ab' and ab'', bc and bc''? It is not defined in the passage..
- Where is the original triangle ABC in the picture? has it collapsed onto a one dimensional line in our viewing prespective?