Economía circular 4.4 Huerta

The previous section developed polyhedral convex cones as a fundamental technique for analyzing a variety of manipulation problems. For spatial problems, the polyhedral convex cones live in the six-dimensional space of wrenches or differential twists. For planar problems, the polyhedral convex cones live in a three-dimensional space of wrenches or differential twists.

But consider Reuleaux’s method for analyzing constraint (section 2.5). It represents polyhedral convex cones in differential twist space, and it requires only two dimensions, not three. It represents a differential twist by the corresponding rotation center, labeled either ⊕ or . The technique of labeling points in the plane is the key to Reuleaux’s method. The space of labeled planar points is called the oriented plane. This section provides a formal definition of the oriented plane, and applies it to represent polyhedral convex cones. Following sections will apply the idea to obtain specific techniques for analyzing planar wrenches and differential twists for planar problems.

w x y positive and negative planes superimposed line at infinity Figure 5.9

The oriented plane.

DEFINITION5.5: Consider all homogeneous coordinate triples(x, y, w) with x, y, and w all real numbers, and not all simultaneously zero. Each such triple determines a directed line through the origin. A point in the oriented plane is a ray of triples:

{(kx, ky, kw) | k > 0} (5.41)

all of which give the same directed line. We distinguish three cases:

w > 0 : For positive w, the ray (kx, ky, kw), k > 0, maps to a point in the Euclidean plane with Euclidean coordinates(x/w, y/w) and with a ⊕ label. Equivalently, we say it maps to the positive plane with coordinates(x/w, y/w).

w < 0 : For negative w, the ray (kx, ky, kw), k > 0, maps to a point in the Euclidean plane with Euclidean coordinates(x/w, y/w) and with a  label. Equivalently, we say it maps to the negative plane with coordinates(x/w, y/w).

w = 0 : For zero w, the ray (kx, ky, kw) is an ideal point. It maps to neither the positive nor the negative plane, but to the ideal line, or line at infinity. For a graphical representation, we can map an ideal point to a point on the unit circle(x, y)/|(x, y)|.

The oriented plane is best understood by the central projections illustrated in figure 5.9. (See the appendix for more on central projection and related material on projective ge- ometry.) If we superimpose the positive and negative planes at w = 1 in homogeneous coordinate space, then we obtain the correct mapping by intersection. An upward pointing

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(w > 0) directed line intersects the positive plane at a point labeled ⊕. A downward point- ing (w < 0) directed line intersects the negative plane at a point labeled . A horizontal (w = 0) directed line intersects neither plane, and is thus an ideal point, or a point at infinity, represented by intersection with the equator.

Thus we envision the oriented plane as two planes plus a circle. The circle is the glue between the two planes. The planes are connected in such a way that a point moving to infinity in one direction, say the+x axis of the positive plane, reappears from the opposite direction, the−x axis of the negative plane. This is readily observed by using the projection of figure 5.9, and letting some ray in homogeneous coordinate space cross the equator.

Just as the projective plane can be regarded as the set of lines through the origin of E3, the oriented plane can be regarded as the set of directed lines through the origin of E3. And, just as the projective plane can be regarded as the sphere S(2) with antipodes identified, the oriented plane can be regarded as the sphere S(2) with antipodes not identified—just a plain sphere. The northern hemisphere is the positive plane, the southern hemisphere is the negative plane, and the equator is again the ideal line.

GEOMETRY AND CONVEXITY

We can do geometry in the oriented plane. For example, two points determine a line, provided they are not antipodes. We can construct the line by working in the homogeneous coordinate space. Each point of the oriented plane is a ray in homogeneous coordinate space. Unless the points are antipodes, the two rays determine a plane. That plane intersects the sphere in a great circle, and intersects the positive and negative planes in lines. Unless, of course, the two given points are ideal points, in which case we obtain the ideal line.

In practice we work directly with labeled points as Reuleaux did, rather than employing the central projection. We draw both planes superimposed, and label points “⊕” or “” depending on whether they are in the positive or negative plane. When it is necessary to consider ideal points we can use a circle.

Here is the rule for constructing the convex hull of two points in the oriented plane. (See figure 5.10.)

• If the two points are antipodes, that is they have the same coordinates in the plane but opposite signs, the convex hull is just the two points.

• If the two points have the same sign, construct the line segment joining them and give it the same sign as the two points.

• If the two points have opposite sign, construct the line through the two points. The two points divide the line into three parts:

Figure 5.10

Line segments on the oriented plane.

–a ray bounded by a “⊕” point – label that ray “⊕”; –a ray bounded by a “” point – label that ray “”;

–a line segment between the “⊕” and “” points – erase that. There is also a point at infinity connecting the two rays.

There are other cases to deal with, involving points at infinity, but it is actually easier to figure out the rules than it is to remember them, or to write them down.

How do we know these rules are right? By carrying out the corresponding operations on the sphere, and examining the projection onto the oriented plane. Given two unique points in the oriented plane, construct the corresponding rays in homogeneous coordinate

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space, take the convex hull of the rays, giving a planar cone or a line, then project the results back to the oriented plane. For example, the rule for antipodal points follows easily from observing that the convex hull of anti-parallel rays is just a line, which projects back to the original points.

Using convex hull of points, we can construct convex polygons in the oriented plane. There are some interesting departures from plane geometry. We have already noted that a set of two antipodal points is convex. Another interesting case is the figure with two edges and two vertices (or is it four?) but nonempty interior. See exercise 5.2 for its construction using convex hull in the oriented plane, and compare it with the central projection of figure 5.5g.

Now for the key observation. The mapping from convex polygons in the oriented plane to polyhedral convex cones in R3is one to one. The mapping is just the central projection we used in figure 5.9. Thus the oriented plane is a very practical tool for representing polyhedral convex cones in R3. The rest of this chapter explores variations of this approach. First we will see that Reuleaux’s method and the line of force are both instances of this technique. Then we develop new methods: force-dual and moment-labeling.