Geodesic problems.
You know how to draw geodesics on a surface using adhesive tape. Question: under what conditions can a geodesic drawn on a cone intersect itself?
Take a point on a right circular cone and start a geodesic in a direction perpendicular to one of its generators:
Consider the generator symmetric with respect to the cone's axis of revolution (any cone can always be deformed into a right circular cone without altering the geodesic pattern). In the case illustrated above, we would obtain the following when laying the cone flat:
We know that the angle of the cut then represents the amount of angular curvature concentrated at the cone's apex. The geodesic then becomes a straight line in the plane, since the surface is developable.
We see that for a geodesic to intersect itself, the cut angle must be greater than 180°, meaning the cone must be sufficiently sharp.
When reconstructing our cone, we obtain:
Can a geodesic on a cone "reach the apex"?
Only the cone's generators can do so. No matter how close a geodesic is drawn to the apex, it will always move away from it—even if it appears to be drawn toward it. Simply connect the cone's apex to the point on the geodesic closest to it. The generator will then intersect the geodesic at a right angle. We can make a cut along the opposite geodesic and lay the surface flat.
No matter how sharp our cone is, we will only ever obtain successive intersections.
Can geodesics intersect indefinitely? When developing the cone, it's as if the geodesic "bounces" off the generator joining the apex to the point of intersection.
Above, clearly, the "bounce" sends the two parts of the generator in directions that will never intersect again. To achieve multiple intersections, the cone must be very sharp.
But at each "bounce," the angle opens up and eventually becomes trapped within the sector 2π − q. The number of intersections is therefore finite.
The cone's generators form a very special family. But what exactly do we mean by a cone?
We can consider the object "cone" as depicted on the left. In this case, the geodesic-generators are half-lines.
But we can also consider a cone as the object on the right. In this case, what do we mean by a geodesic? If it is defined as the shortest path joining two points, we might encounter situations like this:
We could adopt a conical structure where each generator extends continuously into a second generator located on the second half-cone, and only one such continuation exists, forming a continuous whole. We can conceive of conical points in three-dimensional space (see article 11 of Geometrical Physics A).
Other types of singularities.
Cuspidal points are singular points. Others can be identified, such as "conical points," where surface inflection points—points of "hairy" appearance.
On the left: a sphere with a conical point. On the right: a point of "hairy" appearance.
We create a conical point using a punch. We could thus call this modification "creation of a conical point" P, and its inverse P⁻¹.
Similarly, the creation of a "hairy" point corresponds to the modification H. In fact, the formation of a hairy point follows that of a conical point. It is a conical point whose apex angle has become zero. Thus, the modification leading to a local hairy point on a surface would be P H, and its inverse: H⁻¹ P⁻¹.
There are other ways to modify a surface—for example, by creating a dihedral. The creation of a dihedral is the modification D. This can be implemented independently, provided it concerns a closed path (on a regular surface). The simplest example is the sphere. We can create a "crease" along its equator, for instance. Incidentally, this crease contains "linear curvature," a topic already treated in the introduction of Geometrical Physics A.
If, on a regular surface, this modification affects a segment, each endpoint will undergo a modification P.
Take a sphere—a soft, deformable sphere. Place ourselves inside with a rigid rod and push it into the sphere. The two ends of the rod begin to make contact with the surface. Effect of the "punch": the appearance of two conical points. We continue pushing. The segment comes into contact with the sphere, but the dihedral has not yet formed. If it is in contact with the sphere, this only means there exists a straight path AB on the sphere. But this does not automatically imply that the sphere has a crease. This is analogous to setting up a camping tent with two poles. We install the poles,
Effect of two P modifications: creation of two conical points A and B.
then stretch a cable between them. But if the interior of the tent is under pressure, the fabric will not sag along the cable to form a crease.
Tension in the cable: the surface acquires a straight segment AB. But if the wind blows and the tent is slightly pressurized, the neighborhood of the segment may still maintain tangent plane continuity along the segment—evident from the tent's appearance viewed from another angle.
If the wind stops blowing, the tent walls will sag under their own weight. As soon as motion begins, the tangent plane continuity breaks. The dihedral appears. Modification D.
What use is this?
Before moving on to practical applications, we must define another modification. Imagine a cone: it has a conical point concentrating "angular curvature." If the conical point does not belong to a "true" cone whose side has no curvature, the surface is locally indistinguishable from a cone near the conical point. This means that at any conical point on a surface, there exists a "tangent cone."
But let's return to our cone. We can easily bring two conical points into proximity. We can even physically construct such a surface from two cuts made in a plane:
The lines emanating from A and B are simply "seams" or "glues." We can remove them:
We can merge the conical points, in which case this "bicone" becomes a single cone whose apex contains an angular curvature equal to the sum of the curvatures concentrated in the two conical points we merged.
q₁ + q₂
Thus, we can consider a new modification: "confluence of conical points":
ConfP
With its inverse:
(ConfP)⁻¹
Now take a sphere and push eight segments into it. This will create eight dihedrals and sixteen conical points. We can then merge the conical points two by two to obtain... a cube.
We now know how to transform a sphere into a cube, and vice versa. Similarly, this will help us construct polyhedral representations of surfaces. Now, "inflate" our cube, just as we inflated the tent fabric earlier:
The edges have disappeared. We obtain a result identical to that achieved by pushing in eight "pegs."