What can we say about the parallel curves of a convex curve?  If the curve is smooth, with no flat points, 
then the parallel curves for a sufficiently small radius will remain convex.  Then they will begin to develop cusps, 
when the radius of the parallel surface is equal to the radius of curvature at a point of the curve.  In general that 
will happen at an even number of points, and after a time the parallel curve will once again be convex.  
	This analysis is correct for most curves, but there can be other examples where the parallel curves exhibit 
more complicated behavior.  One class of such curves is the piecewise circular curves, where a parallel curve will 
have a corner whenever the radius equals the radius of one of the arcs of the curve.  Of course when the curve 
itself is a circle, the only singularity will occur when the radius of the parallel curve is the radius of the circle and 
the parallel curve as a whole degenerates to a single point.
	We would like to investigate a generalization of convexity that relates to curves with cusps as well as to 
smooth curves.  The key notion probably should be called "projective convexity".  We define a mapping from the 
curve to the projective line, sending each point to its tangent line, as opposed to the tangent vector on the unit 
sphere.  Then the notion of convexity is well-defined, with local convexity meaning that there are no cusps in the 
circular image mapping and global convexity meaning that the projective gauss map is a twofold covering 
mapping.  We can ask if every mapping of this sort has a parallel curve that is convex in the Euclidean sense, and 
that might be an interesting question in itself.  
	Now, what about surfaces?  Once again for a hypersurface we can define a mapping to the projective space 
of lines through the origin, and we would hope then that the parallel surfaces of a tight surface will be projectively 
tight, even when the parallel surfaces cease to be embeddings or immersions.  In general we expect the parallel 
surfaces to have singular cuspidal edges along the places where the parallel surfaces intersects one of the evolute 
surfaces.  But in certain cases, for example the taut cyclides of Dupin, the evolute surfaces degenerate into curves.  
The parallel surfaces they must have conical singularities.  What will be the effect of the projective gauss mapping 
on such a singular situation?  For a torus of revolution, the ordinary gauss map is four-to-one.  What happens for 
a spindle cyclide?  The same thing should be true.  
	This does give an alternative approach to the concept of minimal total absolute curvature, which would be 
the only notion that could be applied in the case of a non-orientable domain, and also for a non-orientable range.  
It no longer makes sense to speak of the gauss mapping preserving or reversing orientation after all.  
	Recall that we can also make sense of the Steiner formula for the total volume of a parallel surface if we 
treat the integrand not as an absolute value but rather as a signed quantity.  We have a formula for Xr(u,v,) = 
X(u,v) + r N(u,v), where N gives the gauss map.  Then, as opposed to | Xru x Xrv| dudv, we look at Xru x Xrv¥N 
dudv = dA + 2rHdA + r2KdA.  That way we get the Weyl formula for the volume of a tube even in the case where 
the parallel hypersurface is singular.   
	So it seems that we should start by considering those properties of submanifolds that are preserved under 
taking parallel surfaces and develop this geometry independently.  Then we can look at how the properties change 
when we add on new restrictions, like invariance under Mšbius transformations.  From the point of view of 
Laguerre geometry, we are considering the properties of submanifolds that are independent of translation in the 
vertical direction, that being the direction that determines the radius of a circle.  When we look at a smooth 
hypersurface, we often look in the space of all affine planes, those being the ones that are tangent to the 
hypersurface.  In a geometry based on circles, we look at the collection of osculating circles.  But we might just as 
easily look at the collection of osculating quadrics, to pick up the second order behavior.  Working with planes 
means we want to restrict consideration to transformations that preserve those planes; similarly we can look for 
transformations that preserve second order behavior, and these are more likely to involve projective geometry.  
Anyway in a sense we are studying the geometry of contact elements, but we might not have to be so specific.  
For example if we just specify the family of hyperplanes, we can recapture the original hypersurface as an 
envelope so we don't have to give that information to begin with.  The same should be true for certain kinds of 
hypersurfaces.  We know that the cyclides are the envelopes of a family of spheres tangent to three fixed spheres.  
How does that fit into Laguerre geometry?  The spheres tangent to one sphere correspond to points of a 
hypercone.  Two such hypercones intersect in a pair of quadric sections, and three will apparently intersect in a 
conic section.  As in the case of circles in the plane, if we take a cone (x-a)2 + (y-b)2 + (z-c)2 = (r-d)2 then this 
object gives all quadruples corresponding to spheres tangent to the sphere centeres at (a,b,c) with radius r, where 
the tangency is orientation-preserving.  If we have another such cone (x-a')2 + (y-b')2 + (z-c')2 = (r-d')2, then the 
common points lie in the hyperplane (a - a')(-2x +a + a') +(b-b')(-2y +b + b') + (c-c')(-2z +c + c')=  (d-d')(-2r + 
d + d').  If we bring a third cone into the act (x-a")2 + (y-b")2 + (z-c")2 = (r-d")2, then we get another hyperplane, 
and the intersection of these two will be a plane that cuts the hypercone in a planar conic section, as expected.  
This produces the one-parameter family of spheres that have the cyclide as their envelope.  The centers of those 
spheres lie on a conic section in three-space, and their radii should somehow be related linearly