I Term: generalization of parallel curves to 3D

[nomadreid]

I know that the generalization of parallel lines to curves in 2D is just "parallel curves", but is there any term which generalizes the idea of skew lines to curves? "Skew curves" doesn't work, this term already being co-opted by statistics. Example: if you had two curves coiling around each other in the fashion of the typical simplified diagram for the DNA double helix, where each curve (strand) is separated from the other by a constant distance (length of the base pairs). (For the biologists, please don't write to tell me how the simplified diagram is not accurate. I am not asking a question in biology, but just using it as an example of the form.)

 

[Nidum]

Answer removed .

 

[nomadreid]

Thanks for the input and pretty picture, Nidum, but apparently I expressed my question poorly, since you seem to have missed the point of my question. I was not asking specifically about helices. I was merely giving this as an example. My question was for the general term to characterize the non-intersection of two non-linear curves in 3-D space, as well as one which could characterize a pair of such curves that remained a constant distance from one another.

 

[Nidum]

I understood the question .

 

[Stephen Tashi]

I know that the generalization of parallel lines to curves in 2D is just "parallel curves", but is there any term which generalizes the idea of skew lines to curves?

I don't understand what concept you wish to generalize. I can draw two "skew" lines without worrying much about any relationship between them - I just have to make sure they don't intersect. So if I generalized that idea to curves, then two non-intersecting curves would be "skew curves". However, the notion of "parallel" curves suggests curves that have a more restrictive relation.

 

[nomadreid]

I guess I was actually asking two questions. The main one: it does, indeed, seem natural to call two curves that do not intersect in 3-D "skew curves" as a generalization of "skew lines" , but a google search comes up with two definitions of skew curves :
(a) "a curve in three-dimensional space that does not lie in a single plane" https://www.merriam-webster.com/dictionary/skew curve
(b) page 162 , definition 3, of http://link.springer.com/chapter/10.1007/978-3-319-01736-5_4#page-1
(and, being sloppy, "positive skew curve" from probability/statistics, even through that parses as "positive-skew curve" rather than "positive skew-curve", for "positively skewed distribution").
Therefore it seemed dubious whether one could use "skew curve" in the sense of two non-intersecting curves not lying in the same plane.
The second question is less important, but whether, once one had a term for two non-intersecting curves not lying in the same plane, whether the additional restriction of the two curves being equidistant from one another would have a name as well, but upon reflection, the only two smooth continuous curves that I can think of that would fit that description would be the double helix, no?

 

[Stephen Tashi]

Therefore it seemed dubious whether one could use "skew curve" in the sense of two non-intersecting curves not lying in the same plane.

Are we pursuing a study in lexicography or are we in search of a apt definition for a particular mental concept?

I haven't studied the various definitions of "skew curve", so I don't know there is a "standard" definition for it in mathematics.

The second question is less important, but whether, once one had a term for two non-intersecting curves not lying in the same plane, whether the additional restriction of the two curves being equidistant from one another would have a name as well, but upon reflection, the only two smooth continuous curves that I can think of that would fit that description would be the double helix, no?

Curves don't have to lie in a plane. Are we only interested in those that do?

Defining "equidistant" for two curves is an interesting problem. It's intuitively clear what that means, but formulating a mathematical definition seems difficult. For example, two concentric circles in the same plane are (intuitively) "equidistant" from each other. In that case a clear method of measuring distance from a point on one curve "to the other curve" suggests itself. We measure the distance along a radial ray drawn through the point. But for curves of a more complicated shape, how do we define where to take the measurement?

I suspect the simplest approach to mathematical definition of "parallel curves" is to say that one curve can be transformed into the other by certain types of transformations. However, what types of transformations shall we pick?

 

[nomadreid]

Thanks, Stephen Tashi. Working from the bottom up,

I suspect the simplest approach to mathematical definition of "parallel curves" is to say that one curve can be transformed into the other by certain types of transformations. However, what types of transformations shall we pick?

I would suggest that a translation or an enlargement (but not both) in such a way that, if T is the transformation, for every point P on the original curve, the distance between P and T(P) stays constant. In this way, in the (simplified) DNA double helix, each strand can be translated to the other, so the two helices are parallel, and two concentric circles would also be parallel. But not having considered all possible curves, this would probably need polishing.

Defining "equidistant" for two curves is an interesting problem. It's intuitively clear what that means, but formulating a mathematical definition seems difficult. For example, two concentric circles in the same plane are (intuitively) "equidistant" from each other. In that case a clear method of measuring distance from a point on one curve "to the other curve" suggests itself. We measure the distance along a radial ray drawn through the point. But for curves of a more complicated shape, how do we define where to take the measurement?

Good point. For two parallel curves, I see two possibilities:
(a) to fit in with my suggested definition above, the distance between P and T(P).
(b) the smallest distance between two points on the curve.
But the second one clashes with the first one, so I do not really know the best definition.

Curves don't have to lie in a plane. Are we only interested in those that do?

No, as my example with the DNA helices indicate.

Are we pursuing a study in lexicography or are we in search of a apt definition for a particular mental concept?

Somewhere half-way between. I am translating and editing an article for someone who appears to be a bit shaky both on his terminology and on some of the concepts.

 

[mfb]

The second question is less important, but whether, once one had a term for two non-intersecting curves not lying in the same plane, whether the additional restriction of the two curves being equidistant from one another would have a name as well, but upon reflection, the only two smooth continuous curves that I can think of that would fit that description would be the double helix, no?

To phrase that mathematically: Let the first curve be parametrized by A(x) where $x \in [0,1]$, let the second curve be parametrized by B(x) with the same range. Let d(a,b) be the Euclidean distance between two points.
How can we classify all curves that satisfy this?
$$\exists c \in \mathbb{R}: \forall x \in [0,1]: \min_{y \in [0,1]} d(A(x),B(y)) = c \land \min_{z \in [0,1]} d(A(z),B(x)) = c$$
In words, for each point on either curve, the closest point on the other curve has to have a distance of c.

The helix is an obvious example, two parallel lines are an obvious example, two concentric circles are an obvious examples, two "parallel circles" (like two car wheels) are an obvious example but we can also combine those shapes. We can have two lines that are straight and parallel for a while, then curve like two concentric circles, get straight again, curve like two parallel circles, ... and we can probably mix and combine those patterns in many complex ways.