What is the rigorous definition of "Touch" in mathematics?

Question

This is irking me a lot :How do we define "touch" rigorously in mathematics. I've literally wasted hours because of this,one example would be a question in my textbook asking to solve the equation of circle which touches a ceratin point and a line of certain equation passes through this point.I assumed it to be tangent line and solved for it but the question assumed that line to be a secant(without stating) and due to this I wasted hours pondering over what I am doing wrong.

• Does it mean a tangent?
• Does it mean a secant?
• Or does a touch satisfy the definitiont hat some "geometrical object" withing $\epsilon$ under ceratint constraints
• Does touch is used in the sense of only once or twice(like secant) or many times(imagine a sinusoidal curve folded around a circle.Thus 'touching' it many times)

Why do I ask this?

Most of the mathematics books use the words :touch" throughout the world without a rigorous definition and same for the questions,they don't state what kind of 'touch' they mean and under what conditions.

So is there any agreed upon definition of 'touch' reached upon consensus by mathematical community with rigors and well-founded reasoning behind it?

FYI: I added certain tags which may not seem directly applicable but by use of those I want to highlight that this question should be answered in general[1 ]context applicable to those fields(since the things like point and 'touch' are closely intermingled with those[ofcourse when applying a certain geometric perspective, to be exact])

[1]I wonder there maybe different definitions of point and touching in different fields like my teacher once gave a brief light intro into different definitions of 'curve' in light of/w.r.t to different fields such as differential geometry ,topology and then analytic one,although his explanation gave intuiton(which might be wrong) that these defintion kinds of hint at same thing though seen from different light.

So I wonder if there's a general definition?(since generality is seen as 'Beauty' in mathematics) but I am skeptic there might be some exception out there under certain constraint?

Answer 1

This is not a complete, systematic answer, but an extended musing along the lines of Justice Potter Stewart's definition of obscenity. I've never seen an explicit definition of touch, but have seen the term used qualitatively to mean "tangent (at the boundary)" or "intersecting, but with non-overlapping interior". (As you may know, tangere means "touch".)

In the hope of teasing out how the term is used, here's a scattered sampling of actual and prospective usages.

1. If $C$ is the graph of a differentiable function and $L$ is a line, I would understand "$L$ (just) touches $C$ at $p$" to mean "$L$ is tangent to $C$ at $p$". Particularly, $L$ and $C$ might have other points of intersection, some or all of them non-tangencies. (The adverb "just" would signify that $p$ is an isolated point of the intersection $C \cap L$.)

   Generally, I would understand "$L$ crosses $C$ at $p$" to mean $p \in L \cap C$, but $L$ is not the tangent line to $C$ at $p$.

2. If $C$ is a closed plane curve (i.e., the homeomorphic image of a circle in the Euclidean plane), I would understand "a line $L$ touches $C$" to mean $L$ is a supporting line of $C$, namely, "$L \cap C$ is non-empty, and $C$ is contained in one of the closed half-planes determined by $L$".

3. Analogously to the preceding item, if $C$ is the graph of a continuous (presumably non-differentiable) function $f$, i.e., the zero set of the continuous function $F(x, y) = y - f(x)$, I'd tend to interpret "$L$ (just) touches $C$ at $p$" to mean "there exists an $r > 0$ such that in the open ball of radius $r$ about $p$, the function $F$ does not change sign along $L$"; again, "just" would signify that on $L$, the point $p$ is an isolated zero of $F$.

   Particularly, the graph $y = |x|$ just touches the $x$-axis at the origin, the graph $y = |x - 1| + |x + 1|$ touches the $x$-axis along $[-1, 1] \times \{0\}$, a closed convex plane polygon $P$ just touches each line that intersects $P$ in exactly one point, and touches (the line extending) each side.

   Note that according to this criterion, the $x$-axis $y = 0$ and the graph $y = x^{3}$ do not "touch" at the origin, because $y - x^{3}$ changes sign at the origin on the $x$-axis.

4. For two smooth curves or two smooth surfaces in space, I'd assume "touch" means "tangent", while a smooth curve $C$ "touches" a smooth surface $S$ at a point $p$ if the tangent line to $C$ at $p$ lies in the tangent plane to $S$ at $p$.

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One "natural general setting" for the preceding examples would be a smooth manifold, where in speaking of two subsets "touching" we assume at least one is either a smooth submanifold or the closure of an open submanifold bounded by a smooth hypersurface (such as a closed ball in space, but not a closed disk in space). (It sounds odd to me to speak of a line in space touching a closed disk, for example.)

In elementary geometric settings, it's likely there are multiple equivalent "natural" definitions. The bottom line is: When the term arises, pay close attention to context, and try to formulate a definition capturing the author's intent.

Answer 2

A common non rigorous way to look at it.. for touching to occur two lines in a plane cut at two coincident points and carry a common tangent and normal at this double point. The equation has a double root so discriminant vanishes.It is the limiting case when a secant becomes a tangent. For the secant discriminant is non-zero with distinct real roots.

In space we have a common tangential plane of contact and a common normal direction.Tangential direction in space is indeterminate at contact point.