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Calculus books often give the "secant through two points coming closer together" description to give some intuition for tangent lines. They then say that the tangent line is what the curve "looks like" at that point, or that it's the "best approximation" to the curve at that point, and just take it for granted that (1) it's obvious what that means, and (2) that it's visually obvious that such statements are true.

To be fair, it's true that (1) I can sort of see what they mean, and (2) yes, I do have some visual intuition that something like that is correct. But I can't put into words what exactly a tangent line is, all I have is either the formal definition or this unsatisfying vague sense that a tangent "just touches the curve".

Is there a purely geometrical definition of a tangent line to a curve? Something without coordinates or functions, like an ancient Greek might have stated it. As an example, "A line that passes through the curve but does not cut it" is exactly the kind of thing I want, but of course it doesn't work for all curves at all points.

Jack M
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    I think one of the (main?) difficulties lies in the fact that you still have to define what that curve is. The definition of "tangent line" will depend a lot on the definition of "curve". It does not seem easy to me to define a (continuous) curve with purely synthetic geometry notions (i.e., as a set of points, instead of as a point-valued function). And continuous curves themselves are still very far from having a "visually obvious" concept of what a tangent is (think for example to a Peano curve). – Giovanni Mascellani Oct 06 '14 at 10:09
  • @giomasce: I completely agree. Elementary geometry does not deal with such abstract objects as curves, but rather with concrete ones, e.g. straight lines, angles and circles. Therefore one needs to employ other tools like e.g. differential geometry, algebraic geometry or topology. These yield definitions of curve and tangent line which coincide with intuition when considering elementary geometric objects like circles, but also include corner cases not accessible to intuition. – bodo Oct 06 '14 at 10:33

9 Answers9

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A tangent line may or may not cross the curve at the point of tangency, but among all lines through the point of tangency it is always at the boundary between those that cross the curve in one direction at that point and those that cross it in the other direction at that point.

Michael Hardy
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  • Could you clarify what you mean by "one direction" and "other direction"? – Jack M Oct 08 '14 at 22:49
  • @JackM : Suppose for a moment the curve is a graph of an equation $y=f(x)$. Every nonvertical line through the point $(a,f(a))$ is of the form $L(x)=f(a)+m(x-a)$. Suppose that for some $\delta>0$, for every $x\in(a-\delta,a+\delta)$ we have $f(x)<L(x)$ for $x<a$ and $f(x)>L(a)$ for $x>a$. That's crossing the curve in what we can call the increasing direction. Then suppose $f(x)>L(x)$ for $x<a$ and $f(x)<L(x)$ for $x>a$. That's crossing the curve in what we can call the decreasing direction. If $f$ is differentiable at $a$, then there is some number $m_0$ such that the line crosses.... – Michael Hardy Oct 09 '14 at 22:47
  • ...the curve in one direction if $m<m_0$ and in the other direction if $m>m_0$. That number $m_0$ is the slope of the tangent line. – Michael Hardy Oct 09 '14 at 22:48
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I believe there is no good geometric definition of a tangent line: at least, no definition that covers all cases. We need calculus for a good definition. Here is a quote from the book Calculus: Graphical, Numerical, Algebraic by Ross L. Finney et al., page 84.

The problem of how to find a tangent to a curve became the dominant mathematical problem of the early seventeenth century and it is hard to overestimate how badly the scientists of the day wanted to know the answer. Descartes went so far as to say that the problem was the most useful and most general problem not only that he knew but that he had any desire to know.

The textbook then gives the usual calculus definition using limits.

To see the difficulty of finding a definition that always works, try using geometry to explain why the $x$-axis is the tangent line to the curve

$$f(x) = \left\{ {\begin{array}{*{20}{c}} {{x^2}\sin \frac{1}{x},}&{x \ne 0} \\ 0,&{x = 0} \end{array}} \right.$$

at $x=0$.

Rory Daulton
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Let $\mathcal A$ be a set of points, let $P$ belong to $\mathcal A$, and let $\mathcal T$ be a line through $P$. Then $\mathcal T$ is a tangent-line to $\mathcal A$ at $P$ if the following condition is satisfied. If $\epsilon$ is any positive number, there is a positive $\delta$ such that every secant $PX$ for which the distance of $X$ from $P$ is less than $\delta$ makes an angle smaller than $\epsilon$ with $\mathcal T$.

enter image description here

This is Hugh Thurston's answer (in his article On Tangents). Note that this definition is with respect to "a set of points" and not to a "curve". Thurston states that there are two reasons for this. The first is that there is no universally accepted definition of curve. A definition that was in favor for a long time is that a curve is the path of a continuously moving point. After Peano showed that such a curve can completely fill a square, attempts were made to modify the definition, none winning 100% acceptance. Luckily the most interesting curves for the study of tangents are simple arcs, about whose definition there is no controversy. The second reason for not confining our attention to curves is to include tangent lines to graphs of functions. Curves and graphs are by no means the same thing, even in a plane. A circle is a curve, but is not the graph of a function, and on the other hand, the graph of the Dirichlet function is not a curve.

In his work, Thurston also makes a brief but illustrative historical review of the "geometric" concept of the tangent line and discusses other problems related to the existence of tangent lines and their intimate relationship with the concept of derivative.

G al Cubo
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Excluding inflection points, we could state, a tangent line through a given point on a curve is the boundary of a half-plane which contains some segment of the curve to either side of the point, and for which the point itself is on the boundary.

Milo Brandt
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"...the tangent line, or curve, is the limit to all secant lines or curves."
- Smithsonian Contributions to Knowledge VOL. VIII page 249

More specifically we could say: A tangent line is the limit to all secant lines that are parallel to it.

Honest Abe
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We can translate the epsilon-delta definitions into more geometric language. A tangent line to a curve $\gamma$ at a point $x$ is a line $L$ through $x$ that has the following property:

  • Every open truncated cone with vertex $x$ that intersects $L$ also intersects $\gamma$.

Here, "open truncated cone with vertex $x$" means an open set that contains every point between itself and $x$. Unless I've made a mistake, this definition is a shortcut way to state that $L$ is contained in the Bouligand tangent cone to $\gamma$ at $x$.

Chris Culter
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The best geometrical definition of the tangent line to a curve is the one given by Leibniz in his 1784 article providing the foundations of his infinitesimal calculus. The definition of the tangent line is the line through two infinitely close points on the curve.

Mikhail Katz
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One answer to this ancient question is an idea I worked out (but haven't formally published) a year or two ago. I call this the "zoom" definition of differentiability and the tangent line.

Suppose you have an image of the curve and you zoom in ten times, and then a hundred times and so forth. (I mean zooming both in X and Y directions.) The function is differentiable at a point if and only if the zoomed curve converges to a straight line in a specific sense. In fact, you only need the absolute value of the distance between the exact straight line and the curve to converge to zero.

You don't have to calculate any differential quotients at all, which is different to the way the Cauchy-Weierstraß epsilon-delta approach works with limits of (f(x)-f(a))/(x-a) etc. My assertion might sound like it's unlikely to be true, but I am asserting that for all "zoom" levels of magnification, the supremum of the vertical distance between curve and line must converge to zero.

Conclusion: If you are willing to accept the idea of "zooming" as geometric, and the idea of maximum distance between curve and line as geometric, then this is fairly close to a geometric notion of tangent line and differentiability. And of course, you can tilt the axes a little if you wish, either with a rotation or an affine transformation, as long as the function is still a function and the limiting line is not vertical.

PS. The following diagram shows the basic concept. Epsilon must go to zero faster than delta. This is related to the ancient "method of exhaustion" of Eudoxus and Archimedes.

defining tangent line by zooming in

Alan U. Kennington
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  • This is something like the measure-theoretic definition of tangency. – jwg Oct 06 '14 at 07:33
  • Actually it has nothing at all to do with any kind of measure theory. I didn't provide enough information perhaps. The theorem I wrote for this is not published, as far as I know, but it is elementary. If the zoomed-in picture of the function converges the sup-norm to the straight line, then the function is differentiable at the point, and the derivative equals the gradient of the line. The proof is not obvious, but it is obvious when you've proved it! – Alan U. Kennington Oct 06 '14 at 08:29
  • I know that yours does not involve measure theory, however it is similar in spirit. You want none of the curve to lie outside a narrow tube around the tangent. The measure-theoretic notion of tangency needs only a very small amount of (the measure of) the curve to lie outside a similar narrow tube. – jwg Oct 06 '14 at 08:37
  • Yes, you have understood the definition which I am using, which avoids using numbers, and uses geometric distance instead, still with an epsilon and delta, but with no differential quotient. Now if you want to say that a sup-norm is similar to an $L^\infty$ norm, then there is some similarity. But my concept has no measure theory in it at all. – Alan U. Kennington Oct 06 '14 at 08:42
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I'm not sure if this is geometrical enough or too close to the calculus approach, but it may be a helpful visualization for "what the curve "looks like" at that point":

Let's approximate the curve with a polyline (e.g. with equidistant points that all lie on the curve). If we crank up the number of points, zoom in to $x$ and look at the corresponding line segment, we see the tangent. (The line segment is coincident to the tangent line.)

Sebastian Negraszus
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