Mamikon’s Visual Calculus and Hamilton’s Hodograph

[This is a condensed version of an article [5] in Mathematics Today]

A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon’s Theorem states that The area of a tangent sweep of a curve is equal to the area of its tangent cluster’.  We shall illustrate how this theorem can help to solve a range of integration problems.

The area of an annulus

An annulus, the area between two concentric circles of radii ${r}$ and ${R}$.

The inspiration for his theorem came to Mamikon while, as an undergraduate, he examined how to calculate the annular area ${A}$ between two concentric circles, given only the length ${2a}$ of the chord tangent to the inner circle (see Figure). Since ${a^2 = R^2 - r^2}$, it follows that ${A = \pi(R^2-r^2) = \pi a^2}$. For a given ${a}$, this area is independent of the radii, ${r}$ and ${R}$, of the inner and outer circles.

Mamikon considered a segment of length ${a}$ tangent to the inner circle (Figure below, left panel). It is clear that the annulus is the area swept out by the tangent segment as it rotates around the inner circle, the tangent sweep (Figure middle panel). Mamikon realised that this area, comprising the sum of numerous triangular regions, remains unchanged if all the regions are moved parallel to themselves so that all the tangent points coincide, forming the tangent cluster (Figure, right panel). Since the area of the tangent cluster is ${\pi a^2}$, so is that of the tangent sweep, or annulus.

Left: A single tangent segment. Middle: The tangent sweep of a large collection of segments. Right: The tangent cluster, with all segments emanating from a common point.

Integrating Functions

Mamikon showed that the areas of the tangent sweep and tangent cluster are equal in much more general circumstances. The circles forming the annulus may be replaced by any smooth convex curves, closed or open, and the lengths of the tangents do not have to be constant.

Tom Apostol, author of several influential textbooks on calculus, was a strong supporter of Mamikon’s methods [1,2]. In [1] he gave several examples of integral evaluation using Mamikon’s theorem. The exponential function ${y=\exp(x/b)}$ and the parabola ${y = c x^2}$ can easily be integrated in this way. More generally, polynomial functions are easily integrated using Mamikon’s approach.

The area under a cycloid arch is three times the area of the generating circle or three-quarters of the area of the surrounding rectangle. Using Mamikon’s Theorem, we can show that the area can be found by simple geometric reasoning, without any equations or integrations [1]. This was described in an earlier thatsmaths post.

Hamilton’s hodograph

The hodograph is a vector diagram showing how velocity changes with position or time. It was made popular by William Rowan Hamilton who, in 1847, gave an account of it in the Proceedings of the Royal Irish Academy [3]. The underlying idea is very simple: velocity vectors at different times or places are plotted with a common origin, or emanating from a single point. The hodograph is the locus of the arrow-heads. Their varying directions and magnitudes make a pattern that can yield dynamical information in a visually clear way.

In 1609, Kepler published his law of the ellipse, shattering the arguments of the ancient Greeks that circular orbits, being the epitome of perfection, must be found. However, the circle re-emerged some 237 years after Kepler, when Hamilton announced his Law of the Circular Hodograph [3]. In the Figure below, the left panel shows a Kepler orbit with velocity vectors.

Hamilton discovered the remarkable fact that if all velocity vectors are plotted from a common point, they trace out a circle of radius ${\bar v = (v_P+v_A)/2}$ (Figure, right panel). Hamilton’s hodograph is also what Mamikon calls a tangent cluster. Since the area of the hodograph is ${\pi \bar v^2}$, Mamikon’s theorem shows that the region swept out by the vectors around the Kepler orbit (shaded region in Figure, left panel) also has this area.

Left: Kepler orbit with velocity vectors. Right: circular hodograph.

Discussion

Apostol [1] observed that the great contribution of Newton and Leibniz was to demonstrate the relationship between differentiation and integration. He remarked that Mamikon’s method has some of the same ingredients, because it relates moving tangent segments with the areas of the regions swept out by those tangent segments’. Thus, the relationship between differentiation and integration is embedded in Mamikon’s method.

I was informed by Paul Nicholson (Leeds) that Holditch (1868) proved a theorem analogous to Mamikon’s Theorem [4]. However, Mamikon’s method is considerably more general and more powerful.

References

[1] Apostol, Tom M., 2000: A visual approach to calculus problems, Engineering and Science, vol. LXIII, no. 3, pp. 22-31. (A PDF of this article can be found at https://calteches.library.caltech.edu/4007/1/Calculus.pdf ).

[2] Mamikon Mnatsakanian, 1997: Annular rings of equal area. Math Horizons, vol. 5, no. 2, pp. 5-8.

[3] Hamilton, William Rowan, 1847: The hodograph, or a new method of expressing in symbolical language the Newtonian law of attraction. Proc. Royal Irish Academy, vol. 3, pp. 344-353.

[4] Holditch, Rev. Hamnet, 1858: Geometrical Theorem, Quarterly Journal of Pure and Applied Mathematics, Vol 2, London, pg. 38.

[5] Lynch, Peter, 2022: Mamikon’s Visual Calculus and the Hodograph. Mathematics Today, April 2022, 212-214. Proof PDF.

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