Where a strand leaves a body

The picture of a belt drive that everyone has seen has four features: two circles, two straight lines that touch them, and the strand hugging each circle for part of its circumference. Three of those four are decisions somebody made. The straight lines could have been drawn anywhere near the circles and the picture would still read as a belt.

They could not, in fact, and this essay is about why the one right answer is right — because the reason turns out not to be a fact about strands at all.

The construction, and what it is doing #

Take two bodies, give each a wrap sense, and the strand between them is decided.

Along the run, let $\mathbf{u}$ be the direction of travel and $\mathbf{n}$ its left normal. A body of radius $r$ wrapped anticlockwise sits with its centre at $+r\,\mathbf{n}$; wrapped clockwise, at $-r\,\mathbf{n}$. So with $\Delta$ the difference of the two signed radii,

$$\mathbf{c}_2 - \mathbf{c}_1 = L\,\mathbf{u} + \Delta\,\mathbf{n},$$

and the run’s length is $\sqrt{D^2 - \Delta^2}$, its direction the direction between the centres turned by $\arctan(\Delta / L)$, and the tangency points are the centres stepped back along $\mathbf{n}$ by their own radii.

Two circles admit four common tangents and the two senses pick one of them. Both bodies wrapped the same way gives $\Delta = r_2 - r_1$ and one of the outer pair; wrapped opposite ways gives $\Delta = r_2 + r_1$ and one of the two that cross between the circles. On the pair in the hero figure — 44 and 26 mm, centres 170 apart — the outer tangents are 169.044 mm long and the crossed ones 154.919, and the four of them come out of one expression with two signs in it.

The perpendicularity is then not asserted, it is checked. Over two hundred randomly proportioned pairs, open and crossed, the largest departure from a right angle at a tangency — measured as the component of the strand’s direction along the radius — is $2.6 \times 10^{-14}$ mm, and the largest failure of the run to reach from one tangency to the other is $5.7 \times 10^{-14}$. Those are the arithmetic’s noise floor, which is what they should be: the construction is closed-form, so there is nothing to converge and the only thing that can be wrong is the formula.

The refusal is part of the formula #

$D^2 - \Delta^2$ can be negative, and when it is there is no strand.

For a same-sense pair that means one body is inside the other: two pulleys of 40 and 24 mm cannot carry an open belt with their centres 15.99 mm apart, because the second circle is then inside the first and the tangent does not exist. For a crossed pair the threshold is the sum of the radii — the strand has to pass between the two bodies, so they must not overlap. At 64.0001 mm on the same pair the crossed run is 0.10 mm long and both wraps are 359.797°, a mechanism in which the belt is very nearly a pair of complete circles.

The routine refuses at both, and it refuses with the numbers in the message rather than returning a NaN. That matters more here than in most of this site’s libraries, because a NaN in a tangency point propagates into a path that still draws: the polyline simply skips the bad segment, and the figure looks like a belt with an unusual routing.

What the wrap does with the leftovers #

Once both tangencies are known, the wrap is arithmetic: the angular distance from where the strand arrives to where it leaves, taken in the direction the body is wrapped. It is where the asymmetry of a belt drive lives, and it is what the turning number constrains.

An open belt shares its wrap unevenly, and always in the same direction. On 40 and 24 mm pulleys 160 mm apart the large one gets 191.478° and the small one 168.522°. Pull the centres in to 70 mm and the split widens to 206.426° and 153.574°; push them out to 400 mm and it narrows to 184.585° and 175.415°, approaching a half turn each as the strand’s two spans become parallel.

The small pulley always gets less, and the small pulley is usually the driven one. That is the geometric half of a fact every belt drive is designed around, and the other half — how much torque a given wrap can carry before the strand slips — is a force argument and is not here.

Tangency is what shortest looks like #

Now the question this essay exists for. The construction above assumes the strand meets each body at a right angle — it is built into the step along the normal. Why is that the right assumption?

The usual answer is that a taut strand takes the shortest path available to it, and that the shortest path round an obstacle is tangent to it. The second half of that is the part worth checking, and it can be checked without ever mentioning tangency.

Take a strand from one fixed point to another, past a peg it cannot go through. Consider every path of this shape: leave the first point in a straight line for some point on the peg’s surface, follow the surface for a while, then leave in a straight line for the second point. Two numbers describe such a path — where it first touches, and how far round it goes — and nothing in the description says anything about angles. Plot the length of the best such path against where it first touches, and the family has a minimum.

The minimum of the family is 200.64346 mm, and the tangent construction — the one that steps along the normal and never considers any other path — gives 200.64346 mm. The two agree to $3 \times 10^{-7}$ mm, which is the resolution of the grid the family was searched on rather than a disagreement, and the minimising path is exactly the one whose two straights meet the surface at a right angle.

So the right angle is not an assumption about strands that happens to be true. It is what a minimum looks like: a path that met the surface at any other angle would have a corner in it, and a corner can always be cut. Fermat’s argument about light, reached from a different direction and with a body in the way.

Why that is worth having in a library #

A construction that is correct is not the same as a construction that is checked, and the difference is exactly the kind of thing this site keeps finding.

The tangency formula has no residual of its own. It converges nothing, so there is nothing for a solver to report and no number that can come out large when it is wrong. A sign error in the $\Delta$ term would give tangency points on the wrong sides of the bodies — a path that is perfectly smooth, perfectly closed, perfectly plausible and a different mechanism. The minimisation above is the only test in this field that could have caught it without a second implementation, because it asks a question the construction was never given: is this the shortest one?

It also gives the field its vocabulary. Taut means the length constraint is active, which is the same as saying the strand is on the shortest path its route permits. Nothing about tension is needed for that, and nothing about a material: an inextensible line pulled at both ends is a geometry problem, and the moment it goes slack it stops being one, because the shape of a slack strand is decided by things this field does not model.

Three consequences, before the ladder goes on #

The straight parts of a strand are the hull’s edges. If a loop wraps every body the same way, the tangency points are the bodies’ outlines stepped out along the same normal, so the straight runs are the convex hull’s edges pushed outward by the radii. That is why the loop over four equal pegs comes out at exactly the hull’s perimeter plus one circumference — 503.136 plus 87.965, or 591.100 mm, against the 591.100 the run measures.

A body inside the hull is not touched, and it never becomes touched by degrees. A tangent exists between any two circles that are not nested, but a strand only lies on one if the path it makes is the shortest. So the tangency solve answers for pairs of bodies, and whether a body is on the run at all is a question about the set.

Where the tangency is decides what the body feels. The perpendicular distance from a body’s own axis to the strand is its arm: the amount of strand paid out per radian of its rotation. For a circle that distance is its radius, at every angle, which is why nobody notices that it is a separate quantity — and the moment the body is not round, or is not turning about its own centre, the arm and the radius part company.

The crossed tangent is a line somebody else already named #

The two tangents that pass between the circles get much less attention in belt drawings than the outer pair, and they are the more interesting of the four, because a mechanism this site has been drawing since its first field is built on one.

Put two circles on fixed centres, cross a strand between them, and the strand’s perpendicular distance from each axis is that axis’s own radius — by construction, since that is where the tangency was placed. A strand does not slip on what it wraps, so the two bodies must turn in the ratio of those two distances, and that ratio is fixed by the radii alone. Move the centres apart and the strand’s direction changes, its length changes, its tangency points move round both circles — and the ratio does not move at all.

That is the property involute gearing is chosen for, and it is not a fact about the involute curve. It is the fact that a crossed strand’s ratio is the ratio of its arms, which are the two base radii whatever the centre distance is. The essay on the unwound strand makes the connection properly; what belongs here is only that the four tangents are not equally useful. Two of them are how a belt runs, and one of the other two is how a gear pair transmits — which is what makes an involute pair indifferent to its centre distance.

The arc is not the same kind of thing as the run #

Between two tangencies the strand is straight and its length is a formula. On a body it lies along the surface and its length is $r\theta$, and the difference between those two is worth one paragraph because it is where all the mechanism is.

A straight run transmits nothing. It couples the two bodies’ positions — move one and the other must move — but a body could slide along its own tangent and the strand would neither know nor care. The wrap is where a strand meets a body’s rotation: winding the arc on at one end and off at the other is what turns a length constraint into a relation between angles, and it is why the wrap angle appears in every belt-drive calculation and the run length appears in none of them.

The arc also carries this field’s one connection to the rolling constraint. The strand does not slide on the body it wraps, so the contact is a rolling one: the material of the strand at the tangency is instantaneously at rest relative to the surface, exactly as the contact patch of a wheel on a road is. The tangency point travels along both the strand and the surface, at the speed the strand runs, and nothing material moves with it. That is the same sentence the rolling field opens with, and it is the reason a mesh and a belt behave so differently: two gear teeth slide against each other everywhere except at the pitch point, and a strand slides nowhere at all.

The two-pulley case, checked against the book #

None of the above is new, and the last check in this essay is against the textbook. The classical open-belt length is $2C\cos\gamma + (\pi + 2\gamma)r_1 + (\pi - 2\gamma)r_2$ with $\sin\gamma = (r_1 - r_2)/C$, and the crossed one is $2C\cos\gamma + (\pi + 2\gamma)(r_1 + r_2)$ with $\sin\gamma = (r_1 + r_2)/C$. Both were derived by exactly this construction, some time in the nineteenth century, and then written down as formulas so that nobody would have to do it again.

Across forty-two proportions, open and crossed, the run summed from its own tangents and arcs agrees with those formulas to $1 \times 10^{-13}$ mm.

That agreement is the least interesting result in this essay and the most useful one. The formula is exactly right and applies to two round pulleys with a strand round both of them; the construction it came from applies to a tendon over an elbow, a rope round four pegs, a tackle, a chain on two sprockets and a cable on a shaped drum. The general case is not harder — it is the same three lines — and the reason the formula exists at all is that in 1870 somebody had to evaluate $\sqrt{D^2 - \Delta^2}$ by hand and would rather do it once.

The run and the arc are one variational problem #

The essay separates the straight run from the arc — one transmits, the other does not — and mechanically that separation is right. Variationally they are the same object, and saying so explains the right angle rather than merely confirming it.

A taut strand is a shortest path in the plane with the bodies removed. Such a path is straight wherever it is free, because a straight line is the shortest route between two points; and where it is forced onto a body it follows the body’s boundary, because a boundary arc is the shortest route along an obstacle it cannot cross. The whole path is one geodesic with two kinds of piece.

The condition where the pieces meet is what a variational argument calls transversality, and it says the free piece must leave the constrained piece without a corner — tangentially. A path with a corner at the departure can be shortened by rounding the corner, so the minimum has none. Tangent to the circle at the departure point is the same statement as perpendicular to the radius, which is the construction’s right angle.

So the perpendicularity is not a fact about strands and not a fact about circles. It is the first-order condition of a minimisation, and the sampled family in the essay’s measurement finds it because a sampled family finds minima — which is why the family, told nothing about tangency, arrives at 200.64346 mm and so does the construction.

Read that way, the field’s other results line up as consequences of one variational statement rather than as separate constructions. A run is the hull’s edge because a geodesic between two convex bodies is a common tangent. A body inside the hull is untouched because the geodesic never reaches it. And the contact boundary’s quadratic behaviour is what a minimum looks like when the constraint it is against is just becoming active.

That also says what would break the whole apparatus, which a construction on its own cannot. Anything making the strand’s length not the thing being minimised — a strand with bending stiffness, one with friction against the bodies, one whose tension varies along its length — breaks the variational statement, and with it the tangency. Every one of those is on this field’s list of what is not modelled, and now for a single reason rather than four.

What is not modelled #

The strand has no thickness, so a tangency is a point and a wrap is an arc of the body’s own radius; a real belt runs on its pitch line, some way out from the pulley’s surface, and every radius here would be that pitch radius. Nothing here knows about grooves, flanges, or a V-belt’s wedging, all of which change the effective radius and none of which change the geometry of where the strand leaves. The wrap angle is reported and never converted into a capacity: what a given wrap will hold needs a coefficient of friction, and there is not one anywhere in this field.