A catenary is the curve formed by a perfectly flexible, uniform chain hanging under gravity. With the origin at the lowest point (vertex), the standard equation is:
In the seabed-contact case the vertex coincides with the lift-off point on the seabed, so y also equals height above seabed. In the all-lifted case the vertex lies outside the chain span and a shifted form of the equation is used — see Section 6.
Chain sizes used: 8 mm (1.45 kg/m dry), 10 mm (2.3 kg/m dry), 12 mm (3.8 kg/m dry).
The arc length of a catenary from the vertex to a point at height y is:
This formula is used in the table and top graph to find the minimum chain length required to just lift all chain off the seabed at a given water depth.
For a given depth D and catenary parameter a, the minimum chain length that just barely lifts off the seabed at the anchor is found by applying the arc-length formula (Section 3) with y = D:
This threshold divides the calculator into two distinct computation paths:
When there is seabed contact, the chain splits into a flat section on the seabed and a suspended catenary section that rises from the lift-off point to the boat.
The catenary arc is plotted using y = a(cosh(x/a) − 1) for x from 0 to xrise, shifted right by Lseabed in world coordinates.
When the chain is entirely off the seabed, both the anchor end (at height 0, seabed level) and the boat end (at height D) are suspended. The catenary vertex lies outside the chain span, so the standard vertex-origin form must be shifted.
Let p = anchorLocalX / a and q = p + xrise/a be the positions of anchor and boat in the local catenary frame (origin at vertex). The two constraint equations are:
Dividing arc length by height using the product-to-sum identity gives tanh(p + xrise/(2a)) = D/L, and combining with the arc-length equation gives sinh(xrise/(2a)) = sqrt(L² − D²) / (2a). This leads to the three intermediate values computed in the code:
From which the geometry follows directly:
The height above seabed at world distance x from the anchor is:
At x = 0 this gives y = 0 (anchor at seabed). At x = xrise it gives y = D (boat at surface).
The chain tension at the anchor has a vertical component directed upward — it tries to pull the anchor off the seabed:
For each water depth shown in the table, the calculator computes the minimum chain length required to just lift all chain off the seabed using the arc-length formula: Lmin = sqrt( D(D + 2a) ). Scope ratio is L / D. Force down chain and vertical force are also shown for each depth.
Both the main red curve and the reference dashed curve are drawn using identical physics. The origin (0, 0) in both cases corresponds to the anchor position on the seabed.
The x-axis of the chart is scaled to fit whichever curve (main or reference) has the greater horizontal extent.
The take-off angle is the angle the chain makes with the horizontal at the point where it leaves the seabed (or, in the all-lifted case, at the anchor).
A widely used anchoring rule of thumb recommends a minimum chain length of:
where D is water depth in metres. The 20-metre base provides a minimum catenary even in shallow water; the 2×D factor scales the scope with depth to maintain adequate shock absorption and holding power.
The calculator draws this as a dashed reference curve on the chain-curve chart, using the same catenary physics as the main red curve but with L set to 20 + 2D. The reference curve is only shown when the recommended length is less than or equal to the chain length the user has entered — if you have less chain than the recommendation the dashed curve is hidden.