Determine the longitudinal position of the waterplane centroid using waterplane moments or sectional half-breadths with Simpson’s rule.
LCF from AP: — m
LCF as % of LPP from AP: — %
LCF from AP: — m
LCF as % of LPP from AP: — %
The longitudinal centre of flotation is the fore-and-aft position of the geometric centroid of a ship's waterplane area at a stated draft. It describes where the waterplane area is balanced longitudinally.
LCF is a position, not an area or moment. A reported value must therefore state its longitudinal datum, positive direction, length reference and loading condition.
This calculator uses the aft perpendicular (AP) as its origin. Positive distances are measured forward, with x = 0 at AP and x = LPP at the forward perpendicular (FP).
The coordinate of the waterplane centroid is its first longitudinal moment divided by its total area:
xF = ∫x dA ÷ AWP
On this page, xF is reported as LCF measured forward from AP.
| Symbol | Meaning | Usual unit |
|---|---|---|
| xF | Longitudinal coordinate of the centre of flotation | m |
| x | Longitudinal coordinate measured from the selected datum | m |
| dA | Element of waterplane area | m2 |
| ∫x dA | First longitudinal moment of waterplane area about the datum | m3 |
| AWP | Total area of the selected waterplane | m2 |
| LPP | Length between aft and forward perpendiculars | m |
Dividing a first moment in m3 by an area in m2 produces a longitudinal distance in metres.
Datum requirement: the area moment and the reported LCF must use the same origin and positive direction. This calculator assumes a moment about AP with positive x measured forward.
Consider the following waterplane properties:
LCFAP = 139,733.333 ÷ 2,240 = 62.381 m
LCF = 62.381 ÷ 120 × 100 = 51.984% of LPP from AP
The centre of flotation is therefore 62.381 m forward of AP.
| Method | Main inputs | Suitable use |
|---|---|---|
| Waterplane first moment | AWP, ∫x dA and LPP | Known waterplane hydrostatic properties |
| Half-breadth integration | LPP and equally spaced waterline half-breadths | Waterline offsets or preliminary hull geometry |
Use this method when waterplane area and its first longitudinal moment are available from hydrostatic calculations, a hull model or approved vessel particulars.
The two values must refer to the same waterline, draft, trim and datum. A first moment calculated about FP or amidships cannot be entered as though it were taken about AP.
The entered LPP does not alter the centroid obtained from moment divided by area. It is used to express the result as a percentage of the AP–FP length.
For a port-starboard symmetric waterplane, its area can be found from the half-breadth function y(x):
AWP = 2∫y(x) dx
The first longitudinal moment about AP is:
M1 = 2∫xy(x) dx
The centre of flotation is then:
LCF = 2∫xy dx ÷ 2∫y dx = ∫xy dx ÷ ∫y dx
The symmetry factor of two cancels in the centroid ratio. It remains necessary when reporting the complete waterplane area and its full first moment.
For N equally spaced stations over LPP, the spacing is:
Δx = LPP ÷ (N − 1)
Simpson's one-third rule is applied to the half-breadths:
∫y dx ≈ Δx ÷ 3 × [y0 + yn + 4(y1 + y3 + ... + yn−1) + 2(y2 + y4 + ... + yn−2)]
The same Simpson multipliers are applied to the products xiyi to obtain the half-waterplane first moment.
Station requirement: Simpson's one-third rule requires an even number of intervals and therefore an odd number of equally spaced stations. Values must be entered in order from AP to FP.
Consider seven equally spaced stations:
Δx = 120 ÷ (7 − 1) = 20 m
| Station | x from AP (m) | y(x) (m) | xy(x) (m²) | Simpson multiplier |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 |
| 1 | 20 | 6 | 120 | 4 |
| 2 | 40 | 12 | 480 | 2 |
| 3 | 60 | 15 | 900 | 4 |
| 4 | 80 | 14 | 1,120 | 2 |
| 5 | 100 | 8 | 800 | 4 |
| 6 | 120 | 0 | 0 | 1 |
AWP = 2 × 20 ÷ 3 × [0 + 0 + 4(6 + 15 + 8) + 2(12 + 14)]
AWP = 2,240 m2
M1 = 2 × 20 ÷ 3 × [0 + 0 + 4(120 + 900 + 800) + 2(480 + 1,120)]
M1 = 139,733.333 m3
LCF = 139,733.333 ÷ 2,240 = 62.381 m from AP
LCF = 51.984% of LPP from AP
The same physical position may be reported from AP, from FP, as a percentage of length or relative to amidships.
| Convention | Calculation | Reported value |
|---|---|---|
| Forward from AP | xF | 62.381 m from AP |
| Aft of FP | LPP − xF | 57.619 m aft of FP |
| Percentage from AP | xF/LPP × 100 | 51.984% LPP |
| Relative to amidships, positive forward | xF − LPP/2 | 2.381 m forward of amidships |
Hydrostatic tables may instead use amidships as zero and take aft as positive. The datum and sign convention must be read before values are compared.
Do not compare unsigned values: a value stated as 2.4 m aft of amidships cannot be compared directly with 52% of LPP from AP without converting both to the same convention.
LCF is the centroid of the current waterplane. For a sufficiently small change of trim, the change in waterline is commonly represented as a rotation about LCF.
Adding or removing a small weight at LCF produces approximately parallel sinkage or rise because the weight has no longitudinal lever relative to LCF. Adding or removing the same weight away from LCF produces a trimming moment:
Trimming moment = w × d
Here, d is the signed longitudinal distance between the weight and LCF. The corresponding small change of trim requires MCT 1 cm or another equivalent longitudinal stiffness quantity:
Change of trim in centimetres = Trimming moment ÷ MCT 1 cm
Pure moment distinction: a pure trimming couple changes trim regardless of the point at which it is drawn. The no-trim statement applies to adding or removing a small weight at LCF, not to applying a pure moment there.
| Position | Geometric or physical basis | Main use |
|---|---|---|
| LCF | Centroid of waterplane area | Small-trim reference and waterplane geometry |
| LCB | Centroid of displaced underwater volume | Longitudinal position of buoyancy |
| LCG | Centroid of the vessel's total weight | Longitudinal position of gravity |
LCB and LCG are used when considering longitudinal equilibrium. LCF is used when distributing small changes of draft caused by trim.
LCB–LCF separation alone does not determine trim tendency. The relevant equilibrium comparison is between LCG and LCB, while LCF and longitudinal waterplane inertia influence the response to a small trimming moment.
The station tab integrates half-breadths only over the entered LPP span from AP to FP.
If the actual selected waterline extends appreciably forward of FP or aft of AP, those areas and their moments are omitted by this station method. The resulting centroid then describes the entered AP–FP portion rather than the complete waterplane.
For such a hull, use complete waterplane geometry or direct hydrostatic AWP and first-moment data calculated over the full intended waterplane.
LCF belongs to a particular waterplane. As draft changes, bow flare, stern geometry, transom immersion and the length of the waterline may alter the area distribution and move the centroid.
A value calculated for an even-keel condition should not automatically be applied to a substantially trimmed condition.
Heel may also alter the waterplane centroid on flared, chined or asymmetric hulls. The station method on this page assumes a symmetric waterplane represented by one set of half-breadths.
Simpson integration approximates the shape between the supplied ordinates. Accuracy depends on equal spacing and on whether the station series captures rapid changes in the waterline.
Additional geometric detail may be required near:
This calculator does not support unequal station spacing, arbitrary coordinates or mixed Simpson multipliers.
LCF is useful for hydrostatic reporting, small-trim calculations, checking waterplane geometry and locating the longitudinal waterplane centroid.
LCF alone cannot determine:
Result check: the station-integration result should lie within the entered AP–FP span because all half-breadths are non-negative and all coordinates lie between 0 and LPP. An out-of-range result indicates invalid station order, spacing or input data.
The definitions and numerical methods used on this page follow established naval-architecture sources:
NauticalSolver calculators are intended for preliminary engineering, study and independent checking. Use approved hydrostatic particulars, vessel-specific waterplane geometry and the loading computer for operational, contractual or statutory trim calculations.