How to work out the area of convergence from bloodstains, and what the height really is

Measure each stain's width and length; the sine of the impact angle is width over length. Draw a line through each stain's long axis, in the direction it came from, and where the lines cross on the surface is the area of convergence — that part is sound, because nothing can turn a drop sideways in flight. Then, for the height, the textbook raises a line from each stain at its impact angle and reads off where it meets the vertical through the convergence: height = stain height + distance out × tan(angle). That number is not where the source was. Every drop fell while it flew and arrived steeper than the straight line from the source, so every back-projection lands too high. The height is a ceiling, the source was at or below it, and the smallest of the per-stain ceilings is the one to quote — never the average.

The measurements

Three things per stain. Its width and length, with a loupe and a scale, to a tenth of a millimetre; the ratio is what the impact angle comes from, and a stain that is nearly round has an angle that is nearly unknowable, because the arcsine of a ratio near one moves a great deal for a small change in the ratio. Its position on the surface, from a fixed corner. And the direction of its long axis, which is the direction the drop was travelling, read from the tail or the scallops on the leading edge, to about three degrees. Below about twenty degrees of impact the ratio drifts from the ideal, because a real drop is not a sphere when it lands; that is a limit of the measurement, and it is worth saying out loud on the report.

The convergence, and why it can be trusted

Each long axis, extended backwards across the surface, is a line the source lay on in plan. Two of them cross at a point; nine of them cross in a small area, and the least-squares centre of that area is the answer. On the nine stains below it comes out at the origin of the room's coordinates to within a millimetre, and the page reports that anywhere within 0.17 m of it fits the long axes equally well at the three-degree accuracy they were read to. This part of the method is unbiased, and exactly so: neither gravity nor air resistance can turn a drop sideways, so the horizontal direction it was travelling when it struck is the horizontal direction it left on. Where in the room the blow landed is a real answer, from floor stains and wall stains alike.

The height, and why it is only a ceiling

The stringing method takes the impact angle as the angle of the straight line back to the source. It is not. A drop is thrown, and it falls while it flies, so it arrives descending more steeply than the line from the source, and a line run back at that steeper angle overshoots. For a source at height h, a stain at height z a horizontal distance D away, a flight lasting T, arriving at β below the horizontal, the identity is

h = z + D tan(β) − gT²/2

The textbook answer is the first two terms; the third is how far the drop fell during its flight, and it is not in the stain, because the flight time depends on how hard the drop was thrown. What survives is that gT²/2 is never negative, so h ≤ z + D tan(β). Air resistance only makes the arrival steeper still, so the bound holds with drag as well. The number the method gives you is the most the height could have been.

Nine stains, worked

A floor and a north wall, a source that was really at 1.40 m. Distance out is measured from the convergence; the descent is the angle below the horizon the drop arrived at, which on the floor is the impact angle and on the wall is not.

Three things to read off that table. The least ceiling is 1.51 m, from W3, and the source was really at 1.40: the bound holds, and it is tight. The median of the nine is 2.21 m and the mean is 2.81 m, which is what averaging would have said, and both are wrong by the better part of a metre, necessarily: every one of the nine is an upper bound, and the average of a set of ceilings is never a better ceiling than the lowest of them. And the stain that gives the best answer is the one the textbook would have thrown away: W3 struck the wall at 77 degrees, nearly square on, which on a floor would make it worthless — but it was descending at only 4.7 degrees when it arrived, because it was still travelling almost horizontally, and only the descent is in the identity. Shallow descents make tight ceilings; steep ones, like F2 and F4 at nearly sixty degrees, make loose ones, and the page marks them.

What there is no way to get

A lower bound. A single stain is consistent with a source anywhere from the floor up to its own ceiling, and the two obvious ways to close that off both fail when they are tried. Capping the launch speed at something plausible does not work, because worked without drag the speed a drop needed is not reliably an under-estimate of the speed it had. Requiring that no drop went through the ceiling is worse: air resistance flattens the arc, so the drag-free peak is too high, and the test throws away flights that were never that high — on a simulated scene it returned an interval that excluded the true height, stated with complete confidence. A range with an honest top and no bottom is what the measurement contains, and a report that gives more than that has invented it.

What ORIGIN does and does not do

ORIGIN takes the walls, the stains with their positions, widths, lengths and long-axis directions, and the accuracy each was measured to, and works the convergence with its band, each stain's ceiling with the swing the reading accuracy puts on it, the least ceiling as the number to quote, and the median and mean beside it for comparison. It draws the plan and hands back the working. Nothing is uploaded; scene measurements are not facts to hand to a website. The convergence and every ceiling in the table above are checked on every build against the same geometry worked again outside the tool.

It will not give a lower bound, an estimate, or a single height, because the measurements do not contain one. It gives the part that is true.