Tree shadows, modelled over real ground

Section one

Why the geometry has to be right

Overshadowing by trees turns up in three recurring situations: a proposed dwelling sited close to an established canopy, an existing dwelling whose amenity is said to be harmed by a tree, and a garden or open space where the question is how much of it receives sun in March. In each case the argument comes down to a line on a drawing, and to whether a particular window, garden or amenity area falls inside it or outside it.

The established framework for that assessment is the Building Research Establishment's guidance on site layout planning for daylight and sunlight, BR 209. It sets out how to test overshadowing, but it was written with buildings in mind, and the worked examples treat obstructions as simple solid forms on level ground. Applying it to a tree on a bank, with an asymmetric crown, using a single radius and a flat plane, is where most tree shadow work quietly loses its accuracy.

Our approach is to model the geometry properly and then let the framework do its job: cast the shadow from the canopy that was actually surveyed, at the sun position that actually applies on the date and at the location in question, onto the ground surface that actually exists.

What this is not. It is not a daylight and sunlight assessment. The model produces shadow geometry; it does not calculate vertical sky component, annual probable sunlight hours, or any of the numerical criteria in BR 209, and it does not conclude whether a scheme passes or fails. Those are matters for a daylight and sunlight specialist, and this work is intended to give them, or a planning officer, a defensible tree input rather than an assumed one. It is also not a photometric model: it maps where direct sun is geometrically blocked, not how much light reaches a surface through a canopy.

Section two

The standard method, and what we do instead

BS5837:2012 gives a shadow method in a note to clause 5.2.2. A segment is drawn from the centre of the stem with a radius equal to the height of the tree, swept from due north-west round to due east, and taken to indicate the shadow pattern through the main part of the day. It needs nothing but a height and a stem position, it can be drawn by hand, and as a first appraisal it is entirely reasonable. It also carries four assumptions, and none of them holds on the sites where the answer is close.

The BS5837 5.2.2 segment

No canopyThe segment is struck from the centre of the stem. Crown spread, asymmetry and the direction of overhang play no part in it, so two trees of the same height cast the same shadow whether their crowns are five metres across or fifteen.
A radius equal to the tree heightA shadow as long as the tree is tall corresponds to the sun sitting 45 degrees above the horizon. On 21 March, the date daylight and sunlight work is assessed against, the sun does not get near that in England: at Norfolk latitudes it peaks around 37 degrees, so the real midday shadow is already about a third longer than the segment, and by mid-afternoon it is several times longer.
One fixed arcThe same span is used at every latitude and on every date, and it gives no indication of where within that span the shadow lies at any particular time of day.
Level groundA segment drawn in plan presumes the shadow falls on horizontal ground at the base of the stem.

What we model

The surveyed outlineThe shadow is cast from the same canopy polygon that appears on the constraints plan, built from four measured cardinal spreads, so an asymmetric crown casts an asymmetric shadow. Where the direction of overhang matters, which is most boundary cases, this is the difference between a right answer and a plausible one.
The crown as a body, not an outlineSeveral horizontal rings are projected between the crown base and the tip, each tapered towards the top. At low sun angles it is the upper crown that governs the far edge of the shadow, and it reaches well beyond anything a stem-centred radius can describe.
A computed solar positionAltitude and azimuth are calculated for the site's actual coordinates on the actual date and time, so the shadow points where the sun puts it and is as long as the sun makes it, with British Summer Time handled explicitly rather than assumed.
A terrain modelEach shadow ray is followed across a LiDAR terrain surface until it meets the ground, so shadows lengthen downslope and foreshorten upslope.

Section three

Starting from the canopy that was surveyed

The shadow is not cast from a reconstructed idea of the tree. It is cast from the same canopy polygon that is drawn on the tree constraints plan, generated from the four cardinal spread measurements taken in the field and given a naturalised outline rather than a geometric one. Using the drawing's own geometry means the shadow and the canopy can never disagree with each other, which matters when both appear on the same sheet.

Above the crown base, the outline is projected several times over, each ring scaled progressively inwards to reflect the way a crown narrows towards its tip, with the taper set per tree from the surveyed crown form. Every ring, plus the tip itself, is projected along the anti-solar ray; the shadow is the outer boundary of everything that lands.

Set against the BS5837 segment for the same tree, the difference is not a refinement. The segment is a single radius; the modelled shading is a lobe that follows the sun, reaches further as the day opens and closes, and carries whatever shape the crown was recorded with.

The BS5837 segment and the modelled shading, same tree, same day. A tree 14 m high with a 5 m crown radius, plotted on level ground at Norfolk latitude on 21 March. The segment, radius equal to height, sits inside the modelled extent throughout: about a third short at midday, and roughly a third of the modelled reach by late afternoon. On level ground with a symmetrical crown the segment remains a fair first appraisal of direction; it is the distance that it understates.

Why crown depth governs the far edge. The same tree and the same sun, with the crown treated first as a flat outline at one height and then as a body with depth. It is the upper crown that sets the far edge of the shadow, and the gap between the two widens as the sun drops, which is exactly when overshadowing assessments are made.

Section four

Where the sun actually is

Solar altitude and azimuth are computed from the site's coordinates for the exact date and time being assessed, using an established solar position library rather than a nomogram or a standard angle. British Summer Time is resolved explicitly, because an hour's error in March moves a long shadow by several metres and is the single easiest mistake to make in this work.

Shadow length is governed by the cotangent of the solar altitude, which is why the choice of time matters so much more at the ends of the day than in the middle. Between eleven and one the shadow barely moves; in the first and last hours of the assessment window it grows quickly, and small differences in assumed time produce large differences in extent.

shadow_length = tree_height ÷ tan( solar_altitude ) on level ground. Because the rate of change is governed by the cotangent of altitude, the same ten minute step lengthens the shadow far more at 09:00 than at 12:00.

Standard assessment conditions. A batch mode runs the four standard assessment times on 21 March, the equinox date used for daylight and sunlight work, in a single pass and writes each to its own drawing layer. Any other date and time can be run alongside it where a scheme calls for it, for example testing a summer evening amenity space rather than a March window.

Section five

The ground is not flat

This is the part of the method that changes answers rather than refining them. The conventional approach projects the shadow onto a horizontal plane at the base of the stem. On a level site that is correct. On any site with a bank, a cutting, a terrace or a general fall across it, it is not, and the error is systematic rather than random: on falling ground the real shadow reaches substantially further than the flat calculation says, and on rising ground it stops well short.

Instead of projecting onto a plane, each shadow ray is followed outwards across a LiDAR digital terrain model, stepping along the ray and comparing its height against the ground beneath it until the two meet. The crossing point is then refined by successive halving until it is located to within a tenth of a metre. The terrain model is read once into memory, clipped to the survey area with a margin, so a full survey runs in seconds rather than minutes.

Ray marching in cross section. The ray is advanced in fixed steps and its height compared against the terrain at each one. Once it passes below the surface, the crossing is refined by halving the interval until it is fixed to within 0.1 m. Every point on the canopy outline is treated this way, so the resulting shadow follows the shape of the ground rather than a plane.

Show the working: the ray march, and two things that go wrong if you are careless

A ray leaves a canopy point at height z above datum and descends at the solar altitude along the solar azimuth. At a distance t along the ground, its height is:

z( t ) = z_start − t × tan( solar_altitude ) compared at each step against the terrain elevation sampled at that position, by bilinear interpolation of the surrounding cells.

Accumulate the height and it drifts. The obvious implementation adds a fixed drop to a running total at every step. At low sun angles a ray can travel a hundred metres or more, and the accumulated floating point error becomes large enough to miss the terrain crossing entirely. The height is therefore computed analytically from the start point at every step, never accumulated.

A canopy point can start below the ground. Where a tree stands at the foot of a bank, part of its crown may be lower than terrain that is only a few metres away upslope. A naive march from such a point registers an immediate hit against ground it was never going to reach, and produces a shadow collapsed onto the stem. Points in that condition are identified and handled before the march begins.

The march uses a one metre coarse step, refines to 0.1 m by bisection, and abandons a ray beyond 500 m, which is well past any distance at which a tree shadow is material.

Side by side

The clearest way to see the effect is to watch the same tree cast a shadow through a day with terrain awareness on and off. The comparison below tracks the two through the assessment window on sloping ground.

Animation comparing a tree shadow tracked through the
day with and without terrain awareness. The terrain-aware shadow extends further downslope and
bends across the fall of the land, while the flat-plane shadow stays as a straight-edged form. — **The same tree, the same day, with and without terrain.** On falling ground the terrain-aware shadow reaches further and curves with the contour; on rising ground it stops short. The flat-plane result is a straight-edged approximation that happens to be correct only where the site is level.

Test it yourself

The panel below is the two dimensional case: one tree, one ray, a uniform slope, on 21 March at Norfolk latitude. The full model does this in three dimensions against a measured surface and computes the sun for the actual site, but the direction and scale of the effect are the same. Positive slope values represent ground falling away in the direction the shadow travels. The grey marker shows where the BS5837 segment radius, equal to the tree height, would put the edge.

Tree height

Time of day, 21 March

Ground slope

BS5837 segment radius

modelled, flat plane

modelled, across the slope

Show the working: shadow reach on a uniform slope

For a tree of height h, a solar altitude α and a ground slope β measured in the direction the shadow travels, positive downhill:

reach = h ÷ ( tan α − tan β ) reducing to h ÷ tan α where the ground is level. The reach becomes unbounded as the slope approaches the solar altitude, which is the geometrically correct result: ground falling away at the same angle as the incoming ray is never reached by it.

That singularity is the reason a real terrain model matters more than a slope correction. Ground rarely falls at a constant angle, and the point at which a ray finally meets it is determined by the whole profile in between, not by an average gradient. The model marches the actual surface; this panel is the idealised case, shown to make the direction and scale of the effect clear.

Section six

The shadow through a whole day

A single time gives a single shadow. What a designer usually wants is the area a tree shades at any point during the day, so that a terrace, a garden or a run of windows can be placed clear of it. The conventional device for this is a simple arc swept from the stem at a radius set by tree height, which is quick, and which takes no account of crown shape, crown depth or ground.

We produce the arc by casting the real shadow repeatedly through the assessment window and taking the outer envelope of every position. The result is the union of everywhere the tree's shadow reaches during the day, built from the same geometry as the individual time snapshots, so the arc and the snapshots always agree with one another.

The daily envelope. Individual shadow positions are cast at close intervals through the assessment window and their outer boundary taken, so the envelope touches the tip of each position rather than being swept at a nominal radius. Because every position carries the tree's real crown shape and, where terrain is enabled, the real ground surface, the envelope inherits both. The positions drawn here are half-hourly; the envelope itself is built from a much finer interval.

The envelope is interpolated, the snapshots are not. Intermediate positions are computed at reduced vertical resolution and the boundary between them is interpolated, which is appropriate for showing a day's shading on a planning drawing. Where a specific time governs the assessment, use the snapshot for that time, which is computed at full resolution.

Section seven

What you receive

Drawing

Layered CAD output

Each assessed time is written to its own layer in a single drawing file, hatched and set to a transparency that reads clearly over a site plan. A design team can switch times on and off without going back to us for a new drawing.

Coverage

Individual trees, groups and hedges

Groups and hedges are handled as well as individual trees, so a boundary hedge or a woodland edge casts a continuous shadow rather than a row of separate circles with gaps between them.

Record

A stated basis

Date, time, time zone treatment, site coordinates, terrain model source and the tree heights used are all recorded with the output, so the drawing can be reproduced or challenged on its inputs.

Section eight

Limits and caveats

A canopy is not opaque
The model treats the crown as a solid form. Real canopies transmit a variable proportion of direct light depending on species, season, crown density and the angle of the sun, so the shadow shown is the outline of the shaded zone rather than a map of how dark it is. This is the same convention used for buildings in daylight and sunlight work, and it is conservative for a tree.
Seasonality is not modelled
A deciduous tree in leaf and the same tree in February are treated identically. Where the assessment date falls outside the leaf-on period, that has to be argued in the accompanying text rather than read off the drawing.
The terrain model is bare earth
A digital terrain model describes the ground, not what sits on it. Buildings, walls and other trees do not stop or receive the shadow unless they are modelled separately. On a site where a structure clearly intercepts the shadow, the drawing shows where the shadow would fall on open ground.
The daily envelope is interpolated
As set out above, the envelope is built from sampled positions with interpolation between them. Where a single time decides the assessment, use the snapshot for that time.
The envelope boundary is slightly generous
The outer boundary of each shadow position is taken in a way that can very slightly overstate extent for a deeply irregular crown. For overshadowing purposes a conservative boundary is the right way to be wrong.
Tree height is a survey measurement
Shadow length is directly proportional to height, so the accuracy of the output is bounded by the accuracy of the height record. Heights are measured instrumentally, not estimated, and where a shadow question is decisive we say what tolerance the height carries.

Where the model is genuinely decisive. Sloping ground, tall trees near a boundary, low sun angles, and crowns that are markedly one-sided. On a level site with a symmetrical tree and a midday sun, a conventional method and this one will agree closely, and there is no reason to pay for the difference.

Section nine

Sources and version

Building Research Establishment (2022) BR 209: Site layout planning for daylight and sunlight, a guide to good practice, third edition.
BS5837:2012, Trees in relation to design, demolition and construction. Survey extent at 4.4.2.1.
Environment Agency LiDAR composite digital terrain model, Open Government Licence, used as the default terrain surface where no site survey model is supplied.
Solar position computed with the pvlib solar position routines.

Version 1.0 · September 2026 · first publication