Visibility

So you want to walk around the house naked, or sunbathe in the garden. Can your neighbours see you? If this matters, then this note records the process for working out what can be seen from where. We assume that we are trying to be discreet and not secret; we will consider views from windows or standing in adjacent gardens, or the street, we will not worry about people coming up to the boundary and looking over, or peering through knot-holes in fences, that's their problem.

The guiding principle is that light travels in straight lines. For example, if an eye placed at X cannot see any part of a window at Y, then an eye looking through any part of that window cannot see an object at X. We are generally interested in finding out how low can be seen, so in the case of an overlooking window the key line is from the highest part of the window:

ray trace window looking over garden

The diagram shows a window overlooking a garden with a fence about 2m high all round. The dashed line is the lowest that can been seen from the window. The blue zone is likely to be safe for walking about, and the green zone for sunbathing. Adding a 1m screen extends the sunbathing region to anywhere in the garden:

ray trace window looking over garden

Arranging adequate screening is easy. What makes it more interesting is that given that the purpose is sunbathing, it is necessary not to screen the sun. This will limit the places in the garden which can be used, which will vary with time of day, and which may be quite restricted in early and late season. The angle of the sun above the horizon (sun elevation) varies with the seasons. Where I am in the UK (Latitude 52 north, longitude 0), the elevation is 90-52 = 38° at either equinox. The tilt of the earth's axis of 23.5° adds to this at the summer solstice and subtracts at the winter solstice, so very roughly the maximum height of the sun is 14.5° at the worst of winter and 61.5° at midsummer. Better figures can be cribbed from the US military's excellent web pages at http://aa.usno.navy.mil/data/docs/AltAz.html. The table below is a gross simplification:

Approx sun location for England (Lat 52N, Long 0)
timeelevation (deg) for 21st of monthazimuth
UTC/GMTBSTMarch
September
April
August
May
July
June degcompass
08:0009:00   3536120SE by N
09:0010:00 36 4345135SE
10:0011:003243 5153150SE by S
11:0012:003648 5659165S by E
12:0013:003850 5861180S
13:0014:003748 5659195S by W
14:0015:003343 5054210SW by S
15:0016:002736 4246225SW
16:0017:00 28 3437240SW b W
17:0018:00   2528255W by S

It's unlikely to be suitable for sunbathing during the winter, but from the spring to the autumn equinox it quite often is, which means that we need to cope with a sun elevation of only 38 degrees. A typical first floor window might have its upper edge at 5m above ground, and in urban housing might be less than 10m from our sunny spot, that's an elevation of about 27 degrees. In winter, the situation could be like this:

ray trace window looking over garden

where only the green regions are both screened from the window and getting any sun at all. The diagram shows the maximum sun elevation, which occurs around 13:00 BST; it will be lower of course at other times, and much lower for times more than a few hours either side of 13:00. In the summer, with the sun at about 61 degrees, it's a little better:

ray trace window looking over garden

but still not as good as it would be in an ideal world. These diagrams show the worst case, where the sun is in the same direction as the offending window. Of course, if all the windows are to the north, then a screen of any height will not block the sun. Unfortunately, the overlooking windows into my garden are to the south and west.

What height needs to be considered for visibility from a public place? It's quite easy to allow for the tallest pedestrian, and we make the assumption that we don't have to allow for people coming onto our property to have a better look, except possibly for an ungated path to the front door. But do we have to allow for lorry drivers and passengers on the top deck of a double-decker bus?

Another challenging calculation is to allow for the effect of windows acting as mirrors. We don't need to worry about neighbour's windows that we have already screened from, if you cannot be seen from a window, then you cannot be seen in a reflection from that window. But, it is possible to be seen in a reflection from a window on your own property, and it's not generally going to be possible to screen from them. And of course in summer they are likely to be open and reflecting at any angle. However, reflections in windows are usually faint and not often noticed casually; best not to worry about the possibility.

Similar calculations can be made to determine visibility of the inside from outside the house. Obviously it's easy to get complete invisibility when indoors, you could put blinds at every window, but really you want to be as open as possible, bright and able to see out. The ability to see detail looking in through a window from a distance goes down rapidly with depth into the room, so in daylight there's hardly likely to be a problem anyway, but at night with the lights on there obviously could be. It can be a fun exercise to arrange for every window sill to have just enough clutter so that everywhere is just safe.

Geek zone

The above is really more than you need to know. You just pick a sunny spot, lie in it, look around to see where you are overlooked from, ask yourself whether that is a problem, move a few chairs or put up a windbreak to cover it, take your clothes off and enjoy. But while you're soaking up the sun, you might be thinking how could I calculate exactly where the sun and privacy coincide, at any time of day or year. Then perhaps some permanent features could be created that would ensure that the perfect spot was always available. Anyway, I'm going to try.

The first step is to make a numerical model of the garden. Mine's a pocket-handkerchief in an urban estate, about 12m by 10m. It's on a steep slope and is surrounded on three sides by a 1.8-metre fence with a garage to one side and the house to the north. I measured the lawn surface using a laser level rotating about one reference point at the highest point and measured the distance of the surface below the laser at every metre along lines radiating every 15 degrees. The height of the fences was measured from the laser upwards. These data were then plotted as xyz triples using trigonometry. The triples were then converted to a uniform rectangular sample grid at 0.5-metre intervals using interpolation features of MATLAB.

From here, I manually added the fence heights. This gave me a model of the whole garden in the form of a height at every 0.5-metre grid point, with the slightly odd, but acceptable for our calculations, feature that the fences are considered to be 50cm thick. Plotted as a wire-frame model it looks like this:

wire frame of model

The big slab to the right is the back of the house, the gabled roof is the garage, and around the rest of the perimeter is the fence. The bit at the extreme left is a bit of a neighbour's garden included to make a rectangular model. The fences look a bit odd as this plot joins up points with straight lines.

The image below is a contour plot, which shows the view from above. It's mainly to check that the lawn levels make sense. The contours are at about 0.13m intervals so the fences and buildings are shown as an untidy solid mass of contours. In this plot the house is at the bottom.

contour of model

Now there are two more things to add to the model: windows and trees. The locations of overlooking windows will be approximated by either one point at the top middle, or if it's a large one at each of the top corners. The two fair-sized trees and one big shrub that both obscure sightlines and sometimes the sun will be added to the model as blocks.

At this point we switch to a simple mathematical model to test the ray plotting. The garden is now shown in a diagram like the one below. The stars indicate high points such as the fence, trees, house and garage, clear space the low points. This is the same data as in the wireframe model above, except for the trees, on the same 0.5-metre grid.

* * * *                                     
*       * * * *                             
*               * * *                       
*                     * * * *   * * * * * * 
*                             * * *         
*                                   * * * * 
*                                         * 
*                                         * 
*                                         * 
*                                         * 
* * *                                     * 
* * *                                     * 
* * *                                     * 
*                                         * 
*                                         * 
*                                         * 
*                                         * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

Now we start to plot where we can be seen. The method is simple: we assume that for sunbathing we need to be screened up to 50cm above the surface of the lawn. From a point on the lawn, we take a point 50cm above it, then draw a straight line from that point to the first window. If this line hits anything on the way, such as a fence, before it gets to the window, then that point on the lawn is safe from that window. All we have to do is to work this out for every point on the lawn and every window.

As you might expect for a tiny garden in a city, the first result is not promising. There are eleven windows overlooking from six properties, and taken together they can see all of the points indicated by an 'O' on this plot:

* * * * O O O O O O O O O O O O O O O O O O 
* O O   * * * * O O O O O O O O O O O O O O 
* O O O O O O O * * * O O O O O O O O O O O 
* O O O O O O O O O O * * * * O * * * * * * 
* O O O O O O O O O O O O O O * * *     O O 
* O O O O O O O O O O O O O O O O O * * * * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

In a word, everywhere. There is not one spot that cannot be seen from at least one window. But all is not lost. Looking at each window in isolation (too tedious to show here), or by walking around the garden, it can be seen that some parts are overlooked by only one window, so there might be some hope in local screening. The plot below shows a 1-metre high 3-metre long windbreak extending from the shrub at mid left across about a third of the width of the garden.

* * * *                                     
*       * * * *                             
*               * * *                       
*                     * * * *   * * * * * * 
*                             * * *         
*                                   * * * * 
*                                         * 
*                                         * 
*                                         * 
*                                         * 
* * *                                     * 
* * *                                     * 
* * * *                                   * 
*     * *                                 * 
*       * *                               * 
*         * *                             * 
*           * *                           * 
*             * *               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
*                               * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

Now there is a tiny region by the left-hand fence that cannot be seen from anywhere. Hardly enough, but just about big enough for one body:

* * * * O O O O O O O O O O O O O O O O O O 
* O O   * * * * O O O O O O O O O O O O O O 
* O O O O O O O * * * O O O O O O O O O O O 
* O O O O O O O O O O * * * * O * * * * * * 
* O O O O O O O O O O O O O O * * *     O O 
* O O O O O O O O O O O O O O O O O * * * * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * * O O O O O O O O O O O O O O O O O * 
*   O * * O O O O O O O O O O O O O O O O * 
*   O O * * O O O O O O O O O O O O O O O * 
*   O O O * * O O O O O O O O O O O O O O * 
*   O O O O * * O O O O O O O O O O O O O * 
*   O O O O O * * O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

The region can be extended because the bathroom window of the other half of the semi (just left of the bottom left of the plot) is frosted and it's very unlikely that next-door will open it and lean out. Ignoring that window it's better:

* * * * O O O O O O O O O O O O O O O O O O 
* O O   * * * * O O O O O O O O O O O O O O 
* O O O O O O   * * * O O O O O O O O O O O 
* O O O O O O O O O O * * * * O * * * * * * 
* O O O O O O O O O O O O O O * * *     O O 
* O O O O O O O O O O O O O O O O O * * * * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * * O O O O O O O O O O O O O O O O O * 
*     * * O O O O O O O O O O O O O O O O * 
*     O * * O O O O O O O O O O O O O O O * 
*       O * * O O O O O O O O O O O O O O * 
*     O O O * * O O O O O O O O O O O O O * 
*     O O O O * * O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
*     O O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

And as we get on with next-door, we can probably discount his landing window, leaving only the bedroom window:

* * * * O O O O O O O O O O O O O O O O O O 
* O O   * * * * O O O O O O O O O O O O O O 
* O O O O O O   * * * O O O O O O O O O O O 
* O O O O O O O O O   * * * * O * * * * * * 
* O O O O O O O O O O O O O O * * *     O O 
* O O O O O O O O O O O O O O O O O * * * * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * * O O O O O O O O O O O O O O O O O * 
*     * * O O O O O O O O O O O O O O O O * 
*       * * O O O O O O O O O O O O O O O * 
*         * * O O O O O O O O O O O O O O * 
*       O O * * O O O O O O O O O O O O O * 
*       O O O * * O O O O O O O * * * * * * 
*         O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

So now we have an area quite big enough. Does it get any sun? It's easy to add the sun in much the same way we added the windows. The table above gave the sun's position in terms of elevation and azimuth. Rather than use different maths, we imagine a point sun in our coordinate system; if we make it a long way away it will be at a constant angle near enough. Then we draw a line from the height of the lawn to this point sun, and if it hits anything within the model we take that part of the lawn to be in shadow. We do this calculation for each point on the lawn and over various times of year and times of day, hoping to find that our spot is available at any time.

This plot is for half the year from spring equinox to autumn equinox and for 09:00 to 16:00 UTC (10:00 to 17:00 BST). South is nearly straight up in the plots. The 'S' indicates a point that is in shadow at some time in this range.

* * * *                                     
* S S S * * * *                             
* S S S S S S S * * *                       
* S S S S S S S S S S * * * * S * * * * * * 
* S S S S S S S S S S S S S S * * * S S S S 
* S S S S S S S S S S S S S S S S S * * * * 
* S S S S S S S S S S S S S S S S S S S S * 
* S S S S S S S S S S S S S S S S S S S S * 
* S S S S S S S S S S S S S S S S S S S S * 
* S S S S S S S S S S S S S S S S S S S S * 
* * * S S S S S S S S S S S S S S S S S S * 
* * * S S S S S S S S S S S S S S S S S S * 
* * * * S S S S S S S S S S S S S S S S S * 
* S S * * S S S S S S S S S S S S S S S S * 
* S S S * * S S S S S S S S S S S S S S S * 
* S S S S * * S S S S S S S S S S S S S S * 
* S S S S S * * S S S S S S S S S S S S S * 
* S S S S S S * * S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* S S S S S S S S S S S S S S S * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

So we can't have perfection! Going for more realistic times, this plot is for 12:00 to 13:00 UTC in March and September, and for 11:00 to 16:00 UTC for the months in between.

* * * *                                     
* S S S * * * *                             
* S S S S S S S * * *                       
* S S S S S S S S S S * * * * S * * * * * * 
* S S S S S S S S S S S S S S * * * S S S S 
*     S S S S S S S S S S S S S S S * * * * 
* S S S S S S S S S S S S S S S S S S S S * 
* S S S S     S S S S S S S S S S S S S S * 
*         S S S S S S S S S S S S S S S S * 
*     S S S S S S S S S S S S S S S S S S * 
* * * S S S S S S S S S S S S S S S S S S * 
* * * S S S S S S S S S S S S S S S S S S * 
* * * *   S S S S S S S S S S S S S S S S * 
* S S * * S S S S S S S S S S S S S S S S * 
* S S S * *       S S S S S S S S S S S S * 
* S S S S * *   S S S S S S S S S S S S S * 
* S S S S S * * S S S S S S S S S S S S S * 
* S S S S S S * * S S S S S S S * * * * * * 
*     S S S S S S S S S S S S S * * * * * * 
*       S S S S S S S S S S S S * * * * * * 
*           S S S S S S S S S S * * * * * * 
*           S S S S S S S S S S * * * * * * 
*         S S S S S S S S S S S * * * * * * 
*       S S S S S S S S S S S S * * * * * * 
*     S S S S S S S S S S S S S * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

Which by good planning includes the area where we arranged the screening. The shadows at various individual times are plotted here. Now we can put both plots together:

* * * * O O O O O O O O O O O O O O O O O O 
* O O S * * * * O O O O O O O O O O O O O O 
* O O O O O O S * * * O O O O O O O O O O O 
* O O O O O O O O O S * * * * O * * * * * * 
* O O O O O O O O O O O O O O * * * S S O O 
* O O O O O O O O O O O O O O O O O * * * * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* O O O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * O O O O O O O O O O O O O O O O O O * 
* * * * O O O O O O O O O O O O O O O O O * 
* S S * * O O O O O O O O O O O O O O O O * 
* S S S * * O O O O O O O O O O O O O O O * 
* S S S S * * O O O O O O O O O O O O O O * 
* S S S O O * * O O O O O O O O O O O O O * 
*     S O O O * * O O O O O O O * * * * * * 
*         O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*       O O O O O O O O O O O O * * * * * * 
*   O O O O O O O O O O O O O O * * * * * * 
* O O O O O O O O O O O O O O O * * * * * * 
* * * * * * * * * * * * * * * * * * * * * * 

In which points marked 'O' are overlooked, points marked 'S' are not overlooked but are in shadow at some time of day at some time of year. The clear areas are what we aim to find: a spot that has sun at almost all times we are likely to want it, and is not overlooked from any window except for one known neighbour, and then mostly from a frosted window.

And to prove the point I've been using this spot on this superb late April day. [2007-04-28]

There are other useful places. There is an area top-left in the plots that can be screened with the same windbreak from all windows, including the other half of the semi. This spot though gets the shadow of the big tree from mid-afternoon.

From here, the next activity is perhaps to find something more permanent than the windbreak, that looks as if it belongs, and has the same effect. Longer term, it would be nice to find somewhere much more enclosed, or isolated, where it would be possible to walk about.

...

Three years on from writing this note, I am able to use more of the garden. I have documented this with pictures rather than updating the model, see here.

Results

See here.

I have a crude sun hours meter in an upstairs south-facing window. It's a photo-voltaic solar cell driving an electromechanical hours counter. This will read nothing like a proper sun hours meter, traditionally a glass ball with a paper track that gets burnt, but probably does allow one year to be compared with another. These are the readings from it:

Sun hours for each month
 2003200420052006 2007200820092010 2011
Jan 637344 62524950 35
Feb 1088489 941199264 52
Mar 150116134 170142198175 139
Apr 198151166 198158168193 168
May 241189179 149179202188 [1]
Jun 206168206 153164179184  
Jul 209153210 183178178182  
Aug 198178173 171175184173  
Sep 177151166 177153174150  
Oct 116111112 115133125116  
Nov 668778 36525347  
Dec33594828 32302120  
Total 179115091585 1540153516231542  

[1] Meter Failed. Contact fell off one of the solar cells.

 


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