Bris Sextant Theory Patrick Mitchell March 2023 The Bris sextant is a simple but ingenious navigational instrument invented by Swedish yachtsman Sven Yrvind and named after his yacht Bris. It is a stack of three pieces of glass glued together with fixed angles between them. When a light is viewed through it, this arrangement gives multiple virtual images at different fixed angles. It is used with bright heavenly bodies, such as the sun and moon, for navigation, by observing them and timing the moment one of these reflections touches the horizon. The angle of that image is known and therefore the elevation of the body is known. They have the advantage over ordinary sextants that there are no moving parts so associated errors are avoided which means that even without a telescope, they can produce comparably accurate results. The disadvantages are that they cannot be used with dim bodies like stars and that you have to wait for the moment when one of these images touches the horizon. You can't simply take a sighting at any time. Yrvind's original is tiny, measuring about 20 by 10 mm and is easy and cheap to construct so is suitable as a navigational backup instrument. With a couple of modifications, it can be made more practical. Optical Geometry Figure 1 Figure 1 shows a ray of light from a distant source, striking a mirror at an angle a, we'll use the direction of the incident ray as our zero degrees for future angle references. The reflected ray leaves the mirror at the same angle a so has been rotated by a+180+a º Figure 2 In Figure 2 I have added a second mirror whose angle to the first mirror is x, we can work out the angle at which the first reflection strikes the second mirror using the angles in a triangle formula: b=180-a-x º (1) The first reflection has an angle to the incident ray of 180+2a and is turned by the second mirror by 180+ 2b so substituting from 1 it leaves second mirror at an angle to the incident ray of 180+2a+180+2b =180+2a+180+2(180-a-x) =360+2a-360-2a-2x =-2x That the “a”s cancel means that the ray leaves the two mirror system at an angle of -2x to the incident ray and this is independent of the angle at which the light strikes the first mirror. If you have two glasses glued together at a fixed angle, they will always give this relationship between the incident and exiting ray, irrespective of where the incident ray is coming from or the orientation of the mirror. That is why the Bris sextant works. b Figure 3 Figure 3 shows the sextant which uses three pieces of glass with different angles x and y between each. This give 3 pairs of glasses. The centre and left are x º apart, the centre and right are y º apart and the left and right are (x+y)º apart. T his gives three beams leaving the sextant from one incident beam at three different angles: -2x º , -2y º and -2(x+y) º to the incident beam. Those beams are all the result of two reflections and I'll call then the first order virtual images. Figure 4 There are other fainter images that result from more than 2 reflections. One of these from 4 reflections is shown in figure 4. This is one of 8 possible light pathways that have 4 reflections and there are others with 6, 8, 10... etc. reflections. We can count these light pathways for different numbers of reflections but most of these pathways overlap others so do not give unique images. Of the 3 first order images seen in figure 3, one comes from the middle glass and gives 2 second images, one from the middle and one from the front glass as shown in figure 5. Lens of eye Figure 5. A first order image ray from the middle glass is shown in black. The second order image rays arising from it are in grey. Of those 3 first order images seen in figure 3, the other 2 come from the front glass and give 3 each, of which one each again come from the middle glass as shown in figure 6. Figure 6. A first order image ray from the right hand glass is shown in black. The second order image rays arising from it are in grey. In general I will refer to the order of images a n so we have 1 st order (figure 3) 2 nd order (figure 4) ....... n th order. To count images we have to allow for the complication that n th images from the right glass have 3 (n+1) th order images those from the middle glass have 2. Table 1 does that by keeping track of where the images come from. It is written as a spreadsheet with letters for columns, numbers for rows and spread sheet addressing. For example D2 means the number in column D row 2 . Table 1 Column A is the order of the virtual images, where the row order zero is the real image of the sun. The row order 1 is the first set of virtual images that results from 2 reflections, the next order 2 (not shown in this table) results from 4, order 3 from 6, etc. The remaining six columns are used to calculate the number of images on the next line. Column C is the number of images on that line that have come from the right hand glass. Column F is the number that have come from the middle glass. Column D is simply twice column C, and is the number of second reflections from this path that come from the front glass. And column E is simply column C, which is the number of such reflections coming from the middle glass. Comparable calculations are made for columns G and H, and we repeat this process for subsequent lines to calculate how many virtual images there will be, which are 3 for the first set, 8 for the second set, 21 for the third set, and so on: Table 2 Most of these images will be copies by having the same angular deviation as other images. So for example, Figure 7 shows a first order image (left) that has the same angle as a second order image (right). Figure 8 shows two pairs of second order images that have the same angle as each other. Figure 7 Left is a first order image that has the same angle as the second order image on the right. Figure 8. Two pairs of second order images with the same angles. The upshot is that you get five images in the second and all subsequent orders that have not appeared in lower orders. When multiple images coincide they appear brighter. If the two angles x and y are not the same, one will be smaller, in which case twice the smaller one will be less than the sum of the two so the second order of images will contain one that has a smaller angle than the largest angle of the first order set, We can make various observations from the second set of virtual images that apply to all subsequent sets. Firstly, the largest angle deviation image of any set of images is the order of that image set times (-2x-2y), and is -2x-2y below the lowest one of the previous set. There will always be four unique images with lower angles than the previous set, and there will be at least one with an angle that's higher than the lowest one of the previous set. All the rest of the expanding number of virtual images are repeats of angles either within the same set or, from previous sets. Counting visible images If we consider the visible images rather than the order of reflections from which they come they will all be at combinations of whole number multiples of the angles -2x and -2y: -2x, -2x-2y, -2x-4y, -2x-6y ......... -4x, -4x-2y, -4x-4y, -4x-6y ......... -6x, -6x-2y, -6x-4y, -6x-6y ......... : : : If you choose a ratio of x to y of 2 to 3, then they will all be equally spaced. The first order will be 2, 3 and 5. The second will be 4, 6, 7, 8, and 10, and so on. So the 10 th order will be 46, 47, 48, and 50. In reality though, we won't be able to see anything beyond about the 4 th order because subsequent sets become too dim. So now we can calculate what angles to use. We're going to see between about 20 and 25 visible images, and we want these spaced over the range of sun elevations that we want to observe. The maximum elevation the sun can be between the tropics is 90 º . For northern latitudes, the highest sun we'll ever see is 90 plus 23 minus our latitude. So around me, that's never higher than about 60 º Placing images at 3 º intervals is about right, and the intervals are determined by double the angles between the glasses. Therefore, using a ratio of 2 to 3, we would have 3 º and 4.5 º glass angles, giving a first set of virtual images of 6, 9 15 º below the sun. Light Attenuation The proportion of light that plain glass reflects depends on the angle between its surface and the incident ray. When the incident ray is fairly close to 90 º to the surface it reflects about 5%. That means after two reflections, the intensity of the ray is 400 times less than it was from the original and it's 160,000 times less for the 4 reflections etc. However, this proportion of reflected light changes with the incident angle and with light polarisation as shown in figure 9. Figure 9 When the angle a is less than about 70 º the light becomes increasingly polarised with more reflections and the proportion of light reflected rather than transmitted through the glass increases rapidly with angles a under about 30 º so it gets quite high when the angle between the incident ray and the glass is small, an effect that we can exploit. Sun Filter As sextants are usually used to look at the sun, you need a dark glass and Yrvind's design has it on the outer side of the Sextant (figure 10). Figure 10. Sven Yrvind wearing his Bris Sextant Unfortunately that also makes dimmer reflections too dim to see. I modify the design so that the top of the sextant is not blocked with strips of spacing glass to determine the angles, but rather is left open (figure 11). Figure 11 I move the dark glass to the observer side of the sextant and use it only to cover the top half of a larger 75 mm long sextant (figure 12). Figure 12 Figure 13 was taken through the sextant. You can see the dark area above where the glass is blocking out the direct rays of the sun. You can see the three first order reflected suns. Figure 13 The second order images can't be seen, but I can bring them into view by rotating the top of the sextant away from me. This happens because the proportion of reflected light increases as the angle between the incident ray and glass decreases (figure 14). You can see that first order reflections get dimmer because less light is being passed through the glass while second order reflections get brighter because more is being reflected lower down the sextant. This dimming of the transmitted ray means that there is a limit to how bright you can get the lower reflections to be because although the reflected proportion is increasing, the transmitted proportion is decreasing and all images comprise rays that are reflected and transmitted at different points in their paths. This is improved by removing the spacers from the top of the sextant so that the incident ray comes above the outer glasses and hitting the first reflecting glass directly, making more reflections come into view when the sextant is tilted with the top towards the sun (figure 11). Figure 14