Mirrorpair — the series
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One shutter, two views
A photograph with a mirror in it is a stereo pair, and a peculiarly well-behaved one. Its fundamental matrix is skew-symmetric, so both epipoles are the same point; that point is where the camera would see its own lens; and every line joining a mark to its reflection passes through it, to 1.4 × 10⁻¹² px. The baseline is twice the distance to the glass, which is the one number a single view cannot supply and a tape measure can.
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Two matches are enough
A general fundamental matrix has seven free numbers and needs eight correspondences. A mirror pair's has two, and two correspondences fix it — with a straightedge, on a print, by drawing the line from each mark to its reflection and marking where the two cross. Given the same sixteen marks read to four tenths of a pixel, the constrained fit lands 4.8 times closer to the truth than the eight-point algorithm.
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Square to the camera is the worst mirror
A mirror pair's baseline runs along the mirror's normal, so a mirror facing the camera puts the second eye directly behind the first — the forward-motion arrangement, with the epipole in the middle of the frame and the rays to a mark crossing at 23°. Turning it forty-four degrees opens that to 65° and cuts the worst depth error threefold, and the number to watch is not the angle but where the reflected lens sits on the print.
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Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
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A symmetric object is its own stereo pair
A building with a plane of symmetry photographed once gives fourteen correspondences whose joining lines meet at one point to 1.9 × 10⁻¹² pixels, a skew-symmetric matrix, and the object's whole shape to fifteen digits — with no mirror anywhere and no second exposure. What it does not give is the size, and the instrument that decides whether any of it applies is the same meeting point, which opens to 12.7 pixels when the symmetry is half a per cent out.
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Two wide pairs are most of a symmetry
A facade with a symmetric frame and an asymmetric middle is still its own stereo pair. Two pairs whose halves stand wide apart find the symmetry plane to 0.40° with marks read to 0.4 px; twelve pairs find it to 0.26°; two narrow pairs only to 1.35°. The middle that pairs with nothing comes back too, to 0.8 per cent, carried along its rays to the wall the pairs have fixed. The risk is not too little symmetry but a false pair — and a window set wider than its partner is one the standard test cannot see.
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The wall convicts a pair set too wide
A window set wider than its partner's reflection keeps every joining line on the mirror's point, so the one test a symmetric photograph was known to carry cannot see it. The wall the other pairs fix can: it says where the partner should be seen, and a window set ten centimetres too wide misses by 6.0 pixels. With marks read to 0.4 px it convicts a 3-centimetre offset eight or nine times in ten and a 5-centimetre one every time. What no single photograph can see is a move along the camera's own ray to the window.
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A false pair is named by its neighbours, not by itself
Read against the rest of a symmetric facade, a pair whose right half has moved six centimetres is explained exactly by two stories: the right half moved, or the left half moved the mirror way from a pair standing six centimetres over. Predicted from left to right and from right to left, the pair gives one residual twice. What names the half is a mark that shares a line with it — the window's upper corner names a move across ninety-three times in a hundred at two centimetres — and what the readings recover is the move as the camera sees it, blind along its ray.