How pavement art works
You have seen the photographs: a chasm opening in a shopping street, a waterfall down a flight of steps, a person balancing on a ledge that is really flat tarmac. There is always a piece of tape on the ground a few metres back. That tape is the entire trick.
The problem the artist is solving
A painting on a wall faces you. A painting on the ground does not. You look at pavement from above and at a steep angle, so anything drawn on it is compressed towards you and stretched away from you. A circle painted as a circle reads as a squashed oval. This is just foreshortening, and it is the reason ordinary pavement drawings always look like drawings.
The street artist wants the opposite. They want a shape that, seen from one specific standing position, looks like it is standing upright in space rather than lying on the ground. So they have to pre-distort the artwork by exactly the amount the viewing angle will undo.
The method
The traditional way to do it needs no mathematics at all, only patience.
Pick the viewing point first and mark it. Everything depends on it. Set a camera there, at eye height, and leave it. Then take the image you want the viewer to see, print it on a grid, and set up a corresponding grid on the pavement, seen from the camera. The pavement grid is not square. Drawn on the ground it is a splayed trapezoid, wide at the far end and narrow at the near end, because that is the shape that photographs as a square from the camera position.
Now copy the picture square by square. What is in cell C4 of the printed image goes into cell C4 of the pavement grid, which on the ground may be four times larger and lozenge shaped. You are not drawing the picture. You are drawing what the picture has to become in order to arrive correctly at one point.
Artists check constantly through the fixed camera, because on the ground the work looks wrong, and it has to look wrong. Kurt Wenner, who developed the modern form of this in the 1980s, worked out formal geometric constructions for it. Julian Beever and Edgar Mueller made it famous. The taped X is where they put the camera.
Why the photographs look better than being there
A camera has one lens. You have two eyes.
Standing on the X in person, your two eyes each get a slightly different view of the ground, and your brain uses the difference to work out depth. It can tell the tarmac is flat. The illusion is good but you can feel the floor underneath it.
A photograph throws away stereo entirely. There is one viewpoint, exactly the one the artist designed for, and nothing to contradict it. This is why the genre exploded once everyone carried a camera: the work is not really made for the person standing there, it is made for the picture they take.
The same idea, everywhere
Once you know the shape of the trick you start seeing it. Bicycle symbols painted on roads are stretched long, because a driver sees them at a shallow angle. Sponsor logos on a football pitch are drawn enormously elongated so they read correctly to the main broadcast camera, and look absurd to anyone in the stands. Andrea Pozzo painted a dome on the flat ceiling of Sant'Ignazio in Rome in the 1690s, and there is a marble disc set into the floor telling you where to stand for it to work. Step off the disc and the dome buckles.
What our puzzle changes
All of these hand you the viewing point. The tape, the marble disc, the camera position in the photograph: someone has already found the answer and is showing you where to stand.
Our daily puzzle takes the tape away. The fragments are placed by exactly this method, each one pushed along a line from a chosen viewpoint and scaled by the amount it moved, so that from that one direction they assemble into a picture. Then we do not tell you what the direction is. Finding it is the game. The construction is set out in why the puzzle always has exactly one answer, or you can just go and look for it.