One Point Projection

In simplified form, One Point Perspective is a geometric construction in which all the Horizontal Lines are rendered horizontal, all Vertical Lines are rendered vertical, but all Orthogonal Lines reaching into the distance are rendered not as parallel; rather they converge on a single Vanishing Point, as in Sebastiano Serlio's Stage Design for a comic play, produced about 1545 for Francois I of France.  

Sebastiano Serlio, Il primo libro d'Architettura... Il Secondo Libro, Paris, Jehan Barbe, 1545.
https://www.martayanlan.com/pages/books/4507/sebastiano-serlio

The Vanishing Point determines the horizon and actually locates the eye of the viewer in three dimensional space.  We appear to look up at objects above the Vanishing Point, and we look down on objects below the vanishing point. Similarily we can locate our position relative to objects to the right and left.  Not surprisingly then, this is referred to as the Central Vanishing Point. It may be in the centre of the composition, as in Serlio's Scena Comica and Giorgio Martini's Architectural View, but does not have to be, as in Andrea Mantegna's St James.  Wherever it is located, it establishes the point of view of the observer.  This was an extremely important concept in a culture that was moving away from a strictly Deocentric view of the world and beginning to recognize the Individual as a valid starting point for discovery.  It is not by accident that for his Scena Comica, Serlio places his ideal observer in the Royal Box.

     Francesco di Giorgio Martini (attr): Architektonische Vedute, c.1490
      https://kaiser-friedrich-museumsverein.de/architektonische-vedute/

Andrea Mantegna: St James Led to His Execution, ca. 1455
This painting was destroyed on March 11, 1944 by Allied Bombing.
http://www.artchive.com

There are some limitations to this type of rendering: objects further away from the Vanishing Point can start to get quite distorted; and if you clip out a small section, it can look odd without reference to the rest of the composition.  But in a culture that was coming to rely more and more on Science to explain the world, artists needed a method to rationalize space that was based more in Mathematics than Theology.  So regardless of its flaws, One Point Projection continues to be the most widely used form of Perspective.

The architect and engineer Filippo Brunelleschi is usually credited with formulating the mathematical laws of Perspective. Sometime between 1420 and 1434 he created a convincing geometric proof and a demonstration to show how well a one point rendering could reproduce nature.  The drawings for this are lost, and it is known to us only through Leon Battista Alberti's Della Pittura, written in 1435.  Fortunately, Brunelleschi was not the only proponent, and enough early examples survive to make a case that probably Brunelleschi was not completely alone in his discovery.  Masaccio's Trinity for example, probably created about 1425, is by a painter already adept at the method.  It is intended to be viewed from a specific point that corresponds roughly to a normal eye level. If the viewer positions oneself at that point, the effect of looking up into a 3-dimensional space is very convincing and very powerful.

Masaccio: Trinity, 1427
http://www.artchive.com/artchive/M/masaccio/trinity.jpg.html

Crucial to the understanding of One Point Perspective is the concept of a Picture Plane.  The image will likely be painted on a planar surface; a canvas, or a panel.  What Brunelleschi demonstrated is that he could create a rendering and position it in front of the observer so that it perfectly melded with the observed scene.  Subsequent commentators from Alberti onwards have described the Picture Plane as a window through which the artist sees the scene.  A window on which the scene can be rendered.  And various artists have described mechanical solutions to demonstrate the concept.  All these solutions establish a constant relationship between the Observation Point, the Picture Plane, and the Subject.

Albrecht Durer: Treatise on Measurement.1525
Metropolitan Museum of Art

Masaccio's work demonstrates how comfortable he was with Perspective.  His Madonna deals with a standard theme, and he maintains the accepted premise that the holy figures are largest.  But the composition is much more unified, and to our modern eye, more satisfying than earlier Madonnas. It is more convincing and naturalistic because the viewer has a rational spatial relationship to the rendering.  

Masaccio; Throned Madonna with Child and Angels, London, National Gallery,
reproduced in Luciano Berti Umberto Baldini and Rosella Foggi; Masaccio, Florence, Cantini Editore, 1988 p 135

The geometry of One Point Perspective provides a grid to construct the rendering.  In the Giorgio Martini Architectural View the edges of all the floor tiles aim directly at the Central Vanishing Point. However, other implied Vanishing Points can be found by projecting a Diagonal Orthogonal that touches the corners of all the floor tiles.   A diagram from the 1518 Latin edition of Della Pittura better illustrates the concept.  The Diagonal establishes the depth of the composition and provides a progressive foreshortening of the tiles as they recede.  If the floor tiles are square, and the diagonal is 45^o, the distance from the Second Vanishing Point to the Central Vanishing Point will be the same as the distance between the Observer and the Picture Plane.  So now the Observer's point of view can be located in depth as well as laterally and vertically.

Diagonal Orthogonals are used to create a Perspective rendering from a Ground Plan and to measure objects in the rendering. Accurate scale measurements are only possible at the Picture Plane.  For objects in the distance, we can project an Orthogonal to the Picture Plane to be scaled.

        
        Leon Batista Alberti, De Pictura, MS 1448, 13 February 1518, Biblioteca Governativa of Lucca
         https://letteraturaartistica.blogspot.com/2013/11/english-version-rocco-sinisgalli-il.html

So we can accurately add figures or objects to a rendering. And there are subtler ways to use the grid. Note the way the soldier in the bottom right of Mantegna's St. James is stretched out more or less in line with an orthogonal.

In the Arnolfini Wedding Portrait, the Flemish painter Jan van Eyck has created a One Point Projection that appears to be a bit arbitrary. The orthogonals appear at first glance to converge on the mirror. But that is not quite the case. The floor boards converge on one point, the window on another, and the bed on a third. The various vanishing points are all around the mirror, creating the illusion of unity. The image in the mirror of course is the artist. So instead of the focal point of this painting being about marriage, Van Eyck has substituted himself. More about the mirror later.

Van Eyck carried out diplomatic missions for the Duke of Burgundy, so we know he was well travelled. He was almost certainly aware of what the Florentines were doing. But Arnolfini Wedding Portrait was created in 1434, before Alberti published Della Pittura. So it is possible that van Eyck did not fully comprehend the technique and was struggling to rationalize the various vanishing points much as medieval artists had done.

Or, this was neither a struggle nor a misunderstanding. Perhaps it was a conscious rejection of a single unifying point of convergence. The major flaw of One Point Perspective is that it can get distorted at the edges, making objects look larger than is natural. If the point of convergence for van Eyck's window were on the centreline, the window would have been larger, more elongated. Same with the bed. Van Eyck's geometry seems less random if it serves a compositional purpose. This is not uncommon. As early as 20 BCE, Vitruvius advised modifying the perspective of a building to suit the site if it would look better. In his Practical Guide to Scene Painting, written ca. 1863, Frederick Lloyds similarily advocates modifying the perspective where the distortion is unacceptable. Perspective has always been, and continues to be, all about the illusion. Whatever the rules, if it looks right, it is right.

Jan Van Eyck: Arnolfini Wedding Portrait, 1434,
https://www.nationalgallery.org.uk/paintings/jan-van-eyck-the-arnolfini-portrait