Editing Horizon (section)

==Vanishing points==
[[File:TwoPointPerspective.png|thumb|right|210px|Two points on the horizon are at the intersections of the lines extending the segments representing the edges of the building in the foreground. The horizon line coincides here with the line at the top of the doors and windows.]]
{{Main|Vanishing point}}
The horizon is a key feature of the [[picture plane]] in the science of [[graphical perspective]]. Assuming the picture plane stands vertical to ground, and ''P'' is the perpendicular projection of the eye point ''O'' on the picture plane, the horizon is defined as the horizontal line through ''P''. The point ''P'' is the vanishing point of lines perpendicular to the picture. If ''S'' is another point on the horizon, then it is the vanishing point for all lines [[parallel lines|parallel]] to ''OS''. But [[Brook Taylor]] (1719) indicated that the horizon plane determined by ''O'' and the horizon was like any other [[plane (geometry)|plane]]:
:The term of Horizontal Line, for instance, is apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that that Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other plane;…But in this Book I make no difference between the Plane of the Horizon, and any other Plane whatsoever...<ref>{{cite book|author=Taylor, Brook|page=1719|title=New Principles of Perspective}}</ref><ref>{{cite news|author= Anderson, Kirsti|author-link= Kirsti Andersen|date=1991|title= Brook Taylor's Work on Linear Perspective|page= 151|publisher= Springer|isbn=0-387-97486-5}}</ref>
The peculiar geometry of perspective where parallel lines converge in the distance, stimulated the development of [[projective geometry]] which posits a [[point at infinity]] where parallel lines meet. In her book ''Geometry of an Art'' (2007), [[Kirsti Andersen]] described the evolution of perspective drawing and science up to 1800, noting that vanishing points need not be on the horizon. In a chapter titled "Horizon", [[John Stillwell]] recounted how projective geometry has led to [[incidence geometry]], the modern abstract study of line intersection. Stillwell also ventured into [[foundations of mathematics#Projective geometry|foundations of mathematics]] in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by [[Karl von Staudt]] deriving the axioms of a [[field (mathematics)|field]] was deconstructed in the twentieth century, yielding a wide variety of mathematical possibilities. Stillwell states
:This discovery from 100 years ago seems capable of turning mathematics upside down, though it has not yet been fully absorbed by the mathematical community. Not only does it defy the trend of turning geometry into algebra, it suggests that both geometry and algebra have a simpler foundation than previously thought.<ref>{{cite news|author=Stillwell, John|author-link=John Stillwell|date=2006|title= Yearning for the Impossible|url=https://archive.org/details/yearningforimpos00stil|url-access=registration|work= Horizon|pages=[https://archive.org/details/yearningforimpos00stil/page/n60 47]–76|publisher= [[A K Peters, Ltd.]]|isbn= 1-56881-254-X}}</ref>