Vanishing point is below the horizon line, its vanishing lines angle up to the horizon line. If the object is above, they slope down. All vanishing lines end at the horizon line. 1. Projections of two sets of parallel lines lying in some plane appear to converge, i.e. the vanishing point associated with that pair, on a horizon line, or vanishing line formed by the intersection of the image plane with the plane parallel to and passing through the pinhole. Proof: Consider the ground plane , as which is, for the sake of simplicity, orthogonal to the image plane. Also, consider a
Horizon 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 in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by Karl von Staudt