<P> To put it simply, the vanishing line of some plane, say α, is obtained by the intersection of the image plane with another plane, say β, parallel to the plane of interest (α), passing through the camera center . For different sets of lines parallel to this plane α, their respective vanishing points will lie on this vanishing line . The horizon line is a theoretical line that represents the eye level of the observer . If the object 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 . </P> <P> 1 . Projections of two sets of parallel lines lying in some plane π appear to converge, i.e. the vanishing point associated with that pair, on a horizon line, or vanishing line H formed by the intersection of the image plane with the plane parallel to π and passing through the pinhole . Proof: Consider the ground plane π, as y = c which is, for the sake of simplicity, orthogonal to the image plane . Also, consider a line L that lies in the plane π, which is defined by the equation ax + bz = d . Using perspective pinhole projections, a point on L projected on the image plane will have coordinates defined as, </P> <Dl> <Dd> x ′ = f x / z = f d − bz / az </Dd> <Dd> y ′ = f y / z = f c / z </Dd> </Dl> <Dd> x ′ = f x / z = f d − bz / az </Dd>

When was linear perspective with the use of vanishing points first detected in art works