"complex field, the set of points at infinity is a manifold. In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points. Unlike Euclidean and elliptic geometries, each line has two points at infinity: given a line ""l"" and a point ""P"" not on ""l"", the right- and left-limiting parallels converge asymptotically to different points at infinity. All points at infinity together form the Cayley absolute or boundary of a hyperbolic plane."
"trapezoid. This distortion is referred to as foreshortening. Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewer's eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to (and named after) the Earth's horizon. Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer's eye and usually (though not necessarily)"
"the parallel lines forming a side converge at a second point along the horizon (which may be off the drawing paper.) This is a two-point perspective. Converging the vertical lines to a third point above or below the horizon then produces a three-point perspective. Depth can also be portrayed by several techniques in addition to the perspective approach above. Objects of similar size should appear ever smaller the further they are from the viewer. Thus the back wheel of a cart appears slightly smaller than the front wheel. Depth can be portrayed through the use of texture. As the texture"
"(a vanishing point) on the horizon. A drawing has two-point perspective when it contains two vanishing points on the horizon line. In an illustration, these vanishing points can be placed arbitrarily along the horizon. Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or at two forked roads shrinking into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Seen from the corner, one wall of a house would recede towards one vanishing point while the other wall recedes"
"regarded as a ""line"". Thus, two parallel lines meet on a horizon line by virtue of their incorporating the same direction. Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or planes in this regard—those at infinity are treated just like any others. Because a Euclidean geometry is contained within a projective geometry—with projective geometry having a simpler foundation—general results in"
"Parallel projection A parallel projection is a projection of an object in three-dimensional space onto a fixed plane, known as the ""projection plane"" or ""image plane"", where the ""rays"", known as ""lines of sight"" or ""projection lines"", are parallel to each other. It is a basic tool in descriptive geometry. The projection is called ""orthographic"" if the rays are perpendicular (orthogonal) to the image plane, and ""oblique"" or ""skew"" if they are not. A parallel projection is a particular case of ""projection"" in mathematics and ""graphical projection"" in technical drawing. Parallel projections can be seen as the limit of a"
"picture plane in both the modern period and the ""Old Master"" period. Picture plane In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the ""eye point"" (or ""oculus"") and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sightline to the object of interest. In the technique of graphical perspective the picture plane has several features: The horizon frequently features vanishing points of lines appearing parallel in the foreground. The orientation of the picture plane is always"
"and parallel lines perpendicular to the horizon line – similar to the vertical lines used in two-point perspective. One-point, two-point, and three-point perspective are dependent on the structure of the scene being viewed. These only exist for strict Cartesian (rectilinear) scenes. By inserting into a Cartesian scene a set of parallel lines that are not parallel to any of the three axes of the scene, a new distinct vanishing point is created. Therefore, it is possible to have an infinite-point perspective if the scene being viewed is not a Cartesian scene but instead consists of infinite pairs of parallel lines,"
"reduction in object´s size as the objects recede from the observer until they can no more be distinguished (among other optical effects). Parallel lines never intersect in nature, but if sufficiently extended they nearly always seem to intersect in perspective projections and invariantly in human vision. The rare one exception is in the former wherein a plane of a projected object is parallel to the projection plane and thus its edges have no vanishing points. The horizon is depicted in a perspective by a line which designates the infinite limit of a horizontal plane at the observer eyes height. In"
"Picture plane In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the ""eye point"" (or ""oculus"") and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sightline to the object of interest. In the technique of graphical perspective the picture plane has several features: The horizon frequently features vanishing points of lines appearing parallel in the foreground. The orientation of the picture plane is always perpendicular of the axis that comes straight out of your eyes. For"
"Real projective line In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect ""at infinity"". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the ""horizon"" of the visual perspective in the plane, is a real projective line. It is the circle of directions emanating"
"vanishing point are drawn (in grey). These lines give the basic, one point ""railroad tracks"" perspective. The closer it is the horizon line, the farther away it is from the viewer, and the smaller it will appear. The farther away from the viewer it is, the closer it is to being perpendicular to the picture plane. A new point (""the eye"") is now chosen, on the horizon line, either to the left or right of the vanishing point. The distance from this point to the vanishing point represents the distance of the viewer from the drawing. If this point is"
"space. The notion of a projective plane arises out of the idea of perspection in geometry and art: that it is sometimes useful to include in the Euclidean plane an additional ""imaginary"" line that represents the horizon that an artist, painting the plane, might see. Following each direction from the origin, there is a different point on the horizon, so the horizon can be thought of as the set of all directions from the origin. The Euclidean plane, together with its horizon, is called the real projective plane, and the horizon is sometimes called a line at infinity. By the"
"figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or archaically perspector. The figures are said to be perspective from this center. If each of the perspective figures consists of all the points on a line (a range) then transformation of the points of one range to the other is called a ""central perspectivity"". A"
"Convergence (Pollock) Convergence is a 1952 oil painting by Jackson Pollock. The composition was painted on canvas. Convergence is one of the paintings by American artist Jackson Pollock, who is known as a representative of abstract expressionism. The composition was created on a canvas, measuring 93.5 inches by 155 inches. It is an oil painting, which constitutes wide range of colors, lines and shapes, made by method of dripping and pouring paint onto canvas. These lines, spots, circles splattered on a canvas unintentionally convey the artist's emotions. The painting was made during the Cold War with Russia, and it is"
"a single scene there may exist an unlimited number of differing sets of parallel lines which, of course, are not necessarily horizontal defining an unlimited number of planes and an accompanying unlimited number of vanishing points (which correspond to their infinite extensions), which can be on, over or under the horizon. If one would have bilateral vision so as to view 180°, one would see the parallel lines appear to intersect in opposite directions simultaneously even if viewed from between the lines, but this bilateral vision is impossible, both in human vision and in perspective. We must not forget that"
"then the vanishing line of is the horizon line . Anderson notes, ""Only one particular vanishing line occurs, often referred to as the ""horizon"". To put it simply, the vanishing line of some plane, say , is obtained by the intersection of the image plane with another plane, say , parallel to the plane of interest (), passing through the camera center. For different sets of lines parallel to this plane , 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"
"Perspective (geometry) Two figures in a plane are perspective from a point ""O"" if the lines joining corresponding points of the figures all meet at ""O"". Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions. The line which goes through the points where the"
"Equal parallelians point In geometry, the equal parallelians point (also called congruent parallelians point) is a special point associated with a plane triangle. It is a triangle center and it is denoted by X(192) in Clark Kimberling's Encyclopedia of Triangle Centers. There is a reference to this point in one of Peter Yff's notebooks, written in 1961. The ""equal parallelians point"" of triangle ""ABC"" is a point ""P"" in the plane of triangle ""ABC"" such that the three segments through ""P"" parallel to the sidelines of ""ABC"" and having endpoints on these sidelines have equal lengths. The trilinear coordinates of"
"vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection"
"of that circle with the horizon line consists of two ""distance points"". They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground"
"as in the vertical-horizontal illusion where the two lines are exactly the same length. In the Ponzo illusion the converging parallel lines tell the brain that the image higher in the visual field is farther away, therefore, the brain perceives the image to be larger, although the two images hitting the retina are the same size. The optical illusion seen in a diorama/false perspective also exploits assumptions based on monocular cues of depth perception. The M.C. Escher painting ""Waterfall"" exploits rules of depth and proximity and our understanding of the physical world to create an illusion. Like depth perception, motion"
"the line at infinity. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines are parallel. Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are closed curves,"
"Parallel (geometry) In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never"
"Horizon The horizon or skyline is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. The horizon is horizontal. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting intersection of earth and sky is called the ""visible horizon"". When looking at a sea from a shore, the part of the sea closest to the horizon is called the offing. The word ""horizon"" derives from the Greek ""ὁρίζων κύκλος"" ""horizōn kyklos"", ""separating circle"", from"
"Projective Geometry"". Artin posits this challenge to generate algebra (a field ""k"") from geometric axioms: The reflexive variant of parallelism is invoked: parallel lines have either all or none of their points in common. Thus a line is parallel to itself. Axiom 1 requires a unique line for each pair of distinct points, and a unique point of intersection of non-parallel lines. Axiom 2 depends on a line and a point; it requires a unique parallel ""to"" the line and ""through"" the point. Axiom 3 requires three non-collinear points. Axiom 4a requires a translation to move any point to any"
"antisolar point forms the geometric center of several optical phenomena, including subhorizon haloes, rainbows, glories, and heiligenschein. Occasionally, around sunset or sunrise, anticrepuscular rays appear to converge toward the antisolar point near the horizon. However, this is an optical illusion caused by perspective; in reality, the ""rays"" (i.e. bands of shadow) run near-parallel to each other. Also around the antisolar point, the gegenschein is often visible in a moonless night sky away from city lights, arising from the backscatter of sunlight by interplanetary dust. In astronomy, the full Moon or a planet in opposition lies around the antisolar point. During"
"R ⊔ ""point"" is itself the real projective line P(R). Considered as a subset of P(R), it is called ""line at infinity"", whereas is called ""affine plane"", i.e., just the usual plane. The next objective is to make the saying ""parallel lines meet at infinity"" precise. A natural bijection between the plane (which meets the sphere at the north pole and the sphere of the projective plane is accomplished by the gnomonic projection. Each point ""P"" on this plane is mapped to the two intersection points of the sphere with the line through its center and ""P"". These two points"
"a curved line for which objects there fall on corresponding retinal points in the two eyes. This line is called the empirical horizontal horopter. There is also an empirical vertical horopter, which is effectively tilted away from the eyes above the fixation point and towards the eyes below the fixation point. The horizontal and vertical horopters mark the centre of the volume of singleness of vision. Within this thin, curved volume, objects nearer and farther than the horopters are seen as single. The volume is known as Panum's fusional area (it's presumably called an area because it was measured by"
"Parallel coordinates Parallel coordinates are a common way of visualizing high-dimensional geometry and analyzing multivariate data. To show a set of points in an ""n""-dimensional space, a backdrop is drawn consisting of ""n"" parallel lines, typically vertical and equally spaced. A point in ""n""-dimensional space is represented as a polyline with vertices on the parallel axes; the position of the vertex on the ""i""-th axis corresponds to the ""i""-th coordinate of the point. This visualization is closely related to time series visualization, except that it is applied to data where the axes do not correspond to points in time, and"
"up every painting—the ""point"" and the ""line."" He called the physical support and the material surface on which the artist draws or paints the ""basic plane"", or BP. He did not analyze them objectively, but from the point of view of their inner effect on the observer. A point is a small bit of colour put by the artist on the canvas. It is neither a geometric point nor a mathematical abstraction; it is extension, form and colour. This form can be a square, a triangle, a circle, a star or something more complex. The point is the most concise"
"Parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ""Elements"", is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ""If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles."" This postulate does not specifically talk about parallel lines; Euclid gave the definition of parallel lines in Book I, Definition"
"any land. They depict images of clouds viewed from above, suspended in blue sky, with the land below nowhere to be seen; it is the view of clouds regarded at a downward and sideways angle, as from the window of an airplane. These paintings depict a kind of ""pseudo-horizon,"" formed not where land meets sky but where the suspended layer of clouds—a ""pseudo-ground""—meets the empty upper sky. (See the external link below for an image of O'Keeffe's gigantic 1965 aerial cloudscape entitled ""Sky Above Clouds IV"", housed at the Art Institute of Chicago). During this period, O'Keeffe also produced some"