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A vanishing point is a point on the image plane of a perspective rendering where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points. Italian humanist polymath and architect Leon Battista Alberti first introduced the concept in his treatise on perspective in art, De pictura, written in 1435. The vanishing point may also be referred to as the "direction point", as lines having the same directional vector, say D, will have the same vanishing point. Mathematically, let q ≡ (x, y, f) be a point lying on the image plane, where f is the focal length (of the camera associated with the image), and let vq ≡ (x/h, y/h, f/h) be the unit vector associated with q, where h = . If we consider a straight line in space S with the unit vector ns ≡ (nx, ny, nz) and its vanishing point vs, the unit vector associated with vs is equal to ns, assuming both point towards the image plane. When the image plane is parallel to two world-coordinate axes, lines parallel to the axis that is cut by this image plane will have images that meet at a single vanishing point. Lines parallel to the other two axes will not form vanishing points as they are parallel to the image plane. This is one-point perspective. Similarly, when the image plane intersects two world-coordinate axes, lines parallel to those planes will meet form two vanishing points in the picture plane. This is called two-point perspective. In three-point perspective the image plane intersects the x, y, and z axes and therefore lines parallel to these axes intersect, resulting in three different vanishing points.

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Perspective (graphical)

Linear or point-projection perspective () is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, generally on a flat surface, of an image as it is seen by the eye. Perspective drawing is useful for representing a three-dimensional scene in a two-dimensional medium, like paper.

Vanishing point

Projective space

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs.

Perspective Drawing: Techniques and Concepts

Explores the principles of perspective drawing, focusing on vanishing points and different perspective types.

Perspective in Descriptive Geometry

Explores the method of vanishing points in Monge's representation and the perspective images of lines.

Projective Geometry: Birth and Foundations

Explores the birth and foundations of projective geometry, including the vanishing point invention and stereotomy.