: 244–6 She notes, in terms of projective geometry, the vanishing point is the image of the point at infinity associated with L, as the sightline from O through the vanishing point is parallel to L.Īs a vanishing point originates in a line, so a vanishing line originates in a plane α that is not parallel to the picture π. Brook Taylor wrote the first book in English on perspective in 1714, which introduced the term "vanishing point" and was the first to fully explain the geometry of multipoint perspective, and historian Kirsti Andersen compiled these observations. Guidobaldo del Monte gave several verifications, and Humphry Ditton called the result the "main and Great Proposition". Some authors have used the phrase, "the image of a line includes its vanishing point". It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π and its vanishing point. The vanishing point theorem is the principal theorem in the science of 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. 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. Lines parallel to the other two axes will not form vanishing points as they are parallel to 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. If we consider a straight line in space S with the unit vector n s ≡ ( n x, n y, n z) and its vanishing point v s, the unit vector associated with v s is equal to n s, assuming both point towards the image plane. 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 v q ≡ ( x / h, y / h, f / h) be the unit vector associated with q, where h = √ x 2 + y 2 + f 2. 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. A 2D construction of perspective viewing, showing the formation of a vanishing point
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