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Number line assumption. Every line is a set of points which can be put into a one-to-one correspondence with the real numbers. Any point can correspond with 0 (zero) and any other point can correspond with 1 (one). Dimension assumption. Given a line in a plane, there exists at least one point in the plane that is not on the line. Given a plane ...
In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.
Line–plane intersection. The intersection of a line and a plane in general position in three dimensions is a point. Commonly a line in space is represented parametrically ((), (), ()) and a plane by an equation + + =. Inserting the parameter representation into the equation yields the linear equation
A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic.
This image line is perpendicular to every line of the plane which passes through the origin, in particular the original line (point of the projective plane). All lines that are perpendicular to the original line at the origin lie in the unique plane which is orthogonal to the original line, that is, the image plane under the association.
The point is then mapped to a plane by finding the point of intersection of that plane and the line. This produces an accurate representation of how a three-dimensional object appears to the eye. In the simplest situation, the center of projection is the origin and points are mapped to the plane z = 1 {\displaystyle z=1} , working for the ...
Plane orthogonal to line L and including the origin. Point B is the origin. Line L passes through point D and is orthogonal to the plane of the picture. The two planes pass through CD and DE and are both orthogonal to the plane of the picture.
One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through a particular origin point; in this model, lines through the origin are considered to be the "points" of the projective plane, and planes through the origin are considered to be the "lines" in the projective plane. These ...