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Two lines will intersect at a point - even if the point is "at infinity" in the case of parallel lines - where the point of concurrence (i.e. intersection) is the geometric object defining the property; any other line that also intersects at the same point is therefore in the pencil, and conversely those lines that do not are not in the pencil.
Line–line intersection. In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in ...
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
The ten lines involved in Desargues's theorem (six sides of triangles, the three lines Aa, Bb and Cc, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ...
Intersecting chords theorem. In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal.
The Cayley–Bacharach theorem arises for high degree because the number of intersection points of two curves of degree d, namely d 2 (by Bézout's theorem), grows faster than the number of points needed to define a curve of degree d, which is given by. These first agree for d = 3, which is why the Cayley–Bacharach theorem occurs for cubics ...
As any two great circles intersect, there are no parallel lines in elliptic geometry. In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, all perpendiculars to a given line intersect at a single point called the absolute pole of that line. Every point corresponds to an absolute polar line of which it is the ...
Ultraparallel theorem. Poincaré disc: The pink line is ultraparallel to the blue line and the green lines are limiting parallel to the blue line. In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel ...