Search results
Results from the WOW.Com Content Network
Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0. We can represent these two lines in line coordinates as U 1 = (a 1, b 1, c 1) and U 2 = (a 2, b 2, c 2). The intersection P′ of two lines is then simply given by [4]
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 ).
Two straight lines which intersect one another cannot be both parallel to the same straight line. Playfair acknowledged Ludlam and others for simplifying the Euclidean assertion. In later developments the point of intersection of the two lines came first, and the denial of two parallels became expressed as a unique parallel through the given point.
[1]: 300 In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. On a Euclidean plane, a line can be represented as a boundary between two regions.
The Persian mathematician, astronomer, philosopher, and poet Omar Khayyám (1050–1123), attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five principles due to the Philosopher , namely, "Two convergent straight lines intersect and it is impossible for two convergent straight lines to ...
The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. For example, any three skew lines in PG(3,q) are contained in exactly one ...
To show this, Gergonne considered lines through corresponding points of tangency on two of the given circles, e.g., the line defined by A 1 /A 2 and the line defined by B 1 /B 2. Let X 3 be a center of similitude for the two circles C 1 and C 2; then, A 1 /A 2 and B 1 /B 2 are pairs of antihomologous points, and their lines intersect at X 3. It ...
There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are: G1: Every line contains at least 3 points; G2: Every two distinct points, A and B, lie on a unique line, AB. G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).