|
|
| Line 1: |
Line 1: |
| {{redirect|Ideal line|the ideal line in racing|racing line}}
| | I'm Lela and I live in a seaside city in northern Iceland, Fagurholsmyri. I'm 24 and I'm will soon finish my study at Educational Studies.<br><br>My weblog - Oltramare ([http://www.hommages.ch/Defunt/75574/Gabriel_C_Oltramare hommages.ch]) |
| | |
| In [[geometry]] and [[topology]], the '''line at infinity''' is a [[projective line]] that is added to the real (affine) [[plane (geometry)|plane]] in order to give closure to, and remove the exceptional cases from, the [[incidence (geometry)|incidence]] properties of the resulting [[projective plane]]. The line at infinity is also called the '''''ideal line'''''.
| |
| | |
| ==Geometric formulation==
| |
| In projective geometry, any pair of lines always intersect at some point. But [[Parallel (geometry)|parallel]] lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. The point at which the parallel lines intersect depends only on the [[slope]] of the lines, not at all on their [[y-intercept]]. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines is parallel.
| |
| | |
| Every line intersects the line at infinity at some point. The point at which a line intersects the line at infinity determines the slope of the line, but not at all its y-intercept.
| |
| | |
| In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore lines in the projective plane are [[closed curve]], i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical.
| |
| | |
| ==Topological perspective==
| |
| The line at infinity can be visualized as a circle which surrounds the affine plane. However, diametrically opposite points of the circle are equivalent -- they are the same point. The combination of affine plane and line at infinity makes the [[real projective plane]], <math>\mathbb{R}P^2</math><!--''RP<sup>2</sup>''-->.
| |
| | |
| A [[hyperbola]] can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two [[asymptote]]s of the hyperbola. Likewise, a [[parabola]] can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the parabola. If the parabola is cut by its vertex into a symmetrical pair of "horns", then these two horns become more parallel to each other further away from the vertex, and are actually parallel to the axis and to each other at infinity, so that they intersect at the line at infinity.
| |
| | |
| The analogue for the complex projective plane is a 'line' at infinity that is (naturally) a complex [[projective line]]. Topologically this is quite different, in that it is a [[Riemann sphere]], which is therefore a 2-[[sphere]], being added to a complex affine space of two dimensions over ''C'' (so four ''real'' dimensions), resulting in a four-dimensional [[compact space|compact]] [[manifold]]. The result is [[orientable]], while the real projective plane is not.
| |
| | |
| ==History==
| |
| The complex line at infinity was much used in nineteenth century geometry. In fact one of the most applied tricks was to regard a circle as a [[conic]] constrained to pass through two points at infinity, the solutions of
| |
| | |
| :''X''<sup>2</sup> + ''Y''<sup>2</sup> = 0. | |
| | |
| This equation is the form taken by that of any circle when we drop terms of lower order in ''X'' and ''Y''. More formally, we should use [[homogeneous coordinates]]
| |
| | |
| :[''X:Y:Z'']
| |
| | |
| and note that the line at infinity is specified by setting
| |
| | |
| : ''Z'' = 0.
| |
| | |
| Making equations homogeneous by introducing powers of ''Z'', and then setting ''Z'' = 0, does precisely kill off terms of lower order.
| |
| | |
| Solving the equation, therefore, we find that all circles 'pass through' the ''[[circular points at infinity]]''
| |
| | |
| :''I'' = [1:''i'':0] and ''J'' = [1:−''i'':0].
| |
| | |
| These of course are complex points, for any representing set of homogeneous coordinates. Since the projective plane has a large enough [[symmetry group]], they are in no way special, though. The conclusion is that the three-parameter family of circles can be treated as a special case of the [[Linear system of divisors|linear system]] of conics passing through two given distinct points ''P'' and ''Q''. <!--REMOVED UNCITED TEXT: This idea was used so often that a schoolmasterly joke arose, naming the circular points at infinity ''Isaac'' and ''Jacob'', respectively.-->
| |
| | |
| ==See also==
| |
| * [[point at infinity]]
| |
| * [[plane at infinity]]
| |
| * [[hyperplane at infinity]]
| |
| | |
| ==References==
| |
| <!-- NOTE: These need to be wikified -->
| |
| | |
| * Casey, J., ''A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples'', 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888
| |
| | |
| * Kimberling, C., "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998
| |
| | |
| * Lachlan, R., ''An Elementary Treatise on Modern Pure Geometry'', sect. 10. London, Macmillian, p. 6, 1893
| |
| | |
| * Graustein, W. C., ''Introduction to Higher Geometry''. New York, Macmillan, p. 30, 1930
| |
| | |
| * Oldknow, A., "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996
| |
| | |
| * Wells, D., ''The Penguin Dictionary of Curious and Interesting Geometry''. London, Penguin, pp. 141-142, 1991
| |
| | |
| [[Category:Projective geometry]]
| |
| [[Category:Perspective projection]]
| |
| [[Category:Infinity]]
| |
I'm Lela and I live in a seaside city in northern Iceland, Fagurholsmyri. I'm 24 and I'm will soon finish my study at Educational Studies.
My weblog - Oltramare (hommages.ch)