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In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines).
Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their 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 ...
An Infinity cube made of dice being played with An animation showing different moves and states of the Infinity cube (click to animate) An Infinity cube is a kind of mechanical puzzle toy with mathematical principles. Its shape is similar to a 2×2 Rubik's cube. It can be opened and put back together from different directions, thus creating a ...
1 Generally composed of straight line segments. Toggle Generally composed of straight line segments subsection. 1.1 Polygons with specific numbers of sides. 2 Curved.
If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Other mathematicians have devised simpler forms of this property.
The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines is topologically equivalent to the open Möbius strip. [68] One way to see this is to extend the Euclidean plane to the real projective plane by adding one more line, the line at infinity.
If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles. [12]
The line at infinity is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity). The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
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