Infinity, a data geometry term, proves that two parallel lines can be regarded as intersecting at infinity, and all parallel lines intersect at the same infinity.In the spherical polar projection, the point on the complex plane corresponding to the north pole of the complex sphere is infinity.
Two or more lines have the same point, which means one of the following two situations: or there is a point, and all lines pass through it;Or they are parallel.Thus, in elementary geometry, we do not define infinity, but only regard intersection at infinity as another way of saying parallelism.Thus, if P is the infinity on line AB, then PA/PB=1.In elementary geometry, there are many propositions that need to consider the special case of parallelism. The use of infinity avoids these problems.
In the spherical polar projection, the point on the complex plane corresponding to the north pole of the complex sphere is infinity.[1]
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The Proof of the Intersection of Parallel Lines at Infinity
O is the light source, AB is the vertical bar, CD is the horizon, OR//CD, M is the moving point on AB, and the shadow of M is N.
When M gradually rises, N becomes farther and farther, and when M is very close to R, N becomes far beyond imagination.
according to
It can be seen that N is infinite at this time.
N is the intersection point of OM and CD, but OR is parallel to CD, so OR and CD intersect at infinity.
This does not conflict with OR and CD. The infinity isProjective spaceThe "imaginary point" of represents the "direction".In the real world or the general mathematical world, we can not find or think that there is no such point.
The Proof of a Line with Only One Infinite Point
(Add OS//AB to the previous conditions)
I just mentioned that M coincides with R, if M continues to rise, N will reach the left of CD, and will get closer to S with the rise of M.When M moves down and continues to fall through the surface, N gets closer and closer to S from the right.
In this way, when M rises to infinity or falls to infinity, N will reach S. Since one N obviously corresponds to only one M, the "sky", "underground" and "two" infinities are the same.
Line and circle
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The straight line is actually a closed curve. Two points determine a straight line, but three points determine a circle.
In fact, since the infinity point is the "direction", the straight line is the "circle" passing through these two points and the corresponding infinity point.
