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Gear300
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Can anyone provide a visual of a Point at Infinity (or the ideal point)? I'm trying to visualize it and I apparently always end up interpreting every point on the Euclidean plane as an ideal point.
The easiest example to visualize is the infinite points of the extended real line. The real line has the same shape1 as the open interval (0, 1), and the extended real line has the same shape as the closed interval [0, 1]. In this picture, the points [itex]+\infty[/itex] and [itex]-\infty[/itex] of the extended real line correspond to 1 and 0.Gear300 said:Can anyone provide a visual of a Point at Infinity (or the ideal point)? I'm trying to visualize it and I apparently always end up interpreting every point on the Euclidean plane as an ideal point.
Hurkyl said:The easiest example to visualize is the infinite points of the extended real line. The real line has the same shape1 as the open interval (0, 1), and the extended real line has the same shape as the closed interval [0, 1]. In this picture, the points [itex]+\infty[/itex] and [itex]-\infty[/itex] of the extended real line correspond to 1 and 0.
If you curl [0, 1] into a circle (so that 0 and 1 become the same point), you get the projective real line.
The projective complexes have a similar picture: you can wrap the complexes up into a sphere missing a point. Under this picture, complex projective infinity would correspond to that missing point.
1: topologically, at least. There is a homeomorphism between the two spaces. For example, the map [itex]f(x)= (\arctan(x) + \pi / 2) / \pi[/itex]
The point at infinity is a concept in geometry that represents a point that is infinitely far away in all directions. In the Euclidean plane, it is denoted as ∞ and is located at the end of each line, representing the direction of that line extending infinitely.
The point at infinity is usually represented by a dashed or dotted line perpendicular to the lines in the Euclidean plane. This line is called the line at infinity and serves as a visual aid to help understand the concept of the point at infinity.
The point at infinity plays a crucial role in projective geometry, providing a way to represent parallel lines and to extend the concept of distance to include points at infinity. It also helps in understanding the concept of infinity and its relationship to the finite world we live in.
The point at infinity is used in various branches of mathematics, such as projective geometry, complex analysis, and algebraic geometry. It allows for a more unified and elegant approach to solving problems that involve infinity, such as finding the intersection of parallel lines.
No, the point at infinity is a mathematical concept and cannot be visualized in the real world. It is an abstract idea that helps in understanding and solving problems in mathematics, but it does not exist in the physical world.