Real Projective Plane
Boy's surface |
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Boy's surface
- This is Boy's surface, one model of the Real Projective Plane in 3 dimensional space.
Contents
Basic Description
The Real Projective Plane is an abstract , 2 dimensional surface. Nevertheless, many representations of the Real Projective Plane exist:
- The Cross-cap
- The Roman Surface
Like the Mobius Strip, the Real Projective Plane is a non-orientable. Unlike the Mobius strip, however, the Real Projective Plane has no edges, or more technically, boundary. As a surface without boundary, the Real Projective Plane is a true 2 manifold. The presence of an edge in the Mobius strip allows sections of the shape to pass alongside each other, thus avoiding self intersection. Because the Real Projective Plane has no edges, there is no way for the surface to go around itself when it needs to close up. As a result, 3 dimensional models of the Real Projective Plane intersect, or pass through, themselves.
While all of the 3 dimensional models of the Real Projective Plane do intersect themselves, 4 dimensional models can be conceived of, and they don't self intersect. Because 4 is the fewest number of dimensions needed to represent the Real Projective Plane without it intersecting itself, the Projective Plane is said to be 4 embeddable.
The Real Projective Plane is non orientable, which means that, if perpendicular arrows were drawn in the surface to the surface pointing upward, moving them along specific paths in the shape would return them to the starting point as a mirror image of the position they were in when they began. This gives surfaces like the Projective Plane unusual characteristics, such as being one-sided when represented in 3 dimensions.
The Real Projective Plane can be made from the fundamental polygon (Image 1) if the corresponding edges are joined so that the arrows line up. This requires twisting the polygon, and can be confusing to imagine.
A More Mathematical Explanation
The Real Projective Plane (RP^{2}) is the 2 dimensional [[Projective Space|Real Project [...]
The Real Projective Plane (RP^{2}) is the 2 dimensional Real Projective Space. It's geometry describes the result of projecting a shape onto a plane, similar to how a person's shape is projected onto the ground as a shadow by the sun. These are three properties of the Real Projective Plane^{[1]}:
- 1. There are points that lie at infinity, and, when connected, form a line at infinity.
- 2. All lines intersect at exactly one point; parallel lines intersect at points located at infinity.
- 3. Homogeneous coordinates: the point is equivalent to another point if and only if there exists some constant, , such that:
- , meaning
Just as a circle can be made by merging one end of a line segment to the opposite end, the Real Projective Plane can be made by merging each and every point on a sphere to its antipodal point, setting the two as a single point on the Real Projective Plane. In the analogy, If you label the ends of the segment as the same point, but don not connect them, then the figure still represents a circle, but is not a depiction of the circle itself. Likewise, if the points on the sphere are labeled as identical to their antipodal points, but not actually merged with them, we have a model of the Real Projective Plane. This model is called the Projective Sphere.
Each point on the Real Projective Plane is represented by a pair of antipodal points on the Projective Sphere. Consequently, half of the Projective Sphere, the Projective Hemisphere, is also a valid representation of the Real Projective Plane. Each point on the hemisphere represents a single point on the Real Projective Plane, with the exception of the circle around the equator, which is still 2-to-one, as it represents the line at infinity. The line at infinity will be discussed more below.
Image 2 depicts the Projective Sphere, with the Projective Hemisphere shaded yellow. The two antipodal red points identify a single point on the Real Projective Plane where the brown and green parallel lines intersect. The blue line around the equator represents the line at infinity, composed of all the antipodal pairs of points at infinity in each direction.
Non-orientability
As a result of the mapping of a pair of antipodal points on the hemisphere to a single point on RP^{2}, the Real Projective Plane is non-orientable. Here is an example that illustrates the concept of non-orientability and how it arises on the projective plane.
Image 3 shows bird's-eye view of the Projective Hemisphere. On it are two pairs of identified antipodal points at infinity. Let us suppose that a 2 dimensional crab lives within the Real Projective Plane. In Image 4^{[2]}, the fiddler crab is shown walking North-west toward infinity. Notice that his right side is moving toward the green point, while his left side is headed for the red point. The points at infinity in the direction that he is approaching are the same as the corresponding points at infinity directly opposite them, in the South-eastern direction. Thus, once the crab reaches these points at infinity in the North-west, he will emerge from the South-east of the plane. Because the crab's right side went to the green point, and his left to the red point, his right side will emerge from the green dot in the South-east, while his left will emerge from the red dot in the South-east, as in Image 5^{[3]}. This matching has the effect that, after traveling around the Real Projective Plane, the crab's right side has flipped with his left.
Homogeneous Coordinates
The homogeneous coordinate system represents all points , where k is a constant, as the single point on the Real Projective Plane. For instance, , , and are all the same point on the projective plane^{[4]}.
Notice that these points, and indeed all points represented as from the same , , and , form a line that passes through the origin. As a result, each and every lines passing through in 3 dimensional space, or R^{3} maps to a single point on the Real Projective Plane. One can say that the lines are projected as points, hence the name projective plane.
Image 6 illustrates the projection of a line—containing all points for the same , , and —passing through the origin to a single point on the Real Projective Plane. Note that antipodal points on the Projective Sphere represent a single point on RP^{2}.
Why It's Interesting
Not only does the Real Projective Plane have the bizarre property of non-orientability, and one-siededness when modeled in 3 dimensions, it also has many applications in the field of computer vision and graphics.
We see the world as a flat image, a 2 dimensional projection of the 3 dimensional world. Nevertheless, Physics and geometry contribute perspective to the image, meaning that what we see is not merely a simple flattening of what is in front of us. Parallel lines converge at a point on the horizon, things that are closer look bigger than those that are farther away, objects hide things that are behind them in the same line-of-sight, and so forth.
The Real Projective Plane can accommodate many aspects of perspective. For instance, on the Real Projective Plane, parallel lines intersect at a point at infinity, just as they do in human vision. Additionally, the Real Projective Plane's homogeneous coordinate system means that points on the same line through the origin map to the same point on the Real Projective Plane, just as points on the same line of sight appear at the same point in our field of vision.
Additionally, a theorem exists which details how any closed surface "is either homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes"^{[7]}^{[8]}. Follow this link for an explanation: Classification Theorem for Compact Surfaces. This makes the Real Projective Plane a very fundamental and versatile manifold.
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References
- ↑ Real projective plane. June 8, 2011. Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/w/index.php?title=Special:Cite&page=Real_projective_plane&id=433132327. Accessed: June 14, 2011
- ↑ http://dsc.discovery.com/news/2006/10/23/crab_ani_zoom0.html?category=animals&guid=20061023150000
- ↑ http://dsc.discovery.com/news/2006/10/23/crab_ani_zoom0.html?category=animals&guid=20061023150000
- ↑ Projective Space. June 20, 2011. Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/w/index.php?title=Projective_space&oldid=435224643. Accessed June 20, 2011.
- ↑ Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.
- ↑ Surface. May 27, 2011 . Wikipedia, The Free Encyclopedia. ttp://en.wikipedia.org/w/index.php?title=Surface&oldid=431219154. Accessed: July 5, 2011.
- ↑ Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.
- ↑ Surface. May 27, 2011 . Wikipedia, The Free Encyclopedia. ttp://en.wikipedia.org/w/index.php?title=Surface&oldid=431219154. Accessed: July 5, 2011.
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