Real Projective Plane

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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:





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.

Image 1.The Real Projective Plane represented via a fundamental polygon.
Image 1.The Real Projective Plane represented via a fundamental polygon.

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 (RP2) is the 2 dimensional [[Projective Space|Real Project [...]

The Real Projective Plane (RP2) 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 P=(x,y,z) is equivalent to another point P'=(x',y',z') if and only if there exists some constant, k, such that:
kP=P', meaning kx=x'~~~~ky=y'~~~~kz=z'


Image 2. The Projective Sphere with the Projective Hemisphere shaded yellow.
Image 2. The Projective Sphere with the Projective Hemisphere shaded yellow.

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 RP2, 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.

Image 3. A bird's-eye view of the Projective Hemisphere with two pairs of identified antipodal points representing two points at infinity.
Image 3. A bird's-eye view of the Projective Hemisphere with two pairs of identified antipodal points representing two points at infinity.
Image 4. A fiddler crab walking North-west toward infinity.
Image 4. A fiddler crab walking North-west toward infinity.
Image 5. The fiddler crab emerging from the opposite side of the Projective hemishphere, with its right and left sides flipped.
Image 5. The fiddler crab emerging from the opposite side of the Projective hemishphere, with its right and left sides flipped.



Homogeneous Coordinates

Image 6. A line through the origin is mapped to a point on the Real Projective Plane.
Image 6. A line through the origin is mapped to a point on the Real Projective Plane.

The homogeneous coordinate system represents all points (kx,ky,kz), where k is a constant, as the single point (x,y,z) on the Real Projective Plane. For instance, (3,2,1), (6,4,2), and (9,6,3) are all the same point on the projective plane[4].

Notice that these points, and indeed all points represented as (kx,ky,kz) from the same x, y, and z, form a line that passes through the origin. As a result, each and every lines passing through (0,0,0) in 3 dimensional space, or R3 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 (kx,ky,kz) for the same x, y, and z—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 RP2.

The homogeneous coordinates allow us to do represent points as \left (\tfrac{x}{z},\tfrac{y}{z},1 \right ) so long as z\ne0, as I will now prove.

  • We have any fixed point \left (x,y,z \right ).
  • As a result of homogeneous coordinate system, we can multiply these coordinates by any constant and, so long as we multiply them all by the same constant, the point specified will remain the same.
  • Since we are dealing with a specific point, we know that x, y, and z all have specific values. Accordingly, we can multiply x, y, and z by \frac{1}{z} because z has a constant value for this specific point.
  • Multiplying \left (x,y,z \right ) by the constant \frac{1}{z} yields: \left (\tfrac{x}{z},\tfrac{y}{z},1 \right )

Thus, an arbitrary but fixed point can be represented on the Real Projective Plane as \left (\tfrac{x}{z},\tfrac{y}{z},1 \right ) so long as z\ne0. As such, the points are projected onto the plane at z=1, parallel to the xy-plane.

When z=0, points are commonly represented as \left (\tfrac{x}{y},y,0 \right ), for the same reason shown above.

When both y=0 and z=0, points are represented as \left (1,0,0 \right ), as \frac{x}{x}=1.

Image 7 depicts Two parallel lines on the Projective Hemisphere projected to the plane at z=1, or two parallel lines on the plane at z=1 projected onto the Projective Hemisphere. Notice that the points where the two lines meet will be projected to infinity on the plane at z=1, as though the parallel lines were converging to a point on the infinity on the horizon. In fact, any point on the equator, where z=0, represents a point at infinity when projected in this manner. This is why the equator is called the line at infinity.


Image 7. The projections of two parallel lines on the Projective Hemisphere projected to the plane at z=1, or from the plane at z=1 onto the Projective Hemisphere.
Image 7. The projections of two parallel lines on the Projective Hemisphere projected to the plane at z=1, or from the plane at z=1 onto the Projective Hemisphere.


The method of mapping points from the Projective Hemisphere projected to the plane at z=1 has an unintended consequence: because every line that can be represented as (kx,ky,kz) goes through the origin, the origin certainly must be projected to every point on the Real Projective Plane. Fortunately, Mathematicians decided to avoid this problem altogether by omitting the point (0,0,0) from the Real Projective Plane when they initially defined it. Although this may seem inauthentic, remember, the Real Projective Plane is an abstract manifold, and our representations of it do their best to represent its true characteristics, one of which manifests itself as the omission of the origin when we model it.


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"[5][6].




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.
Image 8. The two parallel white lines on either side of the road converge and meet at a point on the horizon.
Image 8. The two parallel white lines on either side of the road converge and meet at a point on the horizon.


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.


Teaching Materials

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References

  1. 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
  2. http://dsc.discovery.com/news/2006/10/23/crab_ani_zoom0.html?category=animals&guid=20061023150000
  3. http://dsc.discovery.com/news/2006/10/23/crab_ani_zoom0.html?category=animals&guid=20061023150000
  4. 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.
  5. Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.
  6. Surface. May 27, 2011 . Wikipedia, The Free Encyclopedia. ttp://en.wikipedia.org/w/index.php?title=Surface&oldid=431219154. Accessed: July 5, 2011.
  7. Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.
  8. Surface. May 27, 2011 . Wikipedia, The Free Encyclopedia. ttp://en.wikipedia.org/w/index.php?title=Surface&oldid=431219154. Accessed: July 5, 2011.

Future Directions for this Page

This page seems to have enough content, however it needs to be more efficiently organized. This requires a knowledge of projective geometry and the order in which the topic is usually taught.


A few comments:

  • Do not delete the crab non-orientability images, or the section as a whole. Not only is it linked to from other pages, but it provides an in depth explanation of how the phenomenon of non-orientability arises in the RPP.
  • Both analytic and synthetic approaches should be used in the organization.
  • A development of the topic of Homogeneous coordinates would be good (possibly a helper page), as well as a clearer explanation of their relation to the RPP.
  • Develop the introduction to projective geometry (possibly a helper page): lines, points, projections, lines at infinity, the origin, etc.




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