The Real Projective Plane

The projective plane is the space of lines through the origin in 3-space. In the projective plane, we have the remarkable fact that any two distinct lines meet in a unique point. Moreover, we may not distinquish between different kinds of smooth conic sections, e.g. between an ellipse and a hyperbola, as illustrated by the animation.

Notice that to each line through the origin correspond two antipodal points of a sphere which is centered at the origin. Instead of looking at the lines in total, we may thus restrict ourselves to the points on the sphere. Here, a line becomes a great circle and it is clear that any two such circles meet in a unique pair of antipodal points. Similarly, we may see that a hyperbola is projectively the same thing as an ellipse. Going back to the definition: They are both just a cone in 3-space; each line on the cone is one projective point of the projective conic section.

This animation was made by Oliver Labs using surfex.

This animation was #3 on our geometric animations advent calendar:

http://www.calendar.algebraicsurface.net

Category:

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Tags:

• geometry
• mathematics
• projective
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• section
• advent
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• bothmer
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• oliverlabs
• saarbruecken
• surfex
• cg

[mb1329]

i just noticed that if you have the video paused and you're reading the description, the image appears to expand in your peripheral. (I paused at :57)

[sjoerdwennekes]

Maybe some sound or explanation would help. Because I don't get it.

[mattrulzhard]

willwak has replied to your comment on The Real Projective Plane:

@mattrulzhard

nerds concieved, designed, built and manage the world duuuuuuuuuuuuude. you're just a consumer.....like a sheep.

Nerds run you like a pimp runs a whore.

[mattrulzhard]

nerds.

[Jbeliski]

So that's what all those flat graphs on graph paper we do in functions and relations are of. There just projections of points from a sphere. That's why pi is in all the equations. IT's just a bloody CIRCLE is all. Why didn't my math teacher tell me that. Goddamit I'm pissed.

[cervantespr]

I guess I get it...

[cervantespr]

After a year studying math, R3 and even logics and more interesting stuff like topology. Well tell you the truth Im still lost. Hehhehe Im sorry I keep more attention to it. My question is more confusing not because is in spanish, rather more than a thought than a question. Or in other words I was thinking and typing at the same time. Well... let me watch the video again...

[razven]

lmfao

[omghitbycar]

ingenios

[AgainstTheRisen]

Its like the star transforms into the light cone?...maybe

[kinglucanitus]

Got'a Love Mathematical Images