Hiya
I was wondering if anyone can help on this one - I'm reading up about Plucker the 19th century mathematician who looked at how to classify curves. One of his fundamental formula was:
through a given point are n(n-1) tangents to a curve. Where n is the order of the curve (e.g y^3 - x^2 = 0 is order 3)
Here he is looking at algebraic curves, and is working in the projective plane. This would mean for a cubic there are 6 tangents to any point, which I'm having a hard time imagining.
Say we have the curve y - x^3 = 0. Well if we put it in homogenous co-ordinates we get yz^2 -x^3 = 0. Does this really have 6 tangents to any point? Any way of plotting this?
Or is there any way of plotting curves in the projective plane (say into a disk model) that would help understanding?
Anyone know about this?
Thanks :)
I was wondering if anyone can help on this one - I'm reading up about Plucker the 19th century mathematician who looked at how to classify curves. One of his fundamental formula was:
through a given point are n(n-1) tangents to a curve. Where n is the order of the curve (e.g y^3 - x^2 = 0 is order 3)
Here he is looking at algebraic curves, and is working in the projective plane. This would mean for a cubic there are 6 tangents to any point, which I'm having a hard time imagining.
Say we have the curve y - x^3 = 0. Well if we put it in homogenous co-ordinates we get yz^2 -x^3 = 0. Does this really have 6 tangents to any point? Any way of plotting this?
Or is there any way of plotting curves in the projective plane (say into a disk model) that would help understanding?
Anyone know about this?
Thanks :)
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