Can anyone help with paramaterization of conics?Im struggling to wrap my head around an example. It considers the conic&nbsp;x2+y2−z2=0&nbsp;then proceeds:Take&nbsp;A=[1,0,1]&nbsp;and the line P(U) defined by&nbsp;x=0. Note that this conic and the point and line are defined over any field since the coefficients are 0 or 1. A point&nbsp;X∈P(U)&nbsp;is of the form&nbsp;X=[0,1,t]&nbsp;or [0, 0, 1] and the map&nbsp;α&nbsp;is α ( [ 0 , 1 , t ] ) = [ B ( ( 0 , 1 , t ) , ( 0 , 1 , t ) ) ( 1 , 0 , 1 ) − 2 B ( ( 1 , 0 , 1 ) , ( 0 , 1 , t ) ) ( 0 , 1 , t ) ] = [ 1 − t 2 , 2 t , 1 + t 2 ] &nbsp;or&nbsp; α ( [ 0 , 0 , 1 ] ) = [ − 1 , 0 , 1 ] . How do I evaluate B(v,v) or B(v,v)(a,b,c) like they have to go from the first line to the second?

Question

Can anyone help with paramaterization of conics?Im struggling to wrap my head around an example. It considers the conic&amp;nbsp;x2+y2−z2=0&amp;nbsp;then proceeds:Take&amp;nbsp;A=[1,0,1]&amp;nbsp;and the line P(U) defined by&amp;nbsp;x=0. Note that this conic and the point and line are defined over any field since the coefficients are 0 or 1. A point&amp;nbsp;X∈P(U)&amp;nbsp;is of the form&amp;nbsp;X=[0,1,t]&amp;nbsp;or [0, 0, 1] and the map&amp;nbsp;α&amp;nbsp;is                            α          (          [          0          ,          1          ,          t          ]          )                                    =          [          B          (          (          0          ,          1          ,          t          )          ,          (          0          ,          1          ,          t          )          )          (          1          ,          0          ,          1          )          −          2          B          (          (          1          ,          0          ,          1          )          ,          (          0          ,          1          ,          t          )          )          (          0          ,          1          ,          t          )          ]                                                        =                      [            1            −                          t                              2                                      ,            2            t            ,            1            +                          t                              2                                      ]                                                &amp;nbsp;or&amp;nbsp;          α          (          [          0          ,          0          ,          1          ]          )          =          [          −          1          ,          0          ,          1          ]          .                    How do I evaluate B(v,v) or B(v,v)(a,b,c) like they have to go from the first line to the second?

Jakayla Hayes · Accepted Answer

For the first question, it is exactly evaluation of a bilinear form, given by the matrix B, on the given vectors: so B(v,w)=vtBw. This gives you a scalar which is understood multipling the following vector.For the second one, when you have a projective line l and two dinstinct points P,Q on it then every other point can be written as λP+μQ, for [λ,μ]∈P1. So you can define t=μλ, if λ≠0, and consider l as the points in the form P+tQ union the remaining point corresponding to λ=0.

Can anyone help with paramaterization of conics? Im struggling

Answered question

Answer & Explanation

New Questions in Integral Calculus