Points at infinity Algebraic extension to pts at infinity: introduction of homogeneous coordianted oo is not algebraic, only a symbol (x,yh if y*0, a finite pt if y=0, the pt at inf Rg the homogeneous coordinates are not unique, up to a scale
11 Algebraic extension to pts at infinity: introduction of homogeneous coordiantes is not algebraic, only a symbol if y 0, the pt at inf. ! ( , ) if 0, a finite pt = y y x x y Points at infinity: Rq: the homogeneous coordinates are not unique, up to a scale
On a plane (x,y,t)h>c,) if t+0, a finite pt if t=o, all pts at inf! Can we see the pts at infinity? The direction d is a pt at infinity: (d,0)
12 The direction d is a pt at infinity: (d,0) On a plane, if t 0, all pts at inf. ! ( , , ) ( , ) if 0, a finite pt = t t y t x x y t Can we see the pts at infinity?
Provisional summary a projective space is an affine space some pts at infinity or a projective space is a space of "homogeneous coordinates
13 a projective space is an affine space + some pts at infinity a projective space is a space of ‘homogeneous coordinates’ or Provisional summary
Formal) definition of projective geometry) Given K-R or C.p can be defined as the nonzero equivalent classes determined by the relation on R-0 D(u,,y,+) If there is non-zero real number such that n+1 Any element (r,,, n+ of the equivalent class will be called the homogeneous coordinates of the point
14 ((Formal) definition of projective geometry) Given K=R or C, can be defined as the nonzero equivalent classes determined by the relation ~ on n P R − 0 n+1 ( , , ) ~ ( , , ) 1 n+1 1 n+1 x x y y If there is non-zero real number such that ( , , ) ( , , ) 1 n+1 = 1 n+1 x x y y Any element of the equivalent class will be called the homogeneous coordinates of the point. ( , , ) 1 n+1 x x
a projective space is nothing but a quotient space(space of equivalent classes) P=R+1-0/ Ipts of pn=tines thru the origin in R tl) a space of homogeneous coordinates Basic structure: linear dependence of points Definition: a pt x is said to be linearly dependent on a set of pts if x=,x+ux,+
15 / ~ n n 1 P = R − 0 + n n 1 pts o f lines thru th e origin in + P = R Definition: a pt x is said to be linearly dependent on a set of pts if ... x = x1 + x2 + • A projective space is nothing but a quotient space (space of equivalent classes): • A space of homogeneous coordinates • Basic structure: linear dependence of points