Two distinct points in RP 2 will determine in line( this is familiar from Euclid

Question

Two distinct points in RP2 will determine in line( this is familiar from Euclidean geometry). The following formula shows how to compute the coefficients of the equation of the line, given homogenous coordinates of the two points - the formula should evoke the cross product of two vectors in R3 . Suppose the two points are (a,b,c) and (d,e,f). Then, the equation of the line will be:

(bf - ce)X + (cd - af)Y + (ae - bd)Z = 0

Question 1: We can think of a point on the line defined by points A = (a,b,c) and B = (d,e,f) as having homogeneous coordinates tA + sB, for real numbers t and s. Show that a point defined in this way will satisfy the equation of the line given above.

Question 2 : Let projective point U be given by (0,3,5) and projective point V be given by (0,0,2). What is the equation of the line defined by these points, and what does it correspond to in R3 ?

Explanation / Answer

1) Every point on a line defined as the set of points with coords tA+sB has coords of the form

(ta+sd, tb+se,tc+sf). We claim that these coords satisfy the given equation. To see that, plug these values for X, Y, Z respectively:
(bf-ce)(ta+sd) + (cd-af)(tb+se) + (ae-bd)(tc+sf) =

abft + bfds + bcdt + cdes + aetc + aefs - acet - cdes - abtf - aefs - bcdt - bdfs = 0.

2) PLugging in th equation above, we have the given line to be 6X = 0, or equivalently, X = 0 which is the Y axis.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.