Motivation
The Veronese surface arises naturally in the study of conics, specifically in formalizing the statement that five points determine a conic. A conic is a degree 2 plane curve, thus defined by an equation:
The pairing between coefficients and variables is linear in coefficients and quadratic in the variables; the Veronese map makes it linear in the coefficients and linear in the monomials. Thus for a fixed point the condition that a conic contains the point is a linear equation in the coefficients, which formalizes the statement that "passing through a point imposes a linear condition on conics". The subtler statement that "five points in general linear position impose independent linear conditions on conics," and thus define a unique conic (as the intersection of five hyperplanes in 5-space is a point) corresponds to the statement that under the Veronese map, points in general position are mapped to points in general position, which corresponds to the fact that the map is biregular (and thus the image of points are in special position if and only if the points were originally in special position).
Read more about this topic: Veronese Surface
Famous quotes containing the word motivation:
“Self-determination has to mean that the leader is your individual gut, and heart, and mind or we’re talking about power, again, and its rather well-known impurities. Who is really going to care whether you live or die and who is going to know the most intimate motivation for your laughter and your tears is the only person to be trusted to speak for you and to decide what you will or will not do.”
—June Jordan (b. 1939)