Projective view of conics and quadrics | Differential Geometry 9 | NJ Wildberger

In this video we introduce projective geometry into the study of conics and quadrics. Our point of view follows Mobius and Plucker: the projective plane is considered as the space of one-dimensional subspaces of a three dimensional vector space, or in other words lines through the origin. In this way we can introduce homogeneous coordinates [X:Y:Z] for the more familiar points [x,y]; the big advantage is that now points at infinity become concrete and accessible: they are simply points of the form [X:Y:0].

A curve like the parabola y=x^2 gets a homogeneous equation YZ=X^2, including now the point at infinity [0:1:0], which corresponds to the direction in the y axis. This gives a uniform view of conics close to Apollonius' view in terms of slices of a cone.

We will see that homogeneous coordinates provide a powerful and useful tool to not only the study of conics and algebraic curves in the plane, but also to quadrics and higher algebraic surfaces in space.

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