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I came across the definition in the book titled Elliptic Curves by Anthony W Knapp, couldn't understand it so looked online, which just confused me more. I'm looking for an explanation in the context of curves in projective plane/space.

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1 Answer

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You get a projective plane from an affine plane if you consider “points at infinity” as regular elements of your plane. This simplifies a number of situations, for example two distinct lines will always intersect in a unique point, the special case of parallel lines vanishes. Parallel lines simply intersect at infinity.

Expressed in coordinates, you add one coordinate. So a point $(x,y)$ in the usual Cartesian plane would be represented as $[x,y,1]$ or any multiple thereof. That's called a homogeneous coordinate vector. So in fact you are no longer dealing in vectors, but strictly speaking in equivalence classes of vectors. Most of the time authors will use the same notation for vectors and for equivalence classes, and rely on context to tell you which is which in those cases where it makes a difference. To convert back, a homogeneous coordinate vector $[x,y,z]$ corresponds to a Cartesian vector $(x/z,y/z)$. If $z=0$, this would be undefined; those are the points at infinity. The vector $[0,0,0]$ has to be excluded, since it would otherwise belong to all equivalence classes. The null vector does not represent any point in the projective plane.

In terms of curves, you might want to make certain that all your equations are homogeneous, i.e. have the same degree in each term (for each geometric object). So for example the equation $7x^2 + 5y = 3$ would not be homogeneous: if you have a vector which solves this, then take twice that vector, you may end up with another representative of the same point which fails the equation. $7x^2 + 5yz = 3z^2$ on the other hand would be a homogeneous equation.

See also Difference between Projective Geometry and Affine Geometry which discusses that difference from a different point of view.

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