ABSTRACT
Algebraic curves in the plane are often 'incomplete', in that certain 'points at infinity' are missing from them. We show here how plane algebraic curves can be extended to projective curves in the projective plane; any additional points obtained in the extension are the points at infinity for the original plane curve. The plane curve becomes an affine view of the projective curve. The asymptotes to the plane curve are tangents to the projective curve at the points at infinity. Different affine views of the projective curve can give further information about the original plane curve. A projective curve has many different affine views; for example, ellipses, hyperbolae, and parabolae are all affine views of the same projective curve. We give a classical defi nition of elements of a projective space as lines through the origin of the appropriate coordinate space. We show that points of the projective plane have a 'coordinate' representation, and use coordinates to obtain subse quent results and for solving examples. The asymptotes of the plane curve can be determined by studying the projective curve. Another application is to determine the boundedness of the plane curve.
