Generalized Loci on Real Inner Product Vector Spaces

Abstract

This paper generalizes the notion of geometric curves such as hyperbolas and ellipses to more general vector spaces with an associated inner product. This is done by generalizing the definition in terms of loci and foci of said curves in Euclidean geometry to a general vector space with a real inner product, through which a norm can be induced. Through this generalization and focusing on the curves that are obtained through linear combinations of norms, we explore some properties of said curves. Specifically, we explore the addition of vectors in the curve, and in what other curves this addition can be found in relation to the original curve. Lastly, we observe the effects of applying the isomorphism to the geometric curve in the vector space onto R^n, and we compare geometric curves obtained with the same definition in different vector spaces with different norms.