Moment curve
In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form
In the Euclidean plane, the moment curve is a parabola, and in three-dimensional space it is a twisted cubic. Its closure in projective space is the rational normal curve.
Moment curves have been used for several applications in discrete geometry including cyclic polytopes, the no-three-in-line problem, and a geometric proof of the chromatic number of Kneser graphs.
Every hyperplane intersects the moment curve in a finite set of at most d points. If a hyperplane intersects the curve in exactly d points, then the curve crosses the hyperplane at each intersection point. Thus, every finite point set on the moment curve is in general linear position.
The convex hull of any finite set of points on the moment curve is a cyclic polytope. Cyclic polytopes have the largest possible number of faces for a given number of vertices, and in dimensions four or more have the property that their edges form a complete graph. More strongly, they are neighborly polytopes, meaning that each set of at most d/2 vertices of the polytope forms one of its faces. Sets of points on the moment curve also realize the maximum possible number of simplices, , among all possible Delaunay triangulations of sets of n points in d dimensions.
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