Projection onto the parabola

TL;DR

This paper derives explicit formulas for projecting points onto parabolas and their higher-dimensional analogs, using critical point analysis of associated polynomials, with special focus on points on the parabola's vertical line.

Contribution

It provides explicit projection formulas onto parabolas and extends these results to higher dimensions, including special cases on the parabola's vertical line.

Findings

Explicit projection formulas for quadratic parabola graphs.

Method based on critical points of related quartic and cubic polynomials.

Extension of projection formulas to higher-dimensional parabolas.

Abstract

In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic polynomial defining its derivative. We also focus on the case where the point we project lies on the vertical line defined by the parabola. Lastly, an explicit formula for the projection onto a higher dimensional parabola is derived.