Here, we introduce the quadratic orthogonal projection (the problem of projecting a point onto a quadratic level set), as an extension of the linear orthogonal projection (the problem of projecting a point onto a hyperplane). As the latter problem is convex and has a closed formula solution, the former one belongs to a special class of non-convex problems. We propose an iterative algorithm for the quadratic orthogonal projection, and test it for distinct quadratic functions, showing its great potential in applications, such as in computer graphics, alternating projections, and orbit projections.