In mathematics, a Hartshorne ellipse is an ellipse in the unit ball bounded by the 4-sphere S<sup>4</sup> such that the ellipse and the circle given by intersection of its plane with S<sup>4</sup> satisfy the Poncelet condition that there is a triangle with vertices on the circle and edges tangent to the ellipse. They were introduced by , who showed that they correspond to k = 2 instantons on S<sup>4</sup>.