We develop two new ideas for interpolation on $$\mathbb {S}^2$$ . In this first part, we will introduce a simple method named Spherical Interpolation of orDER n (SIDER-n) that gives $$C^{n}$$ interpolant given $$n \ge 2$$ The idea generalizes the construction Bézier curves developed {R}$$ second part incorporates ENO philosophy and develops Essentially Non-Oscillatory (SENO) method. When underl...