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This answer is only from the Theravada viewpoint. It's likely that you can find more diverse meditation techniques in Vajrayana.
Shambhavi Mudra is apparently an eyebrow center gazing gesture.
Agochari Mudra is apparently a nose tip gazing gesture.
While these are not found in Theravada Buddhism, the closest thing to it is parimukham. Please read this answer for a detailed analysis on the term parimukham, with respect to the mindfulness of breathing (anapanasati) meditation, in the Pali suttas.
This is often translated as concentrating on the sensation of the breath at the tip of the nose in the mindfulness of breathing (anapanasati) meditation. That does not refer to the tip of the nose at the eyebrow center, but rather, it refers to the nostrils and area around the nose and mouth where the sensation of the breath can be felt.
But the sensation of the breath can also be felt at the diaphragm and abdomen. Please also see this answer.
In any case, this is not about what to do with your eyes during anapanasati. This is about what to concentrate on. The eyes should be kept relaxed and preferably closed, to my understanding.