maximising the angle to view a portrait at the gallery.

How is the angle \theta affected by the distance, x, away from the portrait?

But is this model realistic? What does maximising angle \theta really mean? Does making the angle largest give the clearest view of the picture? would the person standing distance x away from the portrait not tilt his head?

A nice geometry argument claims that the maximum angle occurs when a circle draw that passes the bottom and top of the portrait is tangential to the eye level.

This is because if the circle passing through the 3 points (the eye, bottom and top of the portrait) intersects the eye level at 2 distinct points, the angle \theta can be further maximise when choosing to stand at a point between the 2 distinct points cut by the circle.

It turns out that the above modelling is similar to modelling for the best position of a soccer player to kick the ball from the side line.

maximising the angle to view a portrait at the gallery

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