I'm implementing an application where I need to get the vertex of a parabola, the parabola might be tilted; so it can have an angle with the x-axis not necessarily vertical or horizontal. Can I get the vertex value from 3 points on the parabola?
2026-04-11 23:01:29.1775948489
Rotated parabola 2d vertex
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No.
When we know the parabola' axis is vertical, it takes three points to define a parabola. (See the Lagrange interpolation formula: three points define a 2nd-degree polynomial, which defines a parabola.) Allowing the axis to rotate adds another degree of freedom, so three points are no longer sufficient. Given any three points we can find a parabola in any direction (at least, those where the points are not all in a vertical or horizontal line), so the parabola is not well defined if we allow all directions.
Four points determine a parabola up to a choice of two possibilities.