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Closest Point on Curve Calculator

Find the nearest point on any curve to any point — with visual distance

Given a point P and a curve y = f(x), the closest point on the curve is the point Q where the distance PQ is minimized. This is found by minimizing:

D(x) = \sqrt{(x - p_x)^2 + (f(x) - p_y)^2}

This closest point on curve calculator with graph draws the point, the nearest point on the curve, and the minimum distance segment visually on an interactive graph. Type any function and a point, and the AI finds the exact closest point using numerical optimization. Zoom and pan to explore.

Graph

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FAQ

How is the closest point on a curve found?
The distance from a point (pₓ, pᵧ) to the curve y = f(x) is D(x) = √((x−pₓ)² + (f(x)−pᵧ)²). The closest point minimizes this distance — we find where dD/dx = 0 using numerical methods and check all critical points plus endpoints.
Is the closest point always where the normal line passes through?
Yes! The line from the external point to the closest point on the curve is always perpendicular to the tangent at that point. This is because the shortest path from a point to a curve is along the normal direction.
Can there be multiple closest points?
Yes. For example, the closest point on a circle to its center is every point on the circle (all equidistant). For a parabola, a point on the axis of symmetry may be equidistant from two points on the curve. The calculator finds the global minimum.
Does this work with parametric curves?
Yes. For parametric curves (x(t), y(t)), the calculator minimizes the distance in t-space, finding the parameter value where the curve is closest to your point.
Why use a closest point calculator with a graph?
The graph shows the point, the curve, the closest point, and the minimum distance segment visually. You can immediately see why that point is closest and how the distance relates to the curve's shape. Interactive zoom lets you verify the result.

Related Topics

Tangent Line CalculatorPoint-to-Line DistanceRoot Finder