Regra dos 3 Pontos (parábola)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
Lagrange
About this calculator
The 3-Point Rule (parabola) calculator determines the quadratic polynomial that passes exactly through three given points on the Cartesian plane. Using Lagrange interpolation, it finds the quadratic function that best fits the points, allowing estimation of intermediate values or trend analysis. It is a useful tool for students and professionals working with numerical analysis, curve modeling, and function approximation.
It works based on the Lagrange interpolation formula for three points. The calculator receives coordinates (x1,y1), (x2,y2), and (x3,y3) and builds the polynomial P(x) = y1*L1(x) + y2*L2(x) + y3*L3(x), where each Li(x) is a Lagrange polynomial that equals 1 at its corresponding point and 0 at the others. The result is a parabola that can compute y for any x within or outside the given points' interval, with caution regarding extrapolation.
Use this calculator when you need to find a quadratic function that exactly fits three points, such as in physics (projectile trajectories), economics (quadratic costs), or engineering (calibration curves). It is ideal for interpolating intermediate values between known points, but avoid extrapolating too far beyond the interval, as the parabola may diverge quickly. Remember that Lagrange interpolation can suffer from Runge's phenomenon if points are not well distributed.
Caution: ensure the three points are not collinear (different x values) for a unique solution. If x values are equal, no parabola exists. Also, polynomial interpolation with many points can be unstable; for three points it is safe. Verify that y values do not have large measurement errors, as the curve will pass exactly through them, propagating those errors.
Frequently asked questions
What happens if the three points have the same x value?
If the three points have the same x, it is impossible to determine a parabola, as the function would not be of x (violates the vertical line test). The calculator will return an error.
Can I use this calculator to interpolate values between points?
Yes, interpolation is the main use. The parabola provides an estimate for any x between the smallest and largest x of the given points.
Is the parabola unique for three points?
Yes, given three non-collinear points, there is exactly one parabola (degree 2 polynomial) that passes through them.
What is the difference between Lagrange interpolation and other methods?
Lagrange builds the polynomial directly without solving linear systems, but for many points it can be less numerically stable than methods like Newton's divided differences.
Can I extrapolate using the parabola?
Yes, but extrapolation beyond the interval of the points may be inaccurate, as the parabola can grow rapidly. Use with caution.