Curvature and Osculating Circle

Overview

Curvature measures how sharply a curve bends at a given point. The osculating circle (from the Latin "osculari" meaning "to kiss") is the circle that best approximates the curve at that point. It shares the same tangent, normal, and curvature as the curve itself. For a parametric curve r(t) = (t, a·t²), we can compute the curvature κ, the radius of curvature R = 1/κ, and the center of the osculating circle explicitly.

Mathematical Details

Consider the parabola given by r(t) = (t, a·t²) for some constant a ≥ 0.

Speed: The derivative is r'(t) = (1, 2at). The speed is ||r'(t)|| = √(1 + 4a²t²).
Curvature: Using the formula κ = ||r' × r''|| / ||r'||³, where r''(t) = (0, 2a), we get κ = 2a / (1 + 4a²t²)^3/2.
Radius of Curvature: The radius of the osculating circle is R = 1/κ = (1 + 4a²t²)^3/2 / 2a. If a=0, the curvature is zero and the radius is infinite.
Unit Normal Vector: The principal unit normal N(t) points toward the center of curvature. N(t) = 1/√(1+4a²t²) · (-2at, 1) for a > 0.
Circle Center: The center of the osculating circle is C(t) = r(t) + R(t)N(t).

Worked Example

Let's calculate the osculating circle for the parabola with a=1 at t=0.5. This example is worked out explicitly here.

Evaluate the position: r(0.5) = (0.5, 1(0.5)²) = (0.5, 0.25).
Calculate the curvature: κ = 2(1) / (1 + 4(1)²(0.5)²)^3/2 = 2 / (1 + 1)^3/2 = 2 / 2√2 = 1/√2 ≈ 0.707106.
Find the radius of curvature: R = 1/κ = √2 ≈ 1.41421.
Compute the unit normal: The speed is √(1+1) = √2. N(0.5) = 1/√2 · (-2(1)(0.5), 1) = 1/√2 · (-1, 1).
Determine the center: C = r + R·N = (0.5, 0.25) + √2 · 1/√2 · (-1, 1) = (0.5, 0.25) + (-1, 1) = (-0.5, 1.25).

Experiment

Follow these steps to experiment with the module.

Set Parabola coefficient a to 1 and Curve parameter t to 0. Notice that the radius of curvature at the vertex is 0.5.
Change Curve parameter t to 1. Observe how the curvature decreases and the osculating circle grows larger as you move away from the vertex.
Set Parabola coefficient a to 0. Observe the straight line limit: curvature drops to zero, and the osculating circle is appropriately undefined as its radius approaches infinity.
Set Parabola coefficient a to 0.5 and Curve parameter t to 1. Check point (1, 0.5), radius 2√2 and center (−1, 2.5).
Set Parabola coefficient a to 1 and Curve parameter t to −0.5. The center reflects to (0.5, 1.25), while curvature and radius match t = 0.5.
Clear Parabola coefficient a. Its error is visible and the old circle/results disappear. Restore 1, pan, and choose Reset view; the input values remain unchanged.

You can try this interactive module yourself: Curvature and Osculating Circle

Limitations

The visualization uses standard double-precision arithmetic. At very small non-zero values of a or very large values of t, floating-point inaccuracies might affect the precise drawing of the large circle. The view fits the complete circle and center with a common scale on both axes when their extents are numerically representable. Direction arrows are scaled for visibility. Extremely small positive a can put the mathematically finite circle beyond the floating-point plotting range; that state is explicitly distinguished from the zero-curvature straight line at a = 0. Decimal readouts are rounded, not symbolic exact values.

References

The following reference material provides foundational calculus definitions and theorems.

OpenStax Calculus Volume 3: 3.3 Arc Length and Curvature

Curvature and Osculating Circle