Curvature measures how sharply a curve bends at a point. For a straight line, the curvature is zero; for a tight bend, the curvature is large.
In everyday terms, curvature is what tells you whether a road is almost straight or has a sharp turn. In mathematics and engineering, curvature
is used in path planning, mechanical design, optics, and even in understanding motion along curved trajectories.
For a plane curve written as a function \(y=f(x)\), the most common curvature formula is
\[
\kappa=\frac{|y''|}{\left(1+(y')^2\right)^{3/2}}.
\]
Here \(y'\) is the slope of the tangent line, and \(y''\) describes how quickly that slope is changing. The denominator
\(\left(1+(y')^2\right)^{3/2}\) corrects for the fact that the curve might be tilted: a curve with the same “shape” should not appear to
have different curvature simply because it is viewed at a steeper angle. The absolute value makes \(\kappa\) nonnegative; some textbooks
use a signed curvature that keeps the sign of \(y''\).
The radius of curvature is defined as \(R=1/\kappa\). This is the radius of the circle that best matches the curve at the point.
When \(R\) is small, the curve bends tightly; when \(R\) is large, the curve is almost straight. If \(y''(x_0)=0\) and the curve is smooth
near \(x_0\), then \(\kappa\approx 0\) and \(R\) becomes very large (conceptually infinite), meaning the “best-fitting circle” is so large that
it is practically indistinguishable from a straight line.
That best-fitting circle is called the osculating circle (“kissing” circle). For \(y=f(x)\) at \(x_0\), let \(P=(x_0,f(x_0))\).
A convenient formula for the center of curvature is
\[
C = P + \frac{1+(y')^2}{y''}\,(-y',\,1),
\]
provided \(y''\neq 0\). The vector \((-y',1)\) points in the normal direction (perpendicular to the tangent), and the factor
\(\frac{1+(y')^2}{y''}\) moves you from the point to the center by the correct signed distance. The magnitude of that distance is the radius \(R\).
In practice, \(y'\) and \(y''\) are often approximated numerically using small step sizes \(h\) (finite differences). Central difference formulas
are popular because they are accurate and symmetric around \(x_0\). This calculator uses central stencils to estimate derivatives, then computes
\(\kappa\), \(R\), and (when defined) the osculating circle. By panning and zooming the plot, you can visually compare the curve to its tangent,
normal, and osculating circle and build intuition for what curvature really means.
