Use the Curvature Calculator to instantly determine the curvature ($\kappa$) and radius of curvature ($R$) of the circle passing through three distinct, non-collinear points in a 2D plane.
Curvature Calculator
Curvature Formula
The curvature ($\kappa$) is the reciprocal of the radius of curvature ($R$). When calculating the curvature of a circle defined by three points $P_1(x_1, y_1)$, $P_2(x_2, y_2)$, and $P_3(x_3, y_3)$, the radius $R$ is calculated using the following determinant-based formula derived from the distance formula and triangle area:
$$R = \frac{abc}{2 |A|}$$
Where:
$$a = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
$$b = \sqrt{(x_3-x_2)^2 + (y_3-y_2)^2}$$
$$c = \sqrt{(x_1-x_3)^2 + (y_1-y_3)^2}$$
And $|A|$ is the absolute value of the determinant-based area of the triangle formed by the three points, which relates to the cross product:
$$|A| = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$$
The curvature $\kappa$ is then:
$$\kappa = \frac{1}{R}$$
Formula Source 1: Wolfram MathWorld | Formula Source 2: Wikipedia (Circumradius)Variables
The calculator requires six coordinate inputs to define the three points that determine the circle’s curvature:
- $x_1$: The X-coordinate of the first point.
- $y_1$: The Y-coordinate of the first point.
- $x_2$: The X-coordinate of the second point.
- $y_2$: The Y-coordinate of the second point.
- $x_3$: The X-coordinate of the third point.
- $y_3$: The Y-coordinate of the third point.
What is Curvature?
Curvature is a fundamental concept in geometry that quantifies how sharply a curve bends. A large curvature value indicates a sharp bend, while a small value indicates a gentle, shallow bend. For a straight line, the curvature is zero.
In technical terms, the curvature ($\kappa$) at a specific point on a curve is defined as the reciprocal of the radius of the osculating circle (the circle that “kisses” the curve at that point and shares the same tangent and curvature). The larger the radius of this circle ($R$), the smaller the curvature. This calculator simplifies the concept by finding the curvature of the unique circle passing through the three specified points.
Understanding curvature is essential in fields like physics (path of motion), engineering (design of roads and rails), and computer graphics (smooth curve interpolation).
How to Calculate Curvature (Example)
Let’s find the curvature defined by the points $P_1(1, 0)$, $P_2(0, 1)$, and $P_3(-1, 0)$ (points on a unit circle centered at the origin):
- Calculate Side Lengths ($a, b, c$): Use the distance formula for all three segments.
$$a = \sqrt{(0-1)^2 + (1-0)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414$$
$$b = \sqrt{(-1-0)^2 + (0-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414$$
$$c = \sqrt{(1-(-1))^2 + (0-0)^2} = \sqrt{4 + 0} = 2$$
- Calculate Determinant Area ($|A|$):
$$|A| = \frac{1}{2} |1(1-0) + 0(0-0) + (-1)(0-1)|$$
$$|A| = \frac{1}{2} |1(1) + 0 + (-1)(-1)| = \frac{1}{2} |1 + 1| = 1$$
- Calculate Radius of Curvature ($R$):
$$R = \frac{abc}{2 |A|} = \frac{(\sqrt{2})(\sqrt{2})(2)}{2(1)} = \frac{4}{2} = 2$$
- Calculate Curvature ($\kappa$):
$$\kappa = \frac{1}{R} = \frac{1}{2} = 0.5$$
Frequently Asked Questions (FAQ)
Is curvature always a positive value?
Yes, curvature ($\kappa$) is defined as a non-negative value, representing the magnitude of the bending. In advanced differential geometry, *signed* curvature is used to indicate the direction (left or right) of the bend, but the standard geometric curvature is always positive.
What is the difference between Curvature and Radius of Curvature?
The radius of curvature ($R$) is the radius of the circle that best approximates the curve at a point. Curvature ($\kappa$) is its inverse ($\kappa = 1/R$). They are inversely related: a small radius means high curvature (sharp bend), and a large radius means low curvature (gentle bend).
What happens if my three points are collinear?
If the three points lie on a straight line (collinear), no unique circle can pass through them. In the formula, the determinant area $|A|$ will be zero, leading to an undefined or infinite radius of curvature, and thus zero curvature. The calculator will display an error for this boundary case.
What is the maximum curvature possible?
Mathematically, the radius of curvature ($R$) can approach zero, which means the curvature ($\kappa$) can approach infinity. In practical, physical applications, the maximum curvature is limited by the minimum radius of turn achievable by a system (e.g., how sharp a bend a car can safely take).