Calculator Input
Example Data Table
| x₀ | y₀ | y' | y'' | Radius ρ | Center (h, k) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | (0, 1) |
| 1 | 1 | 2 | 2 | 5.5902 | (-4, 3.5) |
| 2 | 5 | -1 | -0.5 | 5.6569 | (-2, 1) |
Formula Used
Curvature for y = f(x):
κ = |y″| / (1 + (y′)²)3/2
Radius of curvature:
ρ = 1 / κ = (1 + (y′)²)3/2 / |y″|
Center of the osculating circle:
h = x₀ − y′(1 + (y′)²) / y″ and k = y₀ + (1 + (y′)²) / y″
These expressions apply when the curve is smooth and the second derivative is not zero at the evaluation point.
How to Use This Calculator
- Enter the coordinate pair for the exact point where you want the osculating circle.
- Provide the first derivative y′ and second derivative y″ at that same point.
- Choose decimal precision and an optional unit label for cleaner reporting.
- Press Submit to place the result panel above the form.
- Review curvature, radius, center, vectors, and the final circle equation.
- Use the export buttons to save the computed summary as CSV or PDF.
Interpretation Notes
The osculating circle is the circle that best matches the curve near the selected point. A larger radius means gentler bending. A smaller radius means sharper bending. Positive second derivative places the center on the concave-up side. Negative second derivative places the center on the concave-down side. When the second derivative is zero, the curve is locally straight and the radius becomes infinite.
Role of Local Curvature
An osculating circle measures how sharply a curve bends at one chosen point. The calculator converts slope and second derivative data into curvature and radius values that summarize local geometry. High curvature implies tight turning, while low curvature indicates flatter behavior. This makes the tool useful for checking geometric intuition with numerical evidence in exams, tutorials, and practice sessions.
Inputs That Control Accuracy
Reliable output depends on evaluating x0, y0, y′, and y″ at the same location. If those quantities come from inconsistent approximations, the center and radius can drift. A positive second derivative moves the circle toward the concave-up side, and a negative value shifts it opposite. Small derivative errors can noticeably change radius estimates during differentiation and sampled data work directly.
Reading the Radius
The radius of curvature is the reciprocal of curvature, so it grows rapidly when bending weakens. For example, doubling curvature halves radius. This relationship helps students compare curves objectively. A radius near one unit signals tight local turning, while values above five units typically indicate a much gentler bend around the chosen point in plotted neighborhoods.
Center Location and Direction
The center coordinates are not guessed; they come directly from the normal direction of the curve. The calculator uses the point, slope, and concavity to place that center precisely. When curvature is finite, the circle touches the curve and matches its tangent direction locally, creating the best circular approximation near the selected point for engineering-style verification tasks.
Classroom and Applied Uses
This type of calculation appears in differential geometry, motion analysis, optics, and path design. In coursework, it helps connect derivatives with shape interpretation. In applied settings, local curvature supports track transitions, lens profiles, and contour evaluation. The example table included here gives quick benchmark cases for verifying whether inputs produce reasonable outputs across different concavity patterns and slope magnitudes.
Exporting and Comparing Results
The calculator also supports result exports, which helps when comparing several points on one function. By saving curvature, radius, center, and tangent angle, users can build a record for homework, reports, or revision notes. The graph strengthens interpretation by showing the selected point, nearby curve behavior, and the matching circle on one view for faster comparison.
FAQs
1. What does the osculating circle represent?
It is the circle that best matches the curve at one point. It shares the same point, tangent direction, and local curvature there.
2. Why is my radius shown as infinite?
That happens when the second derivative is zero at the selected point. Curvature becomes zero, so the curve is locally straight and no finite osculating circle exists.
3. Do the derivatives need to come from the same point?
Yes. The point coordinates, first derivative, and second derivative must all correspond to the same x-location. Mixed inputs produce misleading radius and center values.
4. What does a larger radius mean?
A larger radius means the curve bends more gently near the chosen point. Smaller radii indicate sharper local turning and higher curvature.
5. Is the graph the exact original function?
The graph uses a local quadratic model built from the point, slope, and second derivative. It is designed to visualize nearby behavior, not the full original function.
6. Can I use this for homework checking?
Yes. It is suitable for checking curvature calculations, comparing worked examples, and exporting clean summaries for revision notes or assignment support.