Calculator
Example Data Table
| Example curve | Inputs | Selected point | Key outputs |
|---|---|---|---|
| Circle | R=5, t0=0.6, t1=0, t2=1 | (x,y) = (5cos0.6, 5sin0.6) | κ = 1/R, ρ = R, tangent and normal lines |
| Parabola | y=0.25x², x0=1.5, x1=0, x2=3 | (x0, y0) = (1.5, 0.5625) | m=dy/dx, κ=|y''|/(1+(y')²)^(3/2) |
| Point list | Five sampled points | Index 2 uses points 1–3 | Total arc length estimate + local curvature estimate |
Formula Used
- Slope (parametric):
dy/dx = (dy/dt)/(dx/dt)whendx/dt ≠ 0. - Speed:
ds/dt = √((dx/dt)² + (dy/dt)²). - Curvature (plane parametric):
κ = |x'y'' − y'x''| / (x'² + y'²)^(3/2). - Radius of curvature:
ρ = 1/κ(whenκ ≠ 0). - Unit tangent:
T = (x', y') / √(x'² + y'²). - Unit normal: rotate
Tby 90°:N = (−Ty, Tx). - Arc length:
s = ∫ √((dx/dt)² + (dy/dt)²) dt(or∫ √(1+(dy/dx)²) dxfor explicit curves).
How to Use This Calculator
- Choose a mode: preset curve, polynomial parametric, or point list.
- Enter the required parameters and the evaluation point (t0 or x0).
- Optional: add an interval (start and end) to compute arc length.
- Press Calculate to show results above the form.
- Use the download buttons to export CSV or PDF reports.
Guide: Understanding Plane-Curve Results
1) What a plane curve means
A plane curve is a path drawn in the x–y plane. This calculator works with parametric curves x(t), y(t), explicit curves y(x), and measured point lists. At a chosen evaluation value (t0 or x0), it reports the position and the derivatives that describe the curve’s local behavior.
2) Choosing the right input mode
Use Preset curves for common shapes like circles, ellipses, cycloids, logarithmic spirals, Lissajous figures, lines, and parabolas. Use Polynomial parametric when your curve is a degree‑5 model: x(t)=Σaxᵢtᶦ and y(t)=Σbyᵢtᶦ. Use Point list for sampled data from a graph or experiment.
In polynomial mode you can enter twelve coefficients, ax0–ax5 and by0–by5, which lets you model smooth paths used in motion planning. In preset mode, arc-length intervals like 0 to 2π are useful for closed curves. The tool reports speed ds/dt for context.
3) Slope and directional vectors
For parametric curves, the slope is dy/dx=(dy/dt)/(dx/dt). When dx/dt is near zero, the tangent becomes vertical and the slope is undefined. The calculator also shows the unit tangent T=(x′,y′)/√(x′²+y′²) and a unit normal N obtained by a 90° rotation of T.
4) Curvature and radius of curvature
Curvature measures how sharply the curve bends: κ=|x′y″−y′x″|/(x′²+y′²)^(3/2). A circle with radius R has constant κ=1/R and radius of curvature ρ=R. When κ is very small, ρ becomes very large, indicating a nearly straight segment.
5) Tangent, normal, and osculating circle
The tangent line is shown in point‑slope form and as y=mx+b when possible. The normal line uses the perpendicular slope, or switches to vertical/horizontal forms when needed. If curvature is defined, the osculating circle is reported using its center of curvature and radius ρ.
6) Arc length with numerical integration
Arc length is optional. For parametric curves it evaluates ∫√(x′²+y′²)dt, and for explicit curves it uses ∫√(1+(dy/dx)²)dx. Simpson’s rule is used with an even step count; more steps generally improve accuracy but increase computation time.
7) Exports and practical uses
After calculating, export a CSV or PDF report for documentation, homework checks, or lab notes. Engineers use curvature to estimate turning tightness, designers use tangents for smooth joins, and data analysts use point‑list arc length to approximate distances from sampled trajectories.
FAQs
1) Should I enter angles in degrees or radians?
Trig-based presets assume radians. If you have degrees, convert using radians = degrees × π / 180 before entering t0, t1, or t2.
2) Why does the slope show “infinite / undefined”?
That happens when dx/dt is zero or extremely small, which creates a vertical tangent. The tangent line is then displayed as x = constant.
3) What does “radius of curvature” tell me?
Radius ρ is the radius of the best‑fitting circle at the point. Small ρ means a tight turn; large ρ means the curve is almost straight locally.
4) How accurate is arc length here?
Arc length is a numerical estimate using Simpson’s rule. Increase “integration steps” for smoother curves or long intervals. Very sharp features may need more steps to stabilize results.
5) How does point-list curvature work?
It uses three consecutive points and fits their circumcircle. The curvature is 4A/(abc), where A is triangle area and a,b,c are side lengths. Collinear points produce near‑zero curvature.
6) Can I export results without recalculating?
Yes. After any calculation, the latest result is kept for the current session. Use the CSV/PDF buttons until you run a new calculation or close the session.