Curvature is a value that measures how curved is the curve at a point on a curve. The most intuitive way to see it is that at any point P on the curve there is a circle of right size that touches P and fits the most. Call this most-fitting circle the osculating circle. The flatter the curve at P, the larger is its osculating circle. The sharper the curve at P, the smaller is its osculating circle. Thus, we can define the value of curvature as 1/r, where r is the radius of the osculating circle. When the osculating circle is large, the curve is flattish, and the curvature 1/r is small.

Tangent circles of a ellipse

Ideally, a line would have curvature of 0 everywhere, and its osculating circle doesn't exist. (i.e. have infinite radius) A Circle of radius R would have a constant curvature of 1/R, and its osculating circle at any point on the circle would be the same as the circle itself.

There are several ways to define osculating circle mathematically. One way is by the limit of the intersetion of two normals at points P1 and P2 on the curve as P1 and P2 approach each other and meet at P. The limit of that intersection is the center C of a osculating circle at P. Thus, distance CP would be the radius of the osculating circle, and 1/distance[C,P] would be the curvature.

Another way to define curvature mathematically is the rate of change of the turning of the tangent line. That is, suppose θ is the angle of the tangent at P and the x-axis. As a P moves on the curve, θ changes. The curvature at P is the rate of change of θ at P. This way, see see that a line has curvature of 0 everywhere since it's tangent doesn't change. And a circle would have constant curvature because its tangent turns in a constant way.


Given a parametric curve {xf[t],yf[t]}, let T[t] be the unit-length tangent vector at t. That is, T[t]:={xf'[t]/,yf'[t]}/s[t], where s[t] is the “speed” defined as s[t]:=Sqrt[xf'[t]^2 + yf'[t]^2]. Let N[t] be the unit-length normal at t, that is, let {t1[t],t2[t]} denoted the coordinates of T[t], then N[t]= {-t2[t],t1[t]}. Now, curvature can be given as:

k[t] := (T'[t] . N[t])/s[t]

Expand it out we have:

k = (x' y'' - x'' y')/(x'^2+y'^2)^(3/2)

(argument suffix [t] dropped for easy reading. xf[t] replaced by x, yf[t] by y.) Note: osculating circle is also called Osculating Circle.

Interesting Examples

Interesting curves involning the curvature concept, see:

For a gallery of artifacts that involves fancy curves, see: Curlicues.

Related Web Sites

Robert Yates: Curves and Their Properties.



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Plane Curves


  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral


  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve


  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature