The following is slightly modified from the differential geometry part of the book *Shape Interrogation for Computer Aided Design and Manufacturing* by Nicholas M. Patrikalakis and Takashi Maekawa of MIT.

A plane curve can be expressed in parametric form as *x* = *x*(*t*); *y* = *y*(*t*); where the coordinates of the point (*x,* *y*) of the curve are expressed as functions of a parameter *t* (time) within a closed interval *t*_{1} ≤ *t* ≤ *t*_{2}. The functions *x*(*t*) and *y*(*t*) are assumed to be continuous with a sufficient number of continuous derivatives. The parametric representation of space curves is: *x* = *x*(*t*); *y* = *y*(*t*); *z* = *z*(*t*); *t*_{1} ≤ *t* ≤ *t*_{2}.

In vector notation the parametric curve can be specified by a vector-valued function **r** = **r**(*t*), where **r** represents the position vector (i.e., **r**(*t*) = (*x*(*t*), *y*(*t*), *z*(*t*)).

Let us consider a segment (displacement) of a parametric curve **r** = **r**(*t*) between two points *P*(**r**(*t*)) and *Q*(**r**(*t*+Δ*t*)) as shown in the figure above. As point *Q* approaches *P* or in other words Δ*t* → 0, the length *s* becomes the differential arc length of the curve:

d*s* = |d**r**/d*t*| d*t* = | **r ^{•}** | d

*t*= √(

**r**•

^{•}**r**) d

^{•}*t*.

Here the dot ** ^{•}** denotes differentiation with respect to the parameter

*t*. Therefore the arc length of a segment of the curve between points

**r**(

*t*

_{0}) and

**r**(

*t*) can be obtained as follows:

*s*(*t*) = ∫ d*s* = ∫ √(**r ^{•}** •

**r**) d

^{•}*t*= ∫ √(

*x*

^{•}^{2}(

*t*) +

*y*

^{•}^{2}(

*t*) +

*z*

^{•}^{2}(

*t*)) d

*t*.

The vector d**r**/d*t* is called the *tangent vector* at point *P*. The magnitude of the tangent vector is

| **r ^{•}** | = d

*s*/d

*t*=

*v*.

Hence the unit tangent vector becomes

**T** = **r ^{•}** / |

**r**| = (d

^{•}**r**/d

*t*) / (d

*s*/d

*t*) = d

**r**/d

*s*≡

**r¹**.

Here the prime **¹** denotes differentiation with respect to the arc length, *s*. We list some useful formulae of the derivatives of arc length *s* with respect to parameter *t* and vice versa:

*v = s*** ^{•}** = d

*s*/d

*t*= |

**r**| = (

^{•}**r**•

^{•}**r**)

^{•}^{1/2},

*a = s*** ^{••}** = d

*s*

**/d**

^{•}*t*= (

**r**•

^{•}**r**

^{•}**) / (**

^{•}**r**•

^{•}**r**)

^{•}^{1/2},

*s*^{•}^{•}** ^{•}** = d

*s*

^{•}**/d**

^{•}*t*= [(

**r**•

^{•}**r**)(

^{•}**r**•

^{•}**r**

^{•}

^{•}**+**

^{•}**r**

^{•}**•**

^{•}**r**

^{•}**) – (**

^{•}**r**•

^{•}**r**

^{•}**)²] / (**

^{•}**r**•

^{•}**r**)

^{•}^{3/2},

*u = t¹* = d*t*/d*s* = 1/| **r ^{•}** | = 1/(

**r**•

^{•}**r**)

^{•}^{1/2},

*b = t¹¹* = d*t¹*/d*s* = – (**r ^{•}** •

**r**

^{•}**) / (**

^{•}**r**•

^{•}**r**)

^{•}^{4/2},

*t¹¹¹* = d*t¹¹*/d*s* = – [(**r ^{•}** •

**r**)(

^{•}**r**•

^{•}**r**

^{•}

^{•}**+**

^{•}**r**

^{•}**•**

^{•}**r**

^{•}**) – 4(**

^{•}**r**•

^{•}**r**

^{•}**)²] / (**

^{•}**r**•

^{•}**r**)

^{•}^{7/2}.

If **r**(*s*) is an arc length parametrized curve, then **r¹**(*s*) is a unit vector, and hence **r¹** • **r¹** = 1. Differentiating this relation, we obtain **r¹** • **r¹¹** = 0, which states that **r¹¹** is orthogonal to the tangent vector, provided it is not a null vector.

The unit vector

**N** = **r¹¹**(*s*)/|**r¹¹**(*s*)| = **T¹**(*s*)/|**T¹**(*s*)|,

which has the direction and sense of **T¹**(*s*) is called the *unit principal normal vector* at *s*. The plane determined by the unit tangent and normal vectors **T**(*s*) and **N**(*s*) is called the *osculating plane* at *s*. The curvature is

*κ* ≡ 1/*ρ* = |**r¹¹**(*s*)|,

and its reciprocal *ρ* is called the *radius of curvature* at *s*. It follows that

**r¹¹** = **T¹** = *κ***N**.

The vector **k** = **r¹¹** = **T¹** is called the *curvature vector*, and measures the rate of change of the tangent along the curve. By definition *κ* is nonnegative, thus the sense of the normal vector is the same as that of **r¹¹**(*s*).

For a space curve, the curvature is

*κ* = |**r ^{•}** ×

**r**

^{•}**| / |**

^{•}**r**|³.

^{•}