Curves for space and time

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 t1tt2. 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); t1tt2.

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)).

Displacement Δr connecting points P and Q on parametric curve r(t).

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

ds = |dr/dt| dt = | r | dt = √(rr) dt.

Here the dot denotes differentiation with respect to the parameter t. Therefore the arc length of a segment of the curve between points r(t0) and r(t) can be obtained as follows:

s(t) = ∫ ds = ∫ √(rr) dt = ∫ √(x2(t) + y2(t) + z2(t)) dt.

The vector dr/dt is called the tangent vector at point P. The magnitude of the tangent vector is

| r | = ds/dt = v.

Hence the unit tangent vector becomes

T = r / | r | = (dr/dt) / (ds/dt) = dr/ds.

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 = ds/dt = | r | = (rr)1/2,

a = s•• = ds/dt = (rr) / (rr)1/2,

s = ds/dt = [(rr)(rr + rr) – (rr)²] / (rr)3/2,

u = t¹ = dt/ds = 1/| r | = 1/(r • r)1/2,

b = t¹¹ = d/ds = – (rr) / (rr)4/2,

t¹¹¹ = dt¹¹/ds = – [(rr)(rr + rr) – 4(rr)²] / (rr)7/2.

If r(s) is an arc length parametrized curve, then (s) is a unit vector, and hence = 1. Differentiating this relation, we obtain 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)| = (s)/|(s)|,

which has the direction and sense of (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 = = κN.

The vector k = r¹¹ = 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.