Curves for space and time, continued

The following is a continuation and revision of the previous post, here.

Based on 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 pdf version in parallel is here.

Let a three-dimensional curve be expressed in parametric form as x = x(t); y = y(t); z = z(t); where the coordinates of the point (x, y, z) of the curve are expressed as functions of a parameter t (time) within a closed interval t₁ ≤ t ≤ t₂. The functions x(t), y(t), and z(t) are assumed to be continuous with a sufficient number of continuous derivatives.

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)).

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

ds = |dr/dt| dt = | r^t | dt = √(r^t • r^t) dt,

where the superscript t denotes differentiation with respect to the arc time parameter t. The vector r^t = dr/dt is called the tangent vector at point A.

Then the arc length, s, of a segment of the curve between points r(t₀) and r(t) can be obtained as follows:

s(t) = ∫ ds = ∫ √(r^t • r^t) dt = ∫ √((dx/dt)² + (dy/dt)² + (dz/dt)²) dt.

The magnitude of the tangent vector is

| r^t | = ds/dt = v.

Hence the unit tangent vector is

T_s = r^t / | r^t | = (dr/dt) / (ds/dt) = dr/ds ≡ r^s,

where the superscript s denotes differentiation with respect to the arc length parameter, s.

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

N_s = r^ss(s) / |r^ss(s)| = T_s^s(s)/|T_s^s(s)|,

which has the direction and sense of r^ss(s) is called the unit principal normal vector at s. The plane determined by the unit tangent and normal vectors T_s(s) and N_s(s) is called the osculating plane at s. The curvature is

κ_s ≡ 1/ρ = |r^ss(s)|,

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

r^ss = T_s^s = κ_s N_s.

The vector k_s = r^ss = T_s^s is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κ_s is nonnegative, thus the sense of the normal vector is the same as that of r^ss(s). For a three-dimensional curve, the curvature is

κ_s = |r^t × r^tt| / | r^t |³.

Let a three-dimensional curve be expressed in parametric form as X = X(s); Y = Y(s); Z = Z(s); where the coordinates of the point (X, Y, Z) of the curve are expressed as functions of a parameter s (length) within a closed interval s₁ ≤ s ≤ s₂. The functions X(s), Y(s), and Z(s) are assumed to be continuous with a sufficient number of continuous derivatives.

In vector notation the parametric curve can be specified by a vector-valued function w = w(s), where w represents the position vector (i.e., w(s) = (X(s), Y(s), Z(s)).

Consider a segment (dischronment) of a parametric curve w = w(s) between two points C(w(s)) and D(w(s+Δs)) as shown in the figure above. As point D approaches C or in other words Δs → 0, the length t becomes the differential arc time of the curve:

dt = |dw/ds| ds = | w^s | ds = √(w^s • w^s) ds,

where w^s = dw/ds, which is called the tangent vector at point C. Then the arc time, t, of a segment of the curve between points w(s₀) and w(s) can be obtained as follows:

t(s) = ∫ dt = ∫ √(w^s • w^s) ds = ∫ √((dX/ds)² + (dY/ds)² + (dZ/ds)²) ds.

The vector w^s = dw/ds is called the tangent vector at point C. The magnitude of the tangent vector is

| w^s | = dt/ds = u.

Hence the unit tangent vector is

T_t = w^s / | w^s | = (dw/ds) / (dt/ds) = dw/dt ≡ w^t.

If w(t) is an arc length parametrized curve, then w^t(t) is a unit vector, and hence w^t • w^t = 1. Differentiating this relation, we obtain w^t • w^tt = T_t • T_t^t = 0, which states that w^tt is orthogonal to the tangent vector, provided it is not a null vector. The unit vector

N_t = w^tt(t) / |w^tt(t)| = T_t^t(t)/|T_t^t(t)|,

which has the direction and sense of w^tt(t) is called the unit principal normal vector at t. The plane determined by the unit tangent and normal vectors T_t(t) and N_t(t) is called the osculating plane at t. The curvature is

κ_t ≡ 1/ρ = |w^tt(t)| = |T_t^t(t)|,

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

w^tt = T_t^t = κ_t N_t.

The vector k_t = w^tt = T_t^t 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 w^tt(t). For a three-dimensional curve, the curvature is

κ_t = |w^s × w^ss| / | w^s |³.

Here are some useful formulae of the derivatives of arc length, s, and the arc time, t:

v = s^t = ds/dt = | r^t | = (r^t • r^t)^1/2 = 1/| w^s | = 1/(w^s • w^s)^1/2,

a = s^tt = ds^t/dt = (r^t • r^tt) / (r^t • r^t)^1/2 = – (w^s • w^ss) / (w^s • w^s)^4/2,

s^ttt = ds^tt/dt = [(r^t • r^t)(r^t • r^ttt + r^tt • r^tt) – (r^t • r^tt)²] / (r^t • r^t)^3/2

= – [(w^s • w^s)(w^s • w^sss + w^ss • w^ss) – 4(w^s • w^ss)²] / (w^s • w^s)^7/2,

u = t^s = dt/ds = 1/| r^t | = 1/(r^t • r^t)^1/2 = | w^s | = (w^s • w^s)^1/2,

b = t^ss = dt^s/ds = – (r^t • r^tt) / (r^t • r^t)^4/2 = (w^s • w^ss) / (w^s • w^s)^1/2,

t^sss = dt^ss/ds = – [(r^t • r^t)(r^t • r^ttt + r^tt • r^tt) – 4(r^t • r^tt)²] / (r^t • r^t)^7/2

= [(w^s • w^s)(w^s • w^sss + w^ss • w^ss) – (w^s • w^ss)²] / (w^s • w^s)^3/2.