pancake.lead Namespace Reference

Classes
class	Segment
class	Lead

Functions
	smoothstep (float x)
Segment	Straight (float length)
Segment	Twist (float angle_deg, float length, float ramp=0.0)
Segment	Bend (float angle_deg, float radius, float ramp=0.0, bool hard=False)
Segment	Release (float length, float kappa_n=0.0, float kappa_g=0.0, float tau=0.0)
np.ndarray	_rhs (np.ndarray y, np.ndarray omega)
np.ndarray	_orthonormalize (np.ndarray y)
np.ndarray	_rk4 (np.ndarray y, Segment seg, float x0, float h)
np.ndarray	_exact (np.ndarray y, np.ndarray omega, float h)

Detailed Description

Lead (terminal) sections defined intrinsically by their strip curvatures.

A lead is a chain of segments.  Each segment prescribes the three curvature
parameters of the strip along its arc length x in [0, L]

    omega(x) = ( tau(x), kappa_g(x), kappa_n(x) )

and the space curve together with its frame follows by integrating the
generalized Frenet-Serret equations for strips (Russenschuck Eq. (19.9))

    r' = T,  T' = kappa_n n - kappa_g b,  n' = -kappa_n T + tau b,  b' = kappa_g T - tau n

from the state at the end of the winding.  Because the frame is *integrated*
rather than derived from v x a, it is defined and continuous on straight
sections, through inflection points and across the transition into the bend.
The derivatives v, a, b of the position are then exact expressions in the
frame and the curvature functions:

    v = T
    a = T' = kappa_n n - kappa_g b
    b = T'' = kappa_n' n + kappa_n n' - kappa_g' b - kappa_g b'

Sign conventions (positive amplitudes):
    kappa_n  turns the tangent towards +n   ("easy way" bend about b)
    kappa_g  turns the tangent towards -b   ("hard way" bend about n)
    tau      rotates n towards +b           (right-handed twist about T)

Each amplitude is multiplied by a profile P(x):

    shape = "plateau": rises smoothly from 0 to 1 over the first `ramp` length
                       units and falls back to 0 over the last `ramp`
                       (ramp = 0: constant curvature -> circular arc / helix / plain twist)
    shape = "fade":    1 -> 0 over the whole segment (used to release the winding curvature)
    shape = "rise":    0 -> 1 over the whole segment

The quintic smoothstep makes the curvatures C^2, so frame, curvature and jerk
are continuous everywhere.  On intervals with constant curvature the frame and
position are advanced with the closed-form matrix exponential (exact to
round-off); on ramps a fourth-order Runge-Kutta scheme with a small step is
used.

Function Documentation

_exact()

np.ndarray _exact ( np.ndarray y, np.ndarray omega, float h )

protected

Closed-form step for constant body-frame Darboux vector omega = (tau, kappa_g, kappa_n).

F(h) = F0 exp(h K),  r(h) = r0 + F0 ( int_0^h exp(x K) dx ) e_1,  K = [omega]_x,
with exp(x K) = I + sin(w x) K^ + (1 - cos(w x)) K^^2  (Rodrigues), w = |omega|.

_orthonormalize()

np.ndarray _orthonormalize ( np.ndarray y )

protected

_rhs()

np.ndarray _rhs ( np.ndarray y, np.ndarray omega )

protected

_rk4()

np.ndarray _rk4 ( np.ndarray y, Segment seg, float x0, float h )

protected

Bend()

Segment Bend	(	float	angle_deg,
		float	radius,
		float	ramp = 0.0,
		bool	hard = False )

Bend the tape by angle_deg with the given bending radius.

hard=False: easy-way bend about b (kappa_n), the tangent turns towards +n for
positive angles.  hard=True: hard-way bend about n (kappa_g), the tangent turns
towards -b for positive angles.  A ramp > 0 blends the curvature in and out
(clothoid-like), which lengthens the segment by `ramp`.

Release()

Segment Release	(	float	length,
		float	kappa_n = 0.0,
		float	kappa_g = 0.0,
		float	tau = 0.0 )

Fade the given curvatures smoothly to zero over `length` (tape leaving the winding).

smoothstep()

smoothstep

(

float

x

)

Quintic smoothstep S(x) = 6x^5 - 15x^4 + 10x^3 on [0, 1] and its derivative.

Straight()

Segment Straight

(

float

length

)

Twist()

Segment Twist	(	float	angle_deg,
		float	length,
		float	ramp = 0.0 )

Twist the tape by angle_deg about its tangent over the given length (base curve straight).