Watching a screw motion¶

from screws import viz

viz.animate_screw(viz.door())          # the door swung about its hinge, with T(θ) = e^[S]θ T(0) beside it
viz.animate_twist(viz.screwdriver())   # the screwdriver under a constant twist: a helix about -z
viz.compare_frames(viz.door())         # V_s beside V_b, and V_s = [Ad_Tsb] V_b underneath
viz.animate_screw(viz.screw_between(T_start, T_end))   # the one screw motion from one frame to another
viz.animate_twist([0, 0, 1, 0, -0.5, 0.1], duration=4).save("mine.gif")   # any twist, any body

A twist is a motion before it is a six-vector. animate_twist plays a constant twist on a body: the screw axis drawn as a line with ŝ and the pitch h, one point's path traced (a helix, or a circle at zero pitch), and purple arrows for the velocities ω × p + v of a few points. animate_screw sweeps a screw axis over θ and prints the line read off it, q = ω × v and h = ωᵀv, with T(θ) changing alongside.

Three pitches. The scenes viz.door(), viz.drawer() and viz.screwdriver() are the three motions of the notes' chapter 3 review: zero, infinite and finite pitch.

The door of notes 3.10 turned about its hinge, with its screw axis, the handle's circle and T(θ)

A drawer sliding out of a cabinet: no axis, every point with the same velocity

A screwdriver driving down: a helix about -z, pitch exaggerated

Space and body twists. compare_frames draws one motion twice. On the left, v_s is the velocity of the body point at the {s} origin: the grey marker is the point that was there at θ = 0, carried off by the motion while the point now at the origin keeps the same velocity. On the right, v_b is the velocity of the point at {b}. Under them, V_s = [Ad_Tsb(θ)] V_b, the adjoint changing and V_s fixed.

$The door's space twist and body twist side by side: v_s at the {s} origin, v_b at the handle$

animate_twist plays the same door in time, V_s = S θ̇:

The door under a constant twist, with the velocities of three of its points

From one frame to another. Every rigid-body displacement is a screw motion (Chasles). viz.screw_between(T_start, T_end) finds the one that carries T_start to T_end, from [S]θ = log(T_end T_start⁻¹) with S a unit screw axis, and animate_screw shows the frame travelling along it, its origin tracing the helix, the start and end frames faded:

A frame carried to another frame with a different origin and orientation by one screw motion

Each call returns an Animation: .save("x.gif") (or .mp4 with screws[video]), it plays inline in a notebook, and .data holds the numbers drawn. viz.Scene, viz.box and viz.prism build your own body. The geometry is screws.screw_line, screw_motion, point_velocity and helix, numpy only. examples/screw_motions.py writes every GIF here.