Helical motion is mathematically defined by an object's position vector across three dimensions, where the distance from the origin is calculated using the Pythagorean theorem and trigonometric identities. The velocity and acceleration are derived as time-dependent vectors, with acceleration linked directly to the horizontal position components. Physically, this path represents the superposition of uniform circular motion in the $x_1-x_2$ plane and constant velocity along the $x_3$ axis, resulting in a trajectory with a uniform radius and constant pitch. A key characteristic identified in the sources is the constancy of the object's speed, which ensures that the total distance travelled, or arc length, is directly proportional to the elapsed time. Computational models and animations further clarify these dynamics by using dynamic tracers and rotating perspectives to demonstrate the real-time progression and steady nature of the helical system.
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%% Proof and Derivation
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