Inertial Resistance on Circular Arc

🧠 Geometric Setup

We model a dancer turning along a circular arc with a radius R centered at the origin (0, 0). The dancer's path and movement are defined as follows:

Arc Position at Angle θ:

arc_point(θ) = (R * cos(θ), R * sin(θ))

Velocity Vector (Tangent to Arc):

velocity_vector(θ) = d/dθ [ arc_point(θ) ]

                   = (-R * sin(θ), R * cos(θ))

Normalized Velocity Direction:

unit_velocity(θ) = velocity_vector(θ) / ||velocity_vector(θ)||

                 = (-sin(θ), cos(θ))   # Since magnitude is R

🚀 Inertial Resistance

We assume the initial motion was in the +z direction, i.e., vector:

initial_direction = (0, 1)

Resistance Vector (difference between original and new velocity):

resistance_vector(θ) = unit_velocity(θ) - initial_direction

                     = (-sin(θ), cos(θ) - 1)

Magnitude of Resistance:

||resistance|| = sqrt( sin²(θ) + (cos(θ) - 1)² )

               = sqrt( sin²(θ) + 1 - 2cos(θ) + cos²(θ) )

               = sqrt(2 - 2cos(θ))

               = 2 * sin(θ / 2)

🎯 Inflection Point Detection

The inflection point is where the direction of travel diverges from the original by 45 degrees (π/4 radians).

Angle Between Tangent and Original Vector:

dot_product = unit_velocity(θ) • (0, 1) = cos(θ)

deviation_angle = acos(cos(θ)) = θ

Inflection is reached when:

θ > π / 4 ≈ 0.7854 radians (≈ 45°)

This implies that the inertial force becomes significant when θ exceeds 45°, and the resistance magnitude is:

2 * sin(π / 8) ≈ 0.765

✅ Summary

• Resistance increases as the deviation from the original direction increases

• *Magnitude is governed by `2 sin(θ/2)`**

• Inflection begins at 45°