There are no curved steps!!

A human cannot execute a genuinely curved step — where their center of mass travels in a continuous arc during a single step — without violating fundamental principles of balance.

💡 Analogy: A Curve is Just a Lot of Straight Lines

A box is a badly drawn circle.
A hexagon is a slightly less badly drawn circle.
A 400-sided polygon is basically a circle.

In dance, every step is a straight line, but if you chain enough small angular steps together, you get the illusion of a smooth curve.

Curved travel is not about bending your path — it’s about changing your direction slightly, over and over again.

Curvature in Dance Travel


### 🔍 Curvature in Dance Travel

A perfect curve can be thought of as the result of infinitely many tiny direction changes.

Mathematically, this is written as:

\(κ = \dfrac{dθ}{ds}\)

Where:
- κ is curvature (how sharply you're turning),
- θ is your heading (direction),
- s is the arc length (distance you're traveling).


rate of angular change per unit distance.
\(κ = \dfrac{dθ}{ds}\)


The smaller your angle change per step (Δθ), the smoother and easier the curve becomes.

If κ = 0, you're traveling in a straight line (Δθ = 0 between steps).

If κ > 0, your direction is changing — you're curving.

A box has huge Δθ (like 90°), so high 'curvature' spikes.

A 400-gon has tiny Δθ, mimicking a smooth \(\dfrac{dθ}{ds}\).


In other words:

You don’t bend the step. You bend the path.

The 'Tango Curving Basic'

Danced as a SSQQS we have 3 curving steps.

But the angle of each step is very large so the effect is 'a curve of 3 straight lines (i.e steps)'

  • The shorter the steps the harder it will be to give the appearance and effect of a curve. It feels horrible too.
  • The longer the steps the easier it will be to give the appearance and effect of a curve. Your partner will love how this feels

Note that a Step is 'Travel'

and that 'Travel' in this case is a vector comprising direction (the angle) and distance (the size of the step) so our Travel Vector model still applies.

3. Model Simplification

From a system modeling perspective (e.g., DanceBot or SBAS):

  • Representing each step as a straight-line vector (T⃗) allows us to calculate poise, torque, and transitions.
  • Curves are formed by chaining discrete straight segments

The Illusion of Curved Steps

While steps can appear curved due to:

  • Rotational movement of the upper body,
  • Sway or shaping,
  • Turning actions across multiple beats,

…the actual center of mass path per step remains linear. The curve is emergent, not atomic.

🚶‍♂️ Requirements for Curved Travel

Any curved travel in partner dancing requires:

  • CBM – Contra Body Movement is essential for initiating rotation and shaping the curve.
  • Sway – A natural side-tilt of the body to maintain balance over a curved path.
    Note: Tango is the exception — it uses no sway.
  • Rise and Fall – Especially in Waltz and Foxtrot, the use of elevation timing amplifies the curvature, often contributing to swing and the illusion of flight. Note: Tango is the exception — it uses no Rise and Fall.

These three elements — CBM, Sway, and Rise & Fall — are biomechanically linked.
You can't curve naturally without them working in sync.

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