The intrinsic equation, expressing the relation between the arc σ (measured from O) and the inclination φ of the tangent at any points to the axis of x, assumes a very simple form. For
dx = cos ½πv²·dv, dy = sin ½πv²·dv;
so that
σ = ∫ √(dx² + dy²) = v, (30),
φ = tan−1(dy/dx) = ½πv² (31).
Accordingly,
φ = ½πσ² (32);
and for the curvature,
dφ / dσ = πσ (33).