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Asymptotic Curve — Definition, Formula & Examples

An asymptotic curve is a curve on a surface along which the normal curvature is zero. At every point of the curve, the surface bends away from its tangent plane only in directions other than the curve's tangent direction.

A curve γ(t)\gamma(t) on a smooth surface SS is called an asymptotic curve if the second fundamental form vanishes along it, i.e., II(γ′(t),γ′(t))=0\mathrm{II}(\gamma'(t), \gamma'(t)) = 0 for all tt. Equivalently, the normal curvature κn\kappa_n in the direction of γ′(t)\gamma'(t) is zero at every point of the curve.

Key Formula

e du2+2f du dv+g dv2=0e\,du^2 + 2f\,du\,dv + g\,dv^2 = 0
Where:
  • e,f,ge, f, g = Coefficients of the second fundamental form of the surface
  • du,dvdu, dv = Differentials of the surface parameters along the curve

How It Works

To find asymptotic curves on a surface, you solve the asymptotic direction equation using the coefficients of the second fundamental form. Given the second fundamental form e du2+2f du dv+g dv2=0e\,du^2 + 2f\,du\,dv + g\,dv^2 = 0, the asymptotic directions at each point are the solutions to this equation. Integrating these direction fields over the surface yields the asymptotic curves. Asymptotic curves exist only where the Gaussian curvature K≤0K \leq 0; on surfaces with K>0K > 0 (like a sphere), there are no real asymptotic directions.

Worked Example

Problem: Find the asymptotic curves on the saddle surface z=xyz = xy.
Step 1: Parametrize the surface as r(u,v)=(u,v,uv)\mathbf{r}(u,v) = (u, v, uv). Compute the second fundamental form coefficients: e=ruu⋅n=0e = \mathbf{r}_{uu} \cdot \mathbf{n} = 0, g=rvv⋅n=0g = \mathbf{r}_{vv} \cdot \mathbf{n} = 0, and f=ruv⋅n=11+u2+v2f = \mathbf{r}_{uv} \cdot \mathbf{n} = \frac{1}{\sqrt{1+u^2+v^2}}.
e=0,g=0,f=11+u2+v2e = 0,\quad g = 0,\quad f = \frac{1}{\sqrt{1+u^2+v^2}}
Step 2: Substitute into the asymptotic equation e du2+2f du dv+g dv2=0e\,du^2 + 2f\,du\,dv + g\,dv^2 = 0. Since e=g=0e = g = 0, this simplifies to 2f du dv=02f\,du\,dv = 0.
2f du dv=02f\,du\,dv = 0
Step 3: Since f≠0f \neq 0, either du=0du = 0 or dv=0dv = 0. This gives two families: u=c1u = c_1 (constant) and v=c2v = c_2 (constant).
u=c1orv=c2u = c_1 \quad \text{or} \quad v = c_2
Answer: The asymptotic curves on z=xyz = xy are the coordinate lines u=constu = \text{const} and v=constv = \text{const}, i.e., lines parallel to the xx-axis and lines parallel to the yy-axis.

Why It Matters

Asymptotic curves reveal where a surface has zero normal curvature, which is essential in structural engineering for designing thin shells and in architecture for ruling surfaces. They also appear in the study of ruled surfaces and minimal surfaces in differential geometry courses.

Common Mistakes

Mistake: Looking for asymptotic curves on surfaces with positive Gaussian curvature (e.g., a sphere).
Correction: Asymptotic directions exist only where K≤0K \leq 0. On elliptic points (K>0K > 0), the second fundamental form is definite and has no real null directions.