Second fundamental form

Key takeaways

Second fundamental form belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Second fundamental form to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Second fundamental form from memory before moving on to harder problems.

Reference excerpt

In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by I I {\displaystyle \mathrm {I\!I} } (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

Surface in R3

Motivation The second fundamental form of a parametric surface S in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:

z = L x 2 2 + M x y + N y 2 2 + higher order terms , {\displaystyle z=L{\frac {x^{2}}{2}}+Mxy+N{\frac {y^{2}}{2}}+{\text{higher order terms}}\,,}

and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form

L d x 2 + 2 M d x d y + N d y 2 . {\displaystyle L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.}

For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 is tangent to S at P, and define the second fundamental form in the same way.

Classical notation The second fundamental form of a general parametric surface is defined as follows. Let r = r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u and v by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product ru × rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

n = r u × r v | r u × r v | . {\displaystyle \mathbf {n} ={\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}\,.}

The second fundamental form is usually written as

I I = L d u 2 + 2 M d u d v + N d v 2 ; {\displaystyle \mathrm {I\!I} =L\,du^{2}+2M\,du\,dv+N\,dv^{2}\,;}

its matrix in the basis {ru, rv} of the tangent plane is

[ L M M N ] . {\displaystyle {\begin{bmatrix}L&M\\M&N\end{bmatrix}}\,.}

The coefficients L, M, N at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:

L = r u u ⋅ n , M = r u v ⋅ n , N = r v v ⋅ n . {\displaystyle L=\mathbf {r} _{uu}\cdot \mathbf {n} \,,\quad M=\mathbf {r} _{uv}\cdot \mathbf {n} \,,\quad N=\mathbf {r} _{vv}\cdot \mathbf {n} \,.}

For a signed distance field of Hessian H, the second fundamental form coefficients can be computed as follows:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Second fundamental form

Start with the simplest possible case. Write down what Second fundamental form claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Second fundamental form before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Second fundamental form ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Second fundamental form

In research: Second fundamental form appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Second fundamental form in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Second fundamental form is common in secondary-school and first-year university syllabi. It links to neighbouring topics Curvature (mathematics), Differential geometry, Differential geometry of surfaces, so understanding it makes those chapters shorter.
In everyday life: Look for Second fundamental form outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Second fundamental form in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Second fundamental form means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Second fundamental form out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Second fundamental form in simple terms?

In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by I I {\displaystyle \mathrm {I\!I} } (read "two"). Together with the first fundamental form, it serves to…

Why does Second fundamental form matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Second fundamental form?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Second fundamental form.