The first fundamental form

Theorem

The first fundamental form

T-106Home MU-306Threads space

Statement

Let \( U \subseteq \mathbb{R}^2 \) be open and let \( \sigma : U \to \mathbb{R}^3 \) be a regular parametrised surface patch, meaning \( \sigma \) is smooth and the two partial derivatives \( \sigma_u = \partial \sigma/\partial u \) and \( \sigma_v = \partial \sigma/\partial v \) are linearly independent at every point of \( U \), so that \( T_p S = \operatorname{span}\{\sigma_u, \sigma_v\} \) is a genuine \(2\)-dimensional tangent plane at each point \( p = \sigma(u,v) \) of the surface \( S = \sigma(U) \). Define \[ E(u,v) = \sigma_u \cdot \sigma_u, \qquad F(u,v) = \sigma_u \cdot \sigma_v, \qquad G(u,v) = \sigma_v \cdot \sigma_v, \] the dot products taken in \( \mathbb{R}^3 \). Then the map \( \mathrm{I}_p : T_pS \times T_pS \to \mathbb{R} \) defined by restricting the Euclidean inner product of \( \mathbb{R}^3 \) to \( T_pS \) is a symmetric positive-definite bilinear form, called the first fundamental form, whose matrix in the basis \( \{\sigma_u,\sigma_v\} \) is \[ \begin{pmatrix} E & F \\ F & G \end{pmatrix}, \] equivalently written as the quadratic differential form \( \mathrm{I} = E\,du^2 + 2F\,du\,dv + G\,dv^2 \); it satisfies \( EG - F^2 \gt 0 \) everywhere on \( U \), and it determines the length of every tangent vector, the arc length of every curve on \( S \), the angle between any two intersecting curves on \( S \), and the area of any region of \( S \), using only data intrinsic to the surface (no reference to how \( S \) sits in the ambient space is needed once \( E, F, G \) are known).

Why it matters

A surface embedded in \( \mathbb{R}^3 \) inherits a Euclidean ruler and protractor from the ambient space, but that ruler is only ever used on tangent vectors, and only through the dot product. The first fundamental form packages exactly that data — the restriction of the ambient inner product to each tangent plane — into three functions \( E, F, G \) of the parameters. This is the single most important reduction in classical differential geometry: it converts a metric question about a curved 2-dimensional object into computable algebra on an open subset of \( \mathbb{R}^2 \).

It also marks the birth of intrinsic geometry. Gauss's Theorema Egregium (a later theorem in this unit) shows that the Gaussian curvature is computable from \( E, F, G \) and their derivatives alone, without reference to the embedding — the first step toward abstract Riemannian metrics, where \( E\,du^2+2F\,du\,dv+G\,dv^2 \) becomes the prototype for a metric tensor on a manifold with no ambient space at all.

Hypotheses

\( \sigma \) is smooth (at least \( C^1 \), and \( C^2 \) is assumed later for curvature computations). Without differentiability, \( \sigma_u, \sigma_v \) need not exist at every point, so \( E, F, G \) are undefined there; e.g. \( \sigma(u,v) = (u,v,|u|) \) has no well-defined tangent plane, hence no first fundamental form, along \( u = 0 \).

Regularity: \( \sigma_u \times \sigma_v \neq 0 \) at every point of \( U \) (equivalently \( \sigma_u, \sigma_v \) linearly independent, equivalently \( EG - F^2 \gt 0 \)). Drop this and the "tangent plane" collapses to a line or a point, so \( \mathrm{I} \) degenerates to a rank-deficient form. Take \( \sigma(u,v) = (u^3, v, 0) \): at \( u=0 \), \( \sigma_u = (0,0,0) \), so \( E = 0 \) there and the matrix \( \begin{pmatrix} E&F\\F&G\end{pmatrix} \) is singular — lengths of \( u \)-direction tangent vectors are reported as zero even though the curve \( v \mapsto \sigma(0,v) \) is a genuine point but the parametrisation is singular, a classic cuspidal-edge failure.

\( U \) is open in \( \mathbb{R}^2 \) and \( \sigma \) is injective with continuous inverse on its image (so \( \sigma \) is a genuine local parametrisation, not merely an immersion). Without injectivity the "surface" \( S = \sigma(U) \) may self-intersect, and \( \mathrm{I} \) is still perfectly well defined pointwise on \( U \) (it only ever sees the map \( \sigma \), not the set \( S \)) — but statements like "the angle between two curves on \( S \) at a point of \( S \)" become ambiguous, since a point of \( S \) may correspond to several \( (u,v) \) with different tangent planes. This is why \( \mathrm{I} \) is best regarded as a form on the parameter domain pulled back through \( \sigma \), not as intrinsically living on the point-set \( S \).

Proof

T_pS = \operatorname{span}\{\sigma_u(u_0,v_0), \sigma_v(u_0,v_0)\} \text{ is a 2-dimensional subspace of } \mathbb{R}^3.

By the regularity hypothesis \( \sigma_u \times \sigma_v \neq 0 \), so \( \sigma_u, \sigma_v \) are linearly independent; they span a plane by definition of linear independence in a vector space (here \( \mathbb{R}^3 \)). A

\mathrm{I}_p(w_1,w_2) := w_1 \cdot w_2 \text{ for } w_1,w_2 \in T_pS \text{ is a well-defined symmetric bilinear form.}

The Euclidean dot product on \( \mathbb{R}^3 \) is itself symmetric and bilinear; restricting a symmetric bilinear form to a linear subspace \( T_pS \subseteq \mathbb{R}^3 \) yields a symmetric bilinear form on that subspace, since bilinearity and symmetry are properties of the restriction of the map, inherited automatically. A

\text{Every } w \in T_pS \text{ can be written uniquely as } w = a\,\sigma_u + b\,\sigma_v, \quad a,b \in \mathbb{R}.

\( \{\sigma_u,\sigma_v\} \) is a basis of \( T_pS \) by Step 1, and coordinates with respect to a basis are unique — the defining property of a basis. A

\mathrm{I}_p(w,w) = a^2 E + 2ab F + b^2 G \quad \text{where } w = a\sigma_u+b\sigma_v.

Expand \( w \cdot w = (a\sigma_u+b\sigma_v)\cdot(a\sigma_u+b\sigma_v) \) by bilinearity of the dot product and collect terms, using \( \sigma_u\cdot\sigma_u = E \), \( \sigma_u\cdot\sigma_v = \sigma_v\cdot\sigma_u = F \), \( \sigma_v\cdot\sigma_v = G \) from Step 2's symmetry. Writing \( du, dv \) for the components \( a, b \) of an infinitesimal displacement gives the classical notation \( \mathrm{I} = E\,du^2+2F\,du\,dv+G\,dv^2 \). B

EG - F^2 = |\sigma_u|^2|\sigma_v|^2 - (\sigma_u\cdot\sigma_v)^2 = |\sigma_u \times \sigma_v|^2 \gt 0.

Lagrange's identity for vectors in \( \mathbb{R}^3 \): \( |a\times b|^2 = |a|^2|b|^2 - (a\cdot b)^2 \), proved directly by expanding both sides in coordinates (or via \( |a\times b| = |a||b|\sin\theta \) and \( a\cdot b = |a||b|\cos\theta \), and \( \sin^2\theta+\cos^2\theta=1 \)). Strict positivity of the right side is exactly the regularity hypothesis \( \sigma_u\times\sigma_v \neq 0 \). C

\mathrm{I}_p \text{ is positive definite: } \mathrm{I}_p(w,w) \gt 0 \text{ for all } w \in T_pS \setminus\{0\}.

\( \mathrm{I}_p(w,w) = |w|^2 \) since \( \mathrm{I}_p \) is literally the restricted dot product (Step 2), and the Euclidean norm is positive definite on \( \mathbb{R}^3 \): \( |w|^2 = 0 \iff w = 0 \) by the axioms of an inner product (definiteness of the standard inner product on \( \mathbb{R}^3 \)). Equivalently, one can verify positive definiteness of the quadratic form \( aE+2abF+b^2G \) directly from \( EG-F^2\gt0 \) and \( E = |\sigma_u|^2 \gt 0 \) via the Sylvester criterion for symmetric \( 2\times2 \) matrices. A

\text{Arc length: for a curve } \gamma(t) = \sigma(u(t),v(t)), \ t\in[a,b], \quad L(\gamma) = \int_a^b \sqrt{E\dot u^2 + 2F\dot u\dot v + G\dot v^2}\ dt.

By the chain rule \( \dot\gamma = \sigma_u\dot u + \sigma_v \dot v \), a tangent vector at each \( t \). Arc length of a curve in \( \mathbb{R}^3 \) is defined as \( \int_a^b |\dot\gamma(t)|\,dt \) (standard definition from vector calculus), and \( |\dot\gamma|^2 = \dot\gamma\cdot\dot\gamma = \mathrm{I}_{\gamma(t)}(\dot\gamma,\dot\gamma) \) by Step 2 applied with \( a=\dot u, b=\dot v \) in Step 4's formula. B

\text{Angle: if two curves through } p \text{ have tangent vectors } w_1 = \sigma_u+k_1\sigma_v,\ w_2 = \sigma_u+k_2\sigma_v \text{ (or any } a\sigma_u+b\sigma_v \text{), then} \quad \cos\theta = \frac{\mathrm{I}_p(w_1,w_2)}{\sqrt{\mathrm{I}_p(w_1,w_1)}\sqrt{\mathrm{I}_p(w_2,w_2)}}.

This is precisely the Euclidean formula \( \cos\theta = \dfrac{w_1\cdot w_2}{|w_1||w_2|} \) for the angle between two vectors in \( \mathbb{R}^3 \) (Cauchy–Schwarz guarantees the right side lies in \( [-1,1] \), so \( \theta \) exists), rewritten using \( \mathrm{I}_p(w_1,w_2)=w_1\cdot w_2 \) from Step 2 and \( \mathrm{I}_p(w_i,w_i)=|w_i|^2 \) from Step 6. In coordinates \( (a_1,b_1),(a_2,b_2) \), the numerator is \( Ea_1a_2+F(a_1b_2+a_2b_1)+Gb_1b_2 \). In particular the parameter curves \( u\mapsto\sigma(u,v_0) \) and \( v\mapsto\sigma(u_0,v) \) meet orthogonally at every point of \( U \) iff \( F\equiv0 \). B

\text{Area: for a region } R \subseteq U, \quad \operatorname{Area}(\sigma(R)) = \iint_R \sqrt{EG-F^2}\ du\,dv.

The area of a parametrised surface patch over \( R \) is defined (from multivariable calculus) as \( \iint_R |\sigma_u\times\sigma_v|\,du\,dv \), approximating \( \sigma(R) \) by the parallelogram spanned by \( \sigma_u\,du,\sigma_v\,dv \) at each point and summing infinitesimal areas \( |\sigma_u\times\sigma_v|\,du\,dv \). By Step 5, \( |\sigma_u\times\sigma_v| = \sqrt{EG-F^2} \), so the area integrand is intrinsic, i.e. expressible through \( E,F,G \) alone. C

Result

\mathrm{I} = E\,du^2 + 2F\,du\,dv + G\,dv^2, \qquad E=\sigma_u\cdot\sigma_u,\ F=\sigma_u\cdot\sigma_v,\ G=\sigma_v\cdot\sigma_v, \qquad EG-F^2\gt0

Reading. The first fundamental form is the dot product of \( \mathbb{R}^3 \), read off in the surface's own \( (u,v) \) coordinates. It is a "ruler and protractor" carried by the surface itself: once \( E, F, G \) are known as functions of \( u,v \), one can compute the length of any tangent vector or curve, the angle between any two curves, and the area of any region, without ever looking back at the embedding map \( \sigma \) directly (only through \( E, F, G \)).

Scope. Applies pointwise to any regular \( C^1 \) parametrised surface patch \( \sigma: U \to \mathbb{R}^3 \); more generally the same construction (a smoothly varying, symmetric, positive-definite bilinear form on tangent spaces) defines a Riemannian metric on any smooth manifold, of which this is the motivating special case for embedded surfaces in \( \mathbb{R}^3 \). It does not by itself determine the embedding (see Corollaries): two very differently bent surfaces can share the same \( E,F,G \).

Corollaries & converses

Intrinsic invariance. Any quantity expressible purely in terms of \( E, F, G \) and their \( u,v \)-derivatives (lengths, angles, areas, and — by the Theorema Egregium — Gaussian curvature) is unchanged by isometric bendings of the surface that do not stretch it, even though such quantities as the ambient shape or the second fundamental form can change drastically.
Isometries preserve \( \mathrm{I} \). If \( \phi: S_1 \to S_2 \) is a local isometry (preserves lengths of all curves) between surfaces with the same parameter domain in matching coordinates, then \( E_1=E_2, F_1=F_2, G_1=G_2 \); this follows from Step 7, since matching arc lengths for all curves forces the integrands, hence \( E,F,G \), to agree.
Converse holds locally. Conversely, if two patches \( \sigma_1,\sigma_2:U\to\mathbb{R}^3 \) have identical \( E,F,G \) on \( U \), then the map \( \sigma_2\circ\sigma_1^{-1} \) is a local isometry of the two surfaces — a standard consequence of Step 7 (equal \( E,F,G \) forces every curve to have the same length under the correspondence \( \sigma_1(u,v)\leftrightarrow\sigma_2(u,v) \)).
The form does not determine the surface up to rigid motion. A flat sheet of paper and the same sheet rolled into a cylinder (without stretching) have identical \( E,F,G\) in suitable coordinates, yet are not related by any rigid motion of \( \mathbb{R}^3 \) — this is the precise sense in which \( \mathrm{I} \) records only intrinsic, not extrinsic, information; distinguishing them needs the second fundamental form.

Fails without

Drop regularity \( (EG-F^2\gt0) \): take \( \sigma(u,v)=(u,0,0) \) on \( U=\mathbb{R}^2 \) (independent of \( v \), a degenerate "surface" collapsing to a line). Then \( G = \sigma_v\cdot\sigma_v = 0 \), so \( \mathrm{I} = du^2 \) is only positive semi-definite: the "curve" \( u\equiv0,\ v(t)=t \) has tangent vector \( \sigma_v=0 \) but nonzero parameter speed, so the notion of angle in Step 8 divides by \( \sqrt{\mathrm{I}(w_2,w_2)}=0\) and is undefined, and area in Step 9 is identically zero for every region — the form can no longer separate distinct directions.
Drop smoothness (only continuous \( \sigma \)): a space-filling curve type construction, or more simply \( \sigma(u,v) = (u,v,f(u,v)) \) with \( f \) continuous but nowhere differentiable (e.g. a Weierstrass-type function extended to two variables), has no tangent plane anywhere, so \( \sigma_u,\sigma_v \) fail to exist and \( E,F,G \) cannot even be written down — the entire construction of \( \mathrm{I} \) requires differentiability as its starting point.
Drop injectivity/local homeomorphism: the standard immersed figure-eight cylinder or a self-intersecting patch (e.g. \( \sigma(u,v) = (\sin 2u, \sin u, v) \) on a suitable domain) still has a well-defined \( \mathrm{I} \) pointwise on \( U \), but at a point of \( S \) where two sheets cross, "the angle between two curves on \( S \) at that point" is ambiguous, since it depends on which preimage in \( U \) is meant — showing \( \mathrm{I} \) is honestly a form pulled back to the parameter domain, not a form living on the point-set image.

Common errors

Writing \( F = \sigma_u \cdot \sigma_v \) but then forgetting the factor of \( 2 \) in the cross term of \( \mathrm{I} = Edu^2+2Fdudv+Gdv^2 \) when computing arc length or angle, effectively halving the mixed contribution.
Confusing the first fundamental form (intrinsic, built from \( \sigma_u,\sigma_v \) only) with the second fundamental form (extrinsic, built from the normal derivative \( \sigma_{uu}\cdot n \) etc.); students often try to read curvature information directly off \( E,F,G\) without invoking the Theorema Egregium machinery.
Assuming \( F = 0 \) automatically for "nice" parametrisations; orthogonality of parameter curves is a special property (Step 8) that must be checked, not assumed — polar-type parametrisations of a sphere have \( F=0\) but a generic parametrisation need not.
Using the area formula \( \iint\sqrt{EG-F^2}\,du\,dv \) over the wrong domain, e.g. integrating over the image \( \sigma(R) \) instead of the parameter region \( R \subset U \); the integral must always be taken in \( (u,v) \)-coordinates.
Treating \( E,F,G\) as constants of the surface rather than functions of \( (u,v) \), leading to nonsense when substituting a specific point's values into a formula meant to hold at every point.

Discussion

The first fundamental form is where "intrinsic" geometry is born as a rigorous idea. Gauss's 1827 Disquisitiones Generales circa Superficies Curvas introduced \( E,F,G \) essentially in this notation and asked which properties of a surface depend only on them — a question answered spectacularly by the Theorema Egregium, which shows Gaussian curvature is one such property. That a bending which preserves \( E,F,G \) (an isometry) automatically preserves curvature, even though curvature is defined via how the surface bends in \( \mathbb{R}^3 \), was Gauss's own "remarkable theorem," and it depends on nothing more than the algebra developed in this proof.

Conceptually, \( \mathrm{I} \) is the restriction of an ambient inner product to a family of subspaces varying smoothly with a base point — exactly the data of a Riemannian metric. Riemann's 1854 habilitation lecture generalised this construction to manifolds with no ambient Euclidean space at all: one simply posits a smoothly varying positive-definite symmetric bilinear form on each tangent space and develops geometry from it directly. The first fundamental form is thus best understood as the concrete, computable prototype that the entire abstract apparatus of Riemannian geometry (metric tensors, geodesics, curvature tensors) was built to generalise.

A recurring pedagogical point is the distinction between the parameter domain \( U \) and the image \( S = \sigma(U) \): \( \mathrm{I} \) technically lives on \( U \) (or, invariantly, on the tangent spaces \( T_pS \) for \( p \in S \)), and different parametrisations of the same surface give different-looking \( E,F,G \) related by the chain rule under a change of parameters — yet the lengths, angles and areas they compute are identical, since these are geometric quantities attached to \( S \) itself, not to the choice of \( (u,v) \).

A subtler point: positive-definiteness of \( \mathrm{I} \) is what makes surface geometry (as treated here) Riemannian rather than merely "pseudo-Riemannian." In relativity, an analogous construction on spacetime — the metric tensor of general relativity — deliberately drops positive-definiteness in favour of Lorentzian signature \( (-,+,+,+) \), so that \( \mathrm{I}(w,w) \) can be zero or negative for nonzero \( w \) (null and timelike vectors). The proof of positive-definiteness given in Step 6 above therefore isolates exactly the hypothesis (a Euclidean, not Lorentzian, ambient inner product) that a Riemannian, as opposed to pseudo-Riemannian, treatment relies on.

Common misconception: that \( E,F,G \) are somehow measuring the surface "from outside," the way curvature does. They are not — they are computed from dot products of tangent vectors, and Corollary 3 shows two patches with identical \( E,F,G \) are locally isometric regardless of how differently they sit in \( \mathbb{R}^3\). The embedding enters only through the definition of \( \sigma_u,\sigma_v \) themselves, not through anything extra once those are fixed.

Worked examples

\text{Sphere of radius } R:\ \sigma(\theta,\phi) = (R\sin\theta\cos\phi,\ R\sin\theta\sin\phi,\ R\cos\theta), \quad \theta\in(0,\pi),\ \phi\in(0,2\pi).

Standard spherical parametrisation with \( \theta \) the polar angle, \( \phi \) the azimuthal angle; excluding the poles and one meridian keeps \( \sigma \) injective and regular. A

\sigma_\theta = (R\cos\theta\cos\phi,\ R\cos\theta\sin\phi,\ -R\sin\theta), \qquad \sigma_\phi = (-R\sin\theta\sin\phi,\ R\sin\theta\cos\phi,\ 0).

Direct partial differentiation of \( \sigma \) with respect to \( \theta \) and \( \phi \). A

E = \sigma_\theta\cdot\sigma_\theta = R^2, \qquad F = \sigma_\theta\cdot\sigma_\phi = 0, \qquad G = \sigma_\phi\cdot\sigma_\phi = R^2\sin^2\theta.

Compute each dot product using \( \cos^2+\sin^2=1 \) repeatedly (definition of \( E,F,G \) from the Statement). B

\text{Length of a meridian } \phi=\phi_0,\ \theta:0\to\pi: \quad L = \int_0^\pi \sqrt{E}\ d\theta = \int_0^\pi R\,d\theta = \pi R.

Apply Step 7's arc-length formula with \( \dot\phi=0 \), so \( \sqrt{E\dot\theta^2}=R|\dot\theta| \), matching the known fact that a meridian is half a great circle of length \( \pi R \). B

\text{Area of the whole sphere (limiting } \theta,\phi \text{ to full ranges):} \quad \operatorname{Area} = \int_0^{2\pi}\!\!\int_0^\pi \sqrt{EG-F^2}\ d\theta\,d\phi = \int_0^{2\pi}\!\!\int_0^\pi R^2\sin\theta\ d\theta\,d\phi.

Apply Step 9's area formula with \( \sqrt{EG-F^2}=\sqrt{R^2\cdot R^2\sin^2\theta} = R^2\sin\theta \) (since \( \sin\theta\ge0 \) on \( (0,\pi) \)). B

\int_0^{2\pi}\!\!\int_0^\pi R^2\sin\theta\,d\theta\,d\phi = 2\pi R^2\big[-\cos\theta\big]_0^\pi = 2\pi R^2(1-(-1)) = 4\pi R^2

Reading. The first fundamental form of the sphere, computed purely from \( \sigma \), reproduces the familiar surface area \( 4\pi R^2 \) without any separate geometric argument.

\text{Right circular cylinder:}\ \sigma(u,v) = (\cos u,\ \sin u,\ v), \quad u\in(0,2\pi),\ v\in\mathbb{R}.

Standard parametrisation of the unit cylinder about the \( z \)-axis. A

\sigma_u = (-\sin u,\ \cos u,\ 0), \qquad \sigma_v = (0,0,1) \quad\Rightarrow\quad E = 1,\ F = 0,\ G = 1.

Differentiate and take dot products as in the Statement; note \( \mathrm{I} = du^2+dv^2 \) is exactly the flat Euclidean form on the \( (u,v) \)-plane. B

\text{Helix } \gamma(t) = \sigma(t, ct),\ t\in[0,2\pi],\ c\gt0 \text{ constant.}\quad \dot u = 1,\ \dot v = c.

A curve on the cylinder wound at constant pitch \( c \); parametrised through \( u(t)=t,\ v(t)=ct \). A

L(\gamma) = \int_0^{2\pi} \sqrt{E\dot u^2+2F\dot u\dot v+G\dot v^2}\ dt = \int_0^{2\pi}\sqrt{1+c^2}\ dt = 2\pi\sqrt{1+c^2}.

Substitute \( E=1,F=0,G=1,\dot u=1,\dot v=c \) into Step 7's formula; since \( \mathrm{I}=du^2+dv^2 \) equals the flat plane metric, this matches "unrolling" the cylinder into a flat strip and measuring the straight-line diagonal of a \( 2\pi \times 2\pi c \) rectangle, length \( \sqrt{(2\pi)^2+(2\pi c)^2}=2\pi\sqrt{1+c^2} \), confirming the isometry of Corollary 4. C

L(\gamma) = 2\pi\sqrt{1+c^2}

Reading. Because the cylinder's first fundamental form is flat (\(E=G=1,F=0\), the same as the plane), arc length on the cylinder can be computed exactly as if the surface were unrolled — a direct illustration of \( \mathrm{I} \) capturing only intrinsic, bending-invariant information.

Problems

Compute \( E,F,G \) for the plane patch \( \sigma(u,v) = \mathbf{a} + u\mathbf{p} + v\mathbf{q} \), where \( \mathbf{p},\mathbf{q}\in\mathbb{R}^3 \) are fixed linearly independent vectors, and verify \( EG-F^2\gt0 \).
Solution
\( \sigma_u=\mathbf{p},\ \sigma_v=\mathbf{q}\) (constants, independent of \(u,v\)), so \( E=\mathbf{p}\cdot\mathbf{p}=|\mathbf{p}|^2,\ F=\mathbf{p}\cdot\mathbf{q},\ G=|\mathbf{q}|^2 \), all constant. By Lagrange's identity (Step 5), \( EG-F^2 = |\mathbf{p}\times\mathbf{q}|^2 \), which is strictly positive precisely because \( \mathbf{p},\mathbf{q}\) are linearly independent, so \( \mathbf{p}\times\mathbf{q}\neq0\). This confirms regularity and shows \( \mathrm{I} \) is a constant quadratic form on a flat plane, as expected.
For the cone \( \sigma(u,v) = (u\cos v, u\sin v, u) \), \( u\gt0,\ v\in(0,2\pi) \), compute \( E,F,G \) and find the length of the circular cross-section \( u=u_0 \) constant, \( v:0\to2\pi \).
Solution
\( \sigma_u=(\cos v,\sin v,1),\ \sigma_v=(-u\sin v,u\cos v,0) \). Then \( E=\cos^2v+\sin^2v+1=2 \), \( F=-u\cos v\sin v+u\sin v\cos v+0=0 \), \( G = u^2\sin^2v+u^2\cos^2v=u^2 \). For the circle \( u\equiv u_0 \), \( \dot u=0,\dot v=1 \), so by Step 7, \( L=\int_0^{2\pi}\sqrt{G}\,dv=\int_0^{2\pi}u_0\,dv=2\pi u_0 \), matching the geometrically obvious circumference of a horizontal circle of radius \(u_0\) on the cone (note the cone's true "radius" in 3-space at height \(u_0\) is \(u_0\), consistent with \(\sigma\)).
Using \( E,F,G \) from Problem 2, find the angle at which the line \( v\equiv v_0 \) (a straight generator, \( u:0\to\infty \)) meets the circle \( u\equiv u_0 \) on the cone.
Solution
The generator has tangent direction \( (\dot u,\dot v)=(1,0)\), the circle has tangent direction \( (0,1) \). By Step 8, \( \cos\theta = \dfrac{\mathrm{I}((1,0),(0,1))}{\sqrt{\mathrm{I}((1,0),(1,0))}\sqrt{\mathrm{I}((0,1),(0,1))}} = \dfrac{F}{\sqrt{E}\sqrt{G}} = \dfrac{0}{\sqrt2\cdot u_0} = 0 \), so \( \theta = \pi/2 \): generators and circles of latitude on a cone always meet at right angles, since \( F\equiv0 \) for this parametrisation.
Show that no regular patch \( \sigma:U\to\mathbb{R}^3 \) with \( E=G=1,\ F=0 \) constant everywhere (a "developable, unit-speed" parametrisation) can have image equal to an open subset of a sphere of radius \( R\neq1 \) containing more than a single point, using only what \( \mathrm{I} \) determines about area.
Solution
If \( E=G=1,F=0\) on a region \( R\subset U\) of parameter-area \(A\), then by Step 9 the corresponding surface area is \( \iint_R\sqrt{EG-F^2}\,du\,dv = \iint_R 1\,du\,dv = A\), the same as the flat parameter area. But any nondegenerate open piece of a sphere of radius \(R\) has strictly positive Gaussian curvature \(1/R^2\) (a later theorem), while \(E=G=1,F=0\) is exactly the flat plane's first fundamental form, forcing (by the Theorema Egregium, which computes curvature from \(E,F,G\) alone) Gaussian curvature \(0\) throughout the patch — contradicting the sphere's curvature \(1/R^2\neq0\). Hence no such patch can parametrise a piece of the sphere; intrinsically flat and intrinsically curved regions can never share \(E,F,G\).
A surface patch has first fundamental form \( \mathrm{I} = du^2 + (1+u^2)\,dv^2 \) (so \(E=1,F=0,G=1+u^2\)). Find the arc length of the curve \( u(t)=t,\ v(t)=t \) for \( t\in[0,1] \), and verify \(EG-F^2\gt0\) throughout.
Solution
\(EG-F^2 = 1\cdot(1+u^2) - 0 = 1+u^2 \gt 0\) for all real \(u\), confirming regularity everywhere. With \(\dot u=\dot v=1\), Step 7 gives \(L=\int_0^1\sqrt{E\dot u^2+2F\dot u\dot v+G\dot v^2}\,dt=\int_0^1\sqrt{1+(1+t^2)}\,dt=\int_0^1\sqrt{2+t^2}\,dt\). Using the standard antiderivative \(\int\sqrt{a^2+t^2}\,dt=\frac{t}{2}\sqrt{a^2+t^2}+\frac{a^2}{2}\ln\!\left(t+\sqrt{a^2+t^2}\right)+C\) with \(a^2=2\): \(L=\left[\frac{t}{2}\sqrt{2+t^2}+\ln\!\left(t+\sqrt{2+t^2}\right)\right]_0^1 = \frac{1}{2}\sqrt{3}+\ln(1+\sqrt3)-\ln(\sqrt2) = \frac{\sqrt3}{2}+\ln\!\left(\frac{1+\sqrt3}{\sqrt2}\right)\).