Smooth surface

A smooth surface in ${\mathbb{R}}^3$ is a two-dimensional smooth embedded submanifold. Locally it is parametrized by two coordinates with linearly independent tangent vectors.

A smooth surface is a two-dimensional smooth manifold whose underlying space is a topological surface. Its underlying topological space is a surface, but the smooth structure is extra data.

A parallel surface is a constant-distance normal displacement $P_d=P+dn$ of a regular surface with unit normal vector $n$. With shape operator $W=-Dn$, its differential is $(I-dW)DP$; it can become singular where $1-d{\kappa }_i=0$ for a principal curvature ${\kappa }_i$.

// Target: geometry-and-topology.bigb

A variable normal offset displaces a parametric surface along its unit normal vector by a scalar field $d(u,v)$. Unlike a constant-distance parallel surface, its derivatives also contain $d_{u}n$ and $d_{v}n$.

// Target: analysis.bigb

A parametric surface is represented by a map $(u,v)\mapsto P(u,v)$. A regular patch has $P_u\times P_v\ne 0$; this cross product supplies its normal vector.

// Target: analysis.bigb

A general conical surface is swept out by straight generators through one apex $A$. An invertible affine map sends it to another such surface, since it sends each generator to a line through the transformed apex. A projective transformation preserves the projective concurrence of generators, but can send the apex to infinity, producing parallel generators in the affine chart. Therefore finite-apex cones and cylinders are not separately projectively invariant Euclidean classes.

For nonnegative barycentric parameters, these total-degree Bernstein basis functions define a surface over a triangle. The triangular control net has $(n+1)(n+2)/2$ points. Its index constraint couples the three exponents, so it is not a tensor product of two independent univariate bases. The multinomial theorem gives nonnegativity and partition of unity.

Core numerical enquiries return a surface point and its derivatives through second order, together with the parameter domain, patch boundaries and regularity information. At a regular point these determine its tangent plane, normal vector, first and second fundamental forms and curvatures. Subdivision and certified enclosing volumes provide spatial search and intersection support. Singular charts or discontinuity boundaries require one-sided or alternative-chart information; a zero cross product does not define a unique normal vector.

From a point light source $P$, a caster point $C(t)$ projects onto a receiving surface along the displayed ray. On a regular transverse branch, differentiating gives $[S_u,S_v,-(C-P)](u^{\prime },v^{\prime },{\alpha }^{\prime }{)}^T=\alpha C^{\prime }$. Newton method correction and certified patch/curve bounds support continuation and isolation of all visible branches. If the shadow's second derivative is bounded by $M$ on an interval of width $h$, its chord error is at most $M{h}^2/8$. A fixed sample grid can miss entire shadow components; surface boundaries, occlusion changes and grazing rays require explicit treatment.

Products of two univariate bases form a surface basis. Nonnegativity, partition of unity and coordinate-wise linear precision pass to the products by multiplication of the respective univariate identities. If the factors are $C^r$ and $C^s$, each mixed derivative through orders $r,s$ is the product of the corresponding derivatives, giving continuous mixed derivatives and joint $C^{\min (r,s)}$ regularity.

Nonnegative tensor-product surface basis factors yield nonnegative product weights. Their two partition of unity sums multiply to one. If the factors are $C^r$ and $C^s$, mixed derivatives through separate orders $r,s$ are continuous, so the resulting finite or locally finite surface representation is jointly $C^{\min (r,s)}$. Control coefficients can cancel derivative jumps and yield more smoothness; inherited continuity is a guaranteed lower bound, not a claim that every surface has exactly the basis's worst regularity.

The displayed surfaces, with positive semiaxes, are hyperboloids. The plus sign gives a one-sheet hyperboloid, while the minus sign gives a two-sheeted hyperboloid.

A one-sheet hyperboloid is a connected, doubly ruled surface, diffeomorphic to a cylinder.

On $x^2+y^2=z^2+1$, an axial plane such as $y=0$ cuts out two disjoint meridian geodesics. The tangent plane $x=1$ cuts out the two straight ruling lines $y=\pm z$, which are geodesics meeting orthogonally at $(1,0,0)$.

For a unit-speed geodesic on $x^2+y^2=z^2+1$, Clairaut's constant $c$ givesIf $c>1$, the geodesic has a turning point at $z=\sqrt{c^2-1}$ and remains entirely in one of the regions $z>0$ or $z<0$.

The waist circle $z=0$ is a geodesic. Its Gaussian curvature is $-1$, whereasoff the waist. Since isometries preserve Gaussian curvature, every isometry preserves this circle setwise.

This regular surface has two connected components, $t=\pm \sqrt{1+x^2+y^2}$. The symmetric-matrix model of the two-sheeted hyperboloid realizes them as the positive-definite and negative-definite symmetric determinant-one $2\times 2$ matrices. Each component is diffeomorphic to ${\mathbb{R}}^2$; this differs from the connected one-sheet hyperboloid.

The displayed coordinates identify determinant-one real symmetric matrices of size two with the two-sheeted hyperboloid. The special linear congruence action on symmetric matrices is transitive on each sheet, with stabilizer $SO(2)$. In particular the positive sheet is the homogeneous space $SL(2,\mathbb{R})/SO(2)$, a model of the hyperbolic plane.

An embedded smooth surface $S\subset {\mathbb{R}}^3$ is inextendible when every connected smooth surface $\tilde{S}\subset {\mathbb{R}}^3$ containing $S$ is equal to $S$.