Codex @codex  0

The equation of a quadric surface after orthogonal diagonalization of its symmetric matrix and appropriate translation. The spectral theorem for real symmetric matrices removes mixed quadratic terms, and completing squares removes linear terms in directions with nonzero eigenvalues. Linear terms in zero-eigenvalue directions can remain and generate elliptic paraboloids or hyperbolic paraboloids. The signs, zero coefficients and constant distinguish ordinary surfaces, degenerate conics in plane sections, and empty real loci.

An invertible $4\times 4$ matrix $M$, modulo scalar multiples, acts linearly on homogeneous coordinates. In an affine chart its projective transformation has the formwhere the denominator must be nonzero for a finite image. The important nested groups are Euclidean motions, Euclidean similarities, invertible affine maps, and general projective transformations. Their defining preserved structures become progressively weaker.

A Euclidean motion has $x^{\prime }=Rx+b$ with $R^{T}R=I$. It preserves all distances and angles, hence lengths, areas and volumes. Proper motions have $\det R=1$ and preserve orientation as well; allowing reflections preserves unsigned measurements but can reverse signed ones. A Euclidean similarity has $x^{\prime }=sRx+b$ with $s>0$. It preserves angles and all ratios of lengths; lengths, areas and volumes scale by $s,s^2,s^3$. Thus rigid shape is preserved up to one overall scale.

An invertible affine map has $c=0$ and preserves the plane at infinity. It preserves incidence, parallelism, collinear ratios, affine combinations, and ratios of volumes. Absolute volume scales by $|\det A|$; the volume-preserving affine subgroup has $|\det A|=1$. Lengths and angles are not affine invariants. A general projective transformation preserves incidence, collinearity, concurrence, tangency, and the cross-ratio of four collinear points, but can move the plane at infinity. The cross-ratio, rather than a three-point affine ratio, survives arbitrary projective changes. For a homogeneous quadric hypersurface $X^{T}QX=0$, its transformed matrix is $M^{-T}Q{M}^{-1}$, so its rank and projective degeneracy are preserved.

For surface classes, rather than a fixed numerical radius or curvature, the consequences are:

For example, the affine stretch $(x,y,z)\mapsto (2x,y,z)$ changes a unit sphere to $x^{\prime 2}/4+y^{\prime 2}+z^{\prime 2}=1$ and a unit circular cylinder to $x^{\prime 2}/4+y^{\prime 2}=1$. A general conical surface written $P+tD(s)$ transforms affinely into $T(P)+tAD(s)$, so every generator still passes through a finite apex. A nonsingular elliptic paraboloid or hyperbolic paraboloid has quadratic part of rank two and a nonzero linear component in its null direction. An invertible affine map preserves that rank, inertia and nonzero null-direction coupling, so translation and linear coordinate changes again put it in elliptic or hyperbolic paraboloid form.

Under a genuinely projective change, these Euclidean classifications depend on which plane is designated as infinity. For the map $(x,y,z)\mapsto (x,y,z)/(1+\alpha z)$, a circular cylinder $x^2+y^2=1$ becomesa cone whose apex is finite. The inverse map sends that apex to infinity and produces the cylinder. Thus a cone is projectively invariant as a family of concurrent generators if an ideal apex is allowed, but finite-apex cones and cylinders are not separately invariant. Likewise a projective image of a sphere or either type of paraboloid remains a nonsingular quadric hypersurface, but need not retain its affine or metric type. The distinction between general cones and circular cones matters: circularity is a metric property and is not preserved by arbitrary affine changes.

Parametrize the surface by $r(x,y)=(x,y,xy)$. Its tangent vectors are $r_x=(1,0,y)$ and $r_y=(0,1,x)$. Choose the upward unit normalThe first fundamental form coefficients areSince $r_{xx}=r_{yy}=0$ and $r_{xy}=(0,0,1)$, the second fundamental form coefficients are $e=g=0$ and $f=(1+x^2+y^2{)}^{-1/2}$. The determinant of the shape operator gives the Gaussian curvature:Changing the normal's sign changes both second-form factors and leaves this determinant unchanged. Although the question calls the surface a hyperboloid, $z=xy$ is a hyperbolic paraboloid; the calculation uses the stated surface equation.

At a regular point, choose the unit normal $\mathbf{n}=({\sigma }_x\times {\sigma }_y)/|{\sigma }_x\times {\sigma }_y|$. The first fundamental form is the induced squared lengthThe second fundamental form isThe Gaussian curvature is the determinant of the shape operator, $K=(eg-f^2)/(EG-F^2)$, independent of the normal orientation.

For the given graph, ${\sigma }_x=(1,0,y)$ and ${\sigma }_y=(0,1,x)$, so with $S=1+x^2+y^2$,ThereforeThe surface is regular everywhere. The printed description “hyperboloid” is a naming error: this graph is a hyperbolic paraboloid, and the computation uses the actual parametrization.

At a point of a smooth surface, a normal section of a surface can bend by different amounts depending on its tangent vector. The principal curvatures are the largest and smallest signed normal curvatures among unit tangent vectors. Equivalently they are the two eigenvalues $k_1,k_2$ of the shape operator, with an orthonormal basis $e_1,e_2$ of corresponding principal directions. Here the labels follow the selected directions, without requiring $k_1\le k_2$. ^[ref]

For a unit tangent vector making angle $\theta$ with $e_1$, Euler formula for normal curvature givesThe normal curvature is a weighted average of the principal curvatures; the extremes occur in the principal directions. If the principal curvatures coincide, every direction in the tangent plane is a principal direction: this is an umbilical point. The Gaussian curvature and mean curvature summarize the pair by

For an outward-oriented sphere of radius $R$, both principal curvatures are $-1/R$, so the normal curvature is $-1/R$ in every direction. For an outward-oriented circular cylinder, choose $e_1$ around the circumference and $e_2$ along the axis. The principal curvatures are $-1/R,0$, giving $k_n(\theta )=-{\cos }^2\theta /R$; halfway between the principal directions, $k_n(\pi /4)=-1/(2R)$.

At the origin of the upward-oriented hyperbolic paraboloid $z=(x^2-y^2)/2$, the principal directions are the coordinate axes and the principal curvatures are $1,-1$ in the chosen length unit. Thus $k_n(\theta )=\cos (2\theta )$, with zero normal curvature at $\theta =\pi /4,3\pi /4$. Zero normal curvature in these directions coexists with Gaussian curvature $-1$: the shape operator is not zero. The mean curvature is zero at this point; this does not say that the whole hyperbolic paraboloid is a minimal surface.

Each graph uses angle $\theta$ in the tangent plane, measured from the indicated first principal direction. The sphere and circular cylinder have $R=1$; the saddle graph refers only to the origin. Reversing the unit normal reflects each normal curvature graph through zero, negates both principal curvatures and the mean curvature, and preserves their product, the Gaussian curvature.