= Solution
An invertible $4\times4$ <matrix> $M$, modulo scalar multiples, acts linearly on <homogeneous coordinates>. In an affine chart its <projective transformation> has the form
$$
x'=\frac{Ax+b}{c^Tx+d},
$$
where 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 $x'=Rx+b$ with $R^TR=I$. 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 $x'=sRx+b$ 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> $X^TQX=0$, its transformed matrix is $M^{-T}QM^{-1}$, so its rank and projective degeneracy are preserved.
For surface \i[classes], rather than a fixed numerical radius or curvature, the consequences are:
|| Transformation class
|| <Sphere>
|| <Circular cylinder>
|| Finite-apex <general conical surface>
|| <Elliptic paraboloid> or <hyperbolic paraboloid>
| <Euclidean motion>
| Yes
| Yes
| Yes
| Yes
| <Euclidean similarity>
| Yes, with scaled radius
| Yes, with scaled radius
| Yes
| Yes, with changed scale
| Invertible <affine map>
| Not generally: an ellipsoid
| Not generally: an elliptic cylinder
| Yes
| Yes, preserving the elliptic/hyperbolic type
| General <projective transformation>
| Not generally
| Not generally
| Concurrence survives, but the apex can move to infinity
| Not generally
For example, the affine stretch $(x,y,z)\mapsto(2x,y,z)$ changes a unit <sphere> to $x'^2/4+y'^2+z'^2=1$ and a unit <circular cylinder> to $x'^2/4+y'^2=1$. 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> $x^2+y^2=1$ becomes
$$
x'^2+y'^2=(1-\alpha z')^2,
$$
a cone whose apex is finite. The inverse map sends that apex to infinity and produces the cylinder. Thus \b[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 \i[circular] cones matters: circularity is a metric property and is not preserved by arbitrary affine changes.
Back to article page
