Quadrics and the Limits of the Blade

ChapterChapter 8 represents a sphere as a single blade, moved by a sandwich product. Chapter 11 represents a plane, a line, and a point the same way, moved by a motor. Both chapters raise the same question in passing and leave it there: what happens to a blade under a transformation neither algebra was built for? This chapter answers it, for the specific case of scaling a sphere unevenly, or shearing it, and for the sharper case of carving a dent into one. The answer turns out to be the same for a circle, a sphere, or an ellipsoid: a transformation either preserves the family of round objects CGA was built to represent, or it doesn't, and knowing which is true of a given matrix is a precise, checkable fact rather than a matter of degree.

13.1 What a Conformal Transformation Preserves FoundationalKnowledge that endures for decades — core principles

TheThe transformations Chapter 8 builds versors for — translation, rotation, reflection, inversion, and uniform scaling — share one property: each preserves the angle between any two curves at the point where they cross. That property has a name, conformal, and it's the actual boundary of what a CGA versor can do, not "rigid motion" as a rough stand-in for it. Uniform scaling changes every length by the same factor and changes no angle at all, which is exactly why it counts as conformal alongside the isometries. In CGA this uniform scaling has its own versor, the dilator:

\[ D = \exp\!\left(\tfrac{1}{2}\ln\rho\,\left(\mathbf{e}_o\wedge\mathbf{e}_\infty\right)\right), \qquad X' = D\,X\,\widetilde{D} \]

applied to Chapter 8's sphere blade \(S = \mathbf{c} + \tfrac{1}{2}(|\mathbf{c}|^2-r^2)\mathbf{e}_\infty + \mathbf{e}_o\), \(D\) sends it to a sphere blade of radius \(\rho r\) centred at \(\rho\mathbf{c}\)^[1][2]. The output is still a sphere blade — the sandwich product never leaves the family of round objects CGA represents. That's the property worth isolating, because the next section shows exactly where it stops holding. same sandwich, same proof obligation as chapter 8's translator

13.2 Why Shear Isn't Conformal FoundationalKnowledge that endures for decades — core principles

TakeTake the two perpendicular vectors \(\mathbf{e}_1=(1,0)\) and \(\mathbf{e}_2=(0,1)\), and shear them with \(M = \begin{pmatrix}1 & 0.5\\ 0 & 1\end{pmatrix}\): \(\mathbf{e}_1\) is unchanged, and \(\mathbf{e}_2\) becomes \((0.5, 1)\). Their dot product, which was \(0\), is now \(1\times0.5 + 0\times1 = 0.5\). Two vectors that met at a right angle now meet at \(\arccos(0.5/\sqrt{1.25}) \approx 63.4^\circ\). Compare that with a uniform scale by \(2\): \(\mathbf{e}_1\) becomes \((2,0)\) and \(\mathbf{e}_2\) becomes \((0,2)\), and their dot product is still \(0\) — the lengths changed, the angle between them didn't. That's the test, stated as a number instead of a description: shear and non-uniform scaling change the angle between some pair of directions, and a conformal versor, by definition, cannot.

Because shear and non-uniform scaling aren't conformal, they have no dilator, rotor, or translator that produces them. The only thing CGA offers for them is the ordinary linear map, applied to a point's Euclidean coordinates before it's re-embedded as a null vector. That recovers the correct transformed points, but the sphere's own blade \(S\) doesn't transform into anything — there is no blade for the shape the points now trace out, because that shape is a general ellipsoid, and an ellipsoid isn't one of the round objects CGA's grade-1 sphere blade was built to hold.

13.3 What Projective GA Can and Can't Carry FoundationalKnowledge that endures for decades — core principles

ChapterChapter 11 already distinguishes PGA from CGA by scope: PGA covers flat objects and rigid motions, CGA adds round ones^[3]. Shear sharpens that distinction rather than crossing it. An affine map \(\mathbf{x}' = A\mathbf{x} + \mathbf{t}\) lifts to PGA cleanly, as an outermorphism acting on every blade at once:

\[ \underline{A}(\mathbf{u}\wedge\mathbf{v}) = A(\mathbf{u})\wedge A(\mathbf{v}) \]

which is precisely why PGA is the natural home for shearing and non-uniformly scaling points, line segments, planes, and meshes^[3][4]: every vertex of a sheared mesh carries through the same formula, correctly, in one pass. But PGA was never built with a sphere blade to begin with — Chapter 11's table of objects stops at planes, lines and points. Applying the outermorphism above to a sampled sphere gives the right sheared points for the same reason it gives the right sheared mesh; it does not thereby produce a native ellipsoid blade, because PGA has no sphere blade for it to have deformed in the first place.

So the two algebras fail the same request for two different reasons. CGA has a sphere blade, but no versor that carries shear. PGA has a versor-like transform for shear, the motor-adjacent outermorphism, but no blade for a round object to carry it on. Neither gap is a missing feature of one algebra that the other happens to have — they're the same boundary, approached from opposite sides. a transform acting on nothing still produces nothing

13.4 Representing the Resulting Quadric Directly FoundationalKnowledge that endures for decades — core principles

AA general ellipsoid in three dimensions is the zero set of

\[ Ax^2+By^2+Cz^2+Dxy+Eyz+Fzx+Gx+Hy+Iz+J=0, \]

nine independent coefficients beyond an overall scale, against a sphere's four (centre and radius). Raising the grade of a CGA blade doesn't add coefficients of this kind: a higher grade in CGA names a different-dimensional object built from more conformal points — a circle from three, a sphere from four — or, dually, a flat object in place of a round one. It was never a slot for the extra cross-terms \(Dxy\), \(Eyz\), \(Fzx\) that a sheared sphere's equation has and a sphere's doesn't. Representing a general quadric as one blade, with the same sandwich-product machinery CGA uses for spheres, needs a larger algebra purpose-built for it. Two are in the literature: Double Conformal Geometric Algebra, which lifts the conformal model a second time to carry quadric surfaces and their intersections^[5], and Quadric Conformal Geometric Algebra, built in \(\mathbb{R}^{9,6}\) specifically so that an arbitrary quadric is the wedge of nine points^[6], the same "object as an outer product of points" idiom Chapter 8 uses for a circle or a sphere, extended to a family neither CGA nor PGA was built to hold.

13.5 The Boundary, Stated Plainly

Capability	CGA	PGA	Quadric GA
Represent a sphere or circle directly	Yes	No	Yes
Translate or rotate it	Yes, by a versor	Yes, by a motor	Yes
Scale it uniformly	Yes, by a dilator	As an affine map, not as a native sphere	Yes
Shear it or scale it unevenly	Points only, not the blade	Points and meshes, correctly	Yes
Represent the resulting ellipsoid as one blade	No	No	Yes

ReadRead the "No" entries carefully: they say that ordinary CGA and PGA have no native blade for the sheared result, not that software built on either algebra can't calculate the transformed points. Both playgrounds below do exactly that calculation — what neither algebra offers is a single object, carried by a sandwich product, standing in for the whole shape the way \(S\) stands in for a sphere.

13.6 Interactive: A Circle Under Scale and Shear

TheThe two-dimensional case has a closed form, so the playground below can print the exact equation live rather than only a picture. A circle's image under \(M = \begin{pmatrix}s_x & \text{shear}\\ 0 & s_y\end{pmatrix}\) is the general conic \(Ax^2+Cxy+By^2=1\), and \(A=B\) with \(C=0\) exactly when \(M\) is a uniform scale with no shear — precisely the case Section 13.1's dilator covers. Move any slider away from that one point and \(C\) becomes nonzero, or \(A\) and \(B\) pull apart: the explanation beneath the diagram names which of the three cases from the table above is on screen.

13.7 Interactive: The Same Story in Three Dimensions

AA sphere under the same kind of map has no equally short closed form — the diagram samples the surface directly, as a grid of points carried through \(M\), rather than solving for one. The classification is identical to the circle's: the explanation beneath the diagram reports whether the current \((s_x, s_y, \text{shear})\) is the one case a CGA dilator reaches, or one of the many it doesn't.

13.8 Key Terms

• Conformal. Preserving the angle between any two curves at the point where they cross. Translations, rotations, reflections, inversions, and uniform scaling are all conformal; shear and non-uniform scaling are not.
• Dilator. The CGA versor for uniform scaling about a point, \(D=\exp(\tfrac12\ln\rho\,(\mathbf{e}_o\wedge\mathbf{e}_\infty))\).
• Outermorphism. The extension of a linear map on vectors to a linear map on every blade, by acting on each factor of a wedge product in turn: \(\underline{A}(\mathbf{u}\wedge\mathbf{v})=A(\mathbf{u})\wedge A(\mathbf{v})\).
• Quadric. The zero set of a general second-degree polynomial in the coordinates — an ellipsoid, paraboloid, or hyperboloid in three dimensions, an ellipse or other conic in two.
• Double Conformal GA / Quadric Conformal GA. Two distinct extensions of the conformal model built specifically to represent general quadric surfaces as single blades, which neither standard CGA nor PGA does.

Related Topics

• CGA Worked Examples: Interval Arithmetic in Conformal Space — the sphere blade and dilator versor this chapter's Section 13.1 builds on directly.
• Projective Geometric Algebra: Points, Planes, and Motors for Graphics — the outermorphism and the CGA/PGA scope boundary Section 13.3 sharpens with the shear case.
• The Ray-Caster's Toolkit: Meet, Join, and Bisection — the implicit-field, inside/outside convention a companion playground example uses to carve a dent into a sphere, a different kind of limit on what a single blade can represent.
• Scalars, Points, Vectors, and Blades Across Dimensions — the grade-counting pattern this chapter's Section 13.4 leans on to explain why a higher grade alone doesn't add quadratic coefficients.