User:Optikos/sandbox

Quadric geometric algebra (QGA) is a geometrical application of the ${\mathcal {R}}^{6,3}$ vector space (i.e., 6 real numbers and 3 imaginary numbers), which is called the ${\mathcal {G}}_{6,3}$ geometric algebra. QGA is also known as the ${\mathcal {C}}\ell _{6,3}$ Clifford algebra. QCGA is also known as the ${\mathcal {C}}\ell _{9,6}$ Clifford algebra. QGA is a super-algebra over ${\mathcal {G}}_{4,1}$ Conformal Geometric Algebra (CGA) and ${\mathcal {G}}_{1,3}$ Spacetime Algebra (STA), which can each be defined within sub-algebras of QGA. In turn, each of QGA and CGA can be defined as a sub-algebra of quadratic conformal geometric algebra (QCGA), which is a geometrical application of the ${\mathcal {R}}^{9,6}$ vector space (i.e., 9 real numbers and 6 imaginary numbers, which is called the ${\mathcal {G}}_{9,6}$ geometric algebra. QGA is also known as the ${\mathcal {C}}\ell _{9,6}$ Clifford algebra.

CGA provides representations of spherical entities (points, spheres, planes, and lines) and a complete set of operations (translation, rotation, dilation, and intersection) that apply to them. QGA extends CGA to also include representations of some non-spherical entities: principal axes-aligned quadric surfaces and many of their degenerate forms such as planes, lines, and points. QCGA reintegrates QGA's ability to operate on axes-aligned quadric surfaces with CGA's ability to rotate, so that the quadric surfaces need not be axis-aligned.

General quadric surfaces are characterized by the implicit polynomial equation of degree 2

$Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fzx+Gx+Hy+Iz+J=0.$

which can characterize quadric surfaces located at any center point and aligned along arbitrary axes. However, QGA includes vector entities that can represent only the principal axes-aligned quadric surfaces characterized by

$Ax^{2}+By^{2}+Cz^{2}+Gx+Hy+Iz+J=0.$

As such, QGA's intuitive expressivity is an advancement over CGA.

A possible performance challenge with using QGA is the increased computation required to utilize its 9D vector space, as compared to the smaller 5D vector space of CGA. A 5D CGA subspace can be utilized in QGA when only CGA entities are involved in computations. Likewise, a possible performance challenge with using QCGA is the increased computation required to utilize its 15D vector space, as compared to the smaller 9D vector space of QGA or the smaller 5D vector space of CGA.

The operations that work correctly with the QGA axes-aligned quadric entities include translation, transposition, dilation, and intersection.

In general, the operation of rotation does not work correctly on non-spherical QGA quadric surface entities, but does on QCGA quadric surface entities. Rotation also does not work correctly on the QGA point entities. Attempting to rotate a QGA quadric surface may result in a different type of quadric surface, or a quadric surface that is rotated and distorted in an unexpected way. Attempting to rotate a QGA point may produce a value that projects as the expected rotated vector, but the produced value is generally not a correct embedding of the rotated vector. The failure of QGA points to rotate correctly also leads to the inability to use outermorphisms to rotate dualized Geometric Outer Product Null Space (GOPNS) entities. To rotate a QGA point, it must be projected to a vector or converted to a CGA point for rotation operations, then the rotated result can be re-embedded or converted back into a QGA point. A quadric surface rotated by an arbitrary angle cannot be represented by any known QGA entity. Representation of general quadric surfaces with useful operations will require an algebra (that appears to be unknown at this time) that extends QGA. As such, QCGA's intuitive expressivity is a notable advancement over both QGA and CGA.

Although rotation is generally unavailable in QGA, the transposition operation is a special-case modification of rotation by $\pi /2$ that works correctly on all QGA Geometric Inner Product Null Space (GIPNS) entities. Transpositions allow QGA GIPNS entities to be reflected in the six diagonal planes $y=\pm x$, $z=\pm x$, and $z=\pm y$.

Entities for all principal axes-aligned quadric surfaces can be defined in QGA. These include ellipsoids, cylinders, cones, paraboloids, and hyperboloids in all of their various forms. Likewise, these quadric surfaces can be defined and operated on in QCGA, but do not need to be aligned to any axis.

A powerful feature of QGA is the ability to compute the intersections of axes-aligned quadric surfaces; a powerful feature of QCGA is the ability to compute the intersections of arbitrarily-axis-unaligned quadric surfaces. With few exceptions, the outer product of QGA GIPNS surface entities represents their surfaces' intersection(s) or lack thereof; likewise with QCGA GIPNS. In both QGA and QCGA, this method of computing intersections works the same as it does in CGA, where only spherical entities are available. In QCGA, this method of computing intersections works the same as it does in QGA, where only axes-aligned quadric surfaces are available.

References

• Julio Zamora-Esquivel. "G6,3 Geometric Algebra; Description and Implementation." Advances in Applied Clifford Algebras 24 (2014), 493-514. Springer Basel.
• http://www.dgcv.nii.ac.jp/Publications/Papers/2018/aaca2018.pdf
• http://vixra.org/pdf/1902.0401v4.pdf
• https://link.springer.com/article/10.1007/s00006-019-0964-1

Category:Clifford algebras Category:Geometric algebra