   Start Calculator Home Introduction Reference Glossary Bugs Bibliography Links  Geometric Calculator - Introduction This calculator differs from most you've met before in that it computes with Clifford numbers rather than the plain old numbers that you're used to seeing. As a consequence, this calculator can compute directly what other calculators compute as an afterthought with extensive subprograms. Clifford Algebras were invented by William K. Clifford in the 19th century. While world reknowned physicists were debating whether vectors or quaternions were the more appropriate medium for expressing physics, Clifford quietly incorporated vectors and quaternions into a system of geometric algebras which worked better than either alone and generalized in ways that neither could. Plain old numbers are, as it happens, part of Clifford's scheme, so as long as you stick to plain old numbers and plain old computations, then this calculator will work just like the other RPN desk calculators that you already know. Plain old numbers are called scalars. Vectors are, as it happens, also part of Clifford's scheme, so you can use the x, y, and z keys on this calculator to enter the unit basis vectors in the x, y, and z directions and implicitly multiply them by the plain old number you've entered. You can add and subtract vectors and multiply them by scalars. In Clifford's geometric scheme of things, scalars represent pure magnitudes and vectors represent oriented lengths. To continue the geometric scheme we need things which represent oriented areas and oriented volumes. These are called bivectors, because two vectors define an area, and trivectors, because three vectors (or a vector and a bivector) define a volume. The unit basis bivectors are xy, yz, and zx each representing a unit area in the plane named by the basis. The unit trivector basis is xyz representing a unit volume. All of which you can enter by simply typing the appropriate keys or by multiplying together the unit basis vectors. Adding all this up, our numbers are multivectors consisting of a scalar part, a vector part, a bivector part, and a trivector part. This may strike you as a bit awkward at first, somewhat like adding apples and oranges and bananas and pomegranates. We have 8 different kinds of fruit to keep straight, each of which is labelled by its unit basis when it appears in a number: x, y, z, xy, yz, zx, and xyz. The eighth basis would be 1, the scalar unit basis, which we don't actually write. There are many interesting consequences of calculating with geometrically defined numbers, which David Hestenes develops in depth. Calculations with the plain old numbers embedded in the geometric numbers work just as they always have. Calculations with the plain old two- and three-dimensional vectors embedded in the geometric numbers work just as they always have. Calculations with numbers of the form a + bxy work very much like complex numbers with xy serving as the imaginary unit. The same is true if yz or zx is used as an imaginary unit. Calculations with numbers of the form s + ixy + jyz + kzx work very much like Hamilton's quaternions. Calculations of areas and volumes are simple and direct, even for complex figures. If a is a unit vector and B is any Clifford number, then -aBa is the Clifford number with all its parts reflected in the plane orthogonal to a. If a and b are unit vectors and C is any Clifford number, then abCba is the Clifford number with all its parts rotated in the ab plane by twice the angle between a and b. If we write q = ab, then the rotation formula becomes qCq*. The plane geometry included within the geometric numbers allows for solution of problems of plane geometry, including terrestrial surveying and navigation problems. The quaternion algebra included within the geometric numbers allows for solution of spherical geometry problems, including spherical trigonometry, celestial navigation, spacecraft attitude, and robot pose problems. In time, this introduction will grow into a more detailed introduction and explanation of why you want to take a Geometric Calculator with you when you leave Earth. Whether you're bound for a life in deep space or a colony on another planet, don't leave home without it.     Roger E Critchlow Jr Last modified: Fri May 16 19:38:36 MDT 2003 elf.org