# Geometry

## 1712 Submissions

[3] **viXra:1712.0642 [pdf]**
*replaced on 2018-04-23 22:13:50*

### Estimation of the Earth's "Unperturbed" Perihelion from Times of Solstices and Equinoxes

**Authors:** James A. Smith

**Comments:** 13 Pages.

Published times of the Earth's perihelions do not refer to the perihelions of the orbit that the Earth would follow if unaffected by other bodies such as the Moon. To estimate the timing of that ``unperturbed" perihelion, we fit an unperturbed Kepler orbit to the timings of the year 2017's equinoxes and solstices. We find that the unperturbed 2017 perihelion, defined in that way, would occur 12.93 days after the December 2016 solstice. Using that result, calculated times of the year 2017's solstices and equinoxes differ from published values by less than five minutes. That degree of accuracy is sufficient for the intended use of the result.

**Category:** Geometry

[2] **viXra:1712.0524 [pdf]**
*submitted on 2017-12-19 13:56:07*

### Projection of a Vector upon a Plane from an Arbitrary Angle, via Geometric (Clifford) Algebra

**Authors:** James A. Smith

**Comments:** 8 Pages.

We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.

**Category:** Geometry

[1] **viXra:1712.0393 [pdf]**
*submitted on 2017-12-11 16:46:30*

### Formulas and Spreadsheets for Simple, Composite, and Complex Rotations of Vectors and Bivectors in Geometric (Clifford) Algebra

**Authors:** James A. Smith

**Comments:** 29 Pages. Formulas and Spreadsheets for Simple, Composite, and Complex Rotations of Vectors and Bivectors in Geometric (Clifford) Algebra

We show how to express the representations of single, composite, and ``rotated" rotations in GA terms that allow rotations to be calculated conveniently via spreadsheets. Worked examples include rotation of a single vector by a bivector angle; rotation of a vector about an axis; composite rotation of a vector; rotation of a bivector; and the ``rotation of a rotation". Spreadsheets for doing the calculations are made available via live links.

**Category:** Geometry