The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying invol... more The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms.
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how... more As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how to calculate the distance and direction between two points on Earth, given the locations' latitudes and longitudes. We validate the results by comparing them to those obtained from online calculators. This example invites a discussion of the benefits of teaching spherical trigonometry (the usual way of solving such problems) at the high-school level versus teaching how to use Geometric Algebra for the same purpose.
To help fill the need for examples of introductory-level problems that have been solved via Geome... more To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which a given vector must be rotated in order that its endpoint be at a given distance d from a specified point P. The three solution methods that are employed here start from a trigonometric equation is derived from GA's formula for rotating vectors. The first two solutions use methods that are "automatic", but produce formulas that are not readily interpreted. In contrast, the third method-which does produce a readily interpreted formula-is based upon an examination of the geometric significance of terms in the initial trigonometric equation.
Because the shortage of worked-out examples at introductory levels is an obstacle to widespread a... more Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we conver...
We show how to express the representation of a composite rotation in terms that allow the rotatio... more We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.
As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to de... more As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenie... more After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.
Published times of the Earth's perihelions do not refer to the perihelions of the orbit that ... more 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.
Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this d... more Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.
Written as somewhat of a "Schaums Outline" on the subject, which is especially useful i... more Written as somewhat of a "Schaums Outline" on the subject, which is especially useful in robotics and mechatronics. Geometric Algebra (GA) was invented in the 1800s, but was largely ignored until it was revived and expanded beginning in the 1960s. It promises to become a "universal mathematical language" for many scientific and mathematical disciplines. This document begins with a review of the geometry of angles and circles, then treats rotations in plane geometry before showing how to formulate problems in GA terms, then solve the resulting equations. The six problems treated in the document, most of which are solved in more than one way, include the special cases that Viete used to solve the general Problem of Apollonius.
As an example for newcomers to GA who may have difficulty applying its identities to real problem... more As an example for newcomers to GA who may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
As an example for newcomers to GA who may have difficulty applying its identities to real problem... more As an example for newcomers to GA who may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
The lives of poor landowners in tropical mountains depend upon their collective capacity to creat... more The lives of poor landowners in tropical mountains depend upon their collective capacity to create and coordinate social preferences derived from their interacting communalistic, hierarchical, and reciprocal exchanges. External actors currently contend for these territories under market rules that are modifying such preferences. We present the design, experimental implementation, and analysis of results of a four-player, land-use board game with stark resource and livelihood limits and coordination/cooperation challenges, as played (separately) by 116 farmers and 108 academics, mainly in the tropical mountains of Chiapas, Mexico. In game session one, we trained and framed players in moral economy, a human core feeling and communalistic norm of solidarity and mutual obligation, which translates into "all players must survive." In session two, we explored to what extent moral economy resisted as a social preference under a hypothetical external monetary incentive scheme unfavorable to it. Using an approach that combines spot game analysis and experimental work, we studied the social preferences that emerged during session two among advantaged and disadvantaged players to deal with inequity in land appropriation and use when imminent "death" approaches. We make comparisons between farmers and academics. Players evolved moral economy, competitive domination, i.e., let competition decide, and coalition, i.e., advantaged players ask the dying to surrender land and die prematurely in exchange for a share of the dismal profits. Farmers basically stuck to the first two preferences in similar proportions whereas academics clearly shifted to coalition, a last-resort choice, which allowed disadvantaged players some final leverage and advantaged players use of liberated resources to improve efficiency. Coalition as strategic cooperation among the unequal is part of the culture in which academics are being educated as sustainability professionals and toward which farmers are being steered. In the stringent social-environmental conditions of this game, the results were a Pareto-superior form of equity, albeit with land surrendering, and many more deaths than other preferences.
We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric... more We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric Algebra, about the product of the lengths of two segments of a single chord. We derive a similar theorem about the product of the lengths of a secant and a chord.
Euclid proved (Elements, Book III, Propositions 20 and 21) proved that an angle inscribed in a ci... more Euclid proved (Elements, Book III, Propositions 20 and 21) proved that an angle inscribed in a circle is half as big as the central angle that subtends the same arc. We present a high-school-level version of Hestenes' GA-based proof ([1]) of that same theorem. We conclude with comments on the need for learners of GA to learn classical geometry as well. "Demonstrate that the inscribed angle φ is is half as large as the central angle θ .
Note: The Appendix to this new version gives an alternate--and much simpler--solution that does n... more Note: The Appendix to this new version gives an alternate--and much simpler--solution that does not use reflections. The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viete did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equa...
We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve t... more We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
We show how to express the representations of single, composite, and ``rotated" rotations in... more 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.
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra (GA), we sho... more As a high-school-level example of solving a problem via Geometric (Clifford) Algebra (GA), we show how to derive equations for the circle formed by the intersection of a plane with a sphere. Among the tasks that we will demonstrate are how to (1) express a plane via GA, (2) calculate the "reject" of a vector from a plane, and (3) express a circle as the rotation of a vector about a point.
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth,... more As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Karman Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying invol... more The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms.
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how... more As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how to calculate the distance and direction between two points on Earth, given the locations' latitudes and longitudes. We validate the results by comparing them to those obtained from online calculators. This example invites a discussion of the benefits of teaching spherical trigonometry (the usual way of solving such problems) at the high-school level versus teaching how to use Geometric Algebra for the same purpose.
To help fill the need for examples of introductory-level problems that have been solved via Geome... more To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which a given vector must be rotated in order that its endpoint be at a given distance d from a specified point P. The three solution methods that are employed here start from a trigonometric equation is derived from GA's formula for rotating vectors. The first two solutions use methods that are "automatic", but produce formulas that are not readily interpreted. In contrast, the third method-which does produce a readily interpreted formula-is based upon an examination of the geometric significance of terms in the initial trigonometric equation.
Because the shortage of worked-out examples at introductory levels is an obstacle to widespread a... more Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we conver...
We show how to express the representation of a composite rotation in terms that allow the rotatio... more We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.
As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to de... more As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenie... more After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres.
Published times of the Earth's perihelions do not refer to the perihelions of the orbit that ... more 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.
Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this d... more Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.
Written as somewhat of a "Schaums Outline" on the subject, which is especially useful i... more Written as somewhat of a "Schaums Outline" on the subject, which is especially useful in robotics and mechatronics. Geometric Algebra (GA) was invented in the 1800s, but was largely ignored until it was revived and expanded beginning in the 1960s. It promises to become a "universal mathematical language" for many scientific and mathematical disciplines. This document begins with a review of the geometry of angles and circles, then treats rotations in plane geometry before showing how to formulate problems in GA terms, then solve the resulting equations. The six problems treated in the document, most of which are solved in more than one way, include the special cases that Viete used to solve the general Problem of Apollonius.
As an example for newcomers to GA who may have difficulty applying its identities to real problem... more As an example for newcomers to GA who may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
As an example for newcomers to GA who may have difficulty applying its identities to real problem... more As an example for newcomers to GA who may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
The lives of poor landowners in tropical mountains depend upon their collective capacity to creat... more The lives of poor landowners in tropical mountains depend upon their collective capacity to create and coordinate social preferences derived from their interacting communalistic, hierarchical, and reciprocal exchanges. External actors currently contend for these territories under market rules that are modifying such preferences. We present the design, experimental implementation, and analysis of results of a four-player, land-use board game with stark resource and livelihood limits and coordination/cooperation challenges, as played (separately) by 116 farmers and 108 academics, mainly in the tropical mountains of Chiapas, Mexico. In game session one, we trained and framed players in moral economy, a human core feeling and communalistic norm of solidarity and mutual obligation, which translates into "all players must survive." In session two, we explored to what extent moral economy resisted as a social preference under a hypothetical external monetary incentive scheme unfavorable to it. Using an approach that combines spot game analysis and experimental work, we studied the social preferences that emerged during session two among advantaged and disadvantaged players to deal with inequity in land appropriation and use when imminent "death" approaches. We make comparisons between farmers and academics. Players evolved moral economy, competitive domination, i.e., let competition decide, and coalition, i.e., advantaged players ask the dying to surrender land and die prematurely in exchange for a share of the dismal profits. Farmers basically stuck to the first two preferences in similar proportions whereas academics clearly shifted to coalition, a last-resort choice, which allowed disadvantaged players some final leverage and advantaged players use of liberated resources to improve efficiency. Coalition as strategic cooperation among the unequal is part of the culture in which academics are being educated as sustainability professionals and toward which farmers are being steered. In the stringent social-environmental conditions of this game, the results were a Pareto-superior form of equity, albeit with land surrendering, and many more deaths than other preferences.
We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric... more We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric Algebra, about the product of the lengths of two segments of a single chord. We derive a similar theorem about the product of the lengths of a secant and a chord.
Euclid proved (Elements, Book III, Propositions 20 and 21) proved that an angle inscribed in a ci... more Euclid proved (Elements, Book III, Propositions 20 and 21) proved that an angle inscribed in a circle is half as big as the central angle that subtends the same arc. We present a high-school-level version of Hestenes' GA-based proof ([1]) of that same theorem. We conclude with comments on the need for learners of GA to learn classical geometry as well. "Demonstrate that the inscribed angle φ is is half as large as the central angle θ .
Note: The Appendix to this new version gives an alternate--and much simpler--solution that does n... more Note: The Appendix to this new version gives an alternate--and much simpler--solution that does not use reflections. The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viete did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equa...
We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve t... more We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
We show how to express the representations of single, composite, and ``rotated" rotations in... more 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.
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra (GA), we sho... more As a high-school-level example of solving a problem via Geometric (Clifford) Algebra (GA), we show how to derive equations for the circle formed by the intersection of a plane with a sphere. Among the tasks that we will demonstrate are how to (1) express a plane via GA, (2) calculate the "reject" of a vector from a plane, and (3) express a circle as the rotation of a vector about a point.
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth,... more As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Karman Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
Using GA’s capacities for formulating refl ections, we solve an interesting
multiple-tangency pro... more Using GA’s capacities for formulating refl ections, we solve an interesting multiple-tangency problem. The solution is obtained in two ways, the easiest of which transforms the relevant refl ections into a single rotation. The solution is validated via a GeoGebra worksheet.
Using geometric algebra (GA), we derive a solution to the classic Snellius-Pothenot problem. We n... more Using geometric algebra (GA), we derive a solution to the classic Snellius-Pothenot problem. We note two types of cases where that solution does not apply, and present a GA-based solution for one of those cases. The points A, B, C and the angles α and β are known. Determine the location of the observer point P .
We show how to use the GA concept of the “rejection” of vectors, and
also the related outer produ... more We show how to use the GA concept of the “rejection” of vectors, and also the related outer product, to derive equations for the location and radius of a triangle’s incircle.
To help fill the need for examples of introductory-level problems that have been solved via Geome... more To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we derive the equation for a plane that is tangent to three given planes. The approach that we use determines the unit bivector of the tangent plane from the interior and exterior products of the vectors that connect the centers of the given spheres. A more-general version of this approach is presented in an appendix.
Make Two 3D Vectors Parallel by Rotating Them Around Separate Axes, 2022
To help fill the need for examples of introductory-level problems that have been solved via Geome... more To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which two unit vectors must be rotated in order to be parallel to each other. Among the ideas that we use are a transformation of the usual GA formula for rotations, and the use of GA products to eliminated variables in simultaneous equations. We will show the benefits of (1) examining an interactive GeoGebra construction before attempting a solution, and (2) considering a range of implications of given information.
Euclid proved (Elements, Book III, Propositions 20 and 21) that an angle inscribed in a circle is... more Euclid proved (Elements, Book III, Propositions 20 and 21) that an angle inscribed in a circle is half as big as the central angle that subtends the same arc. We present a high-school-level version of Hestenes' GA-based proof ([1]) of that same theorem. We conclude with comments on the need for learners of GA to learn classical geometry as well.
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenie... more After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to nd the intersections of spheres with lines, planes, and other spheres.
We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric... more We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric Algebra, about the product of the lengths of two segments of a single chord. We derive a similar theorem about the product of the lengths of a secant and a chord.
After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenie... more After reviewing the sorts of calculations for which Geometric Algebra (GA) is especially convenient, we identify a common theme through which those types of calculations can be used to find the intersections of spheres with lines, planes, and other spheres. "Derive an equation for the circle formed by the intersection of the sphere S with the plane P." 1 Figure 1: An example of a problem statement: "Derive an equation for the circle formed by the intersection of the sphere S with the plane P."
Via Geometric (Clifford) Algebra: Intersection of a Plane with a Sphere, 2019
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra (GA), we sho... more As a high-school-level example of solving a problem via Geometric (Clifford) Algebra (GA), we show how to derive equations for the circle formed by the intersection of a plane with a sphere. Among the tasks that we will demonstrate are how to (1) express a plane via GA, (2) calculate the "reject" of a vector from a plane, and (3) express a circle as the rotation of a vector about a point.
As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how... more As a high-school-level example of solving a problem via Geometric (Clifford) Algebra, we show how to calculate the distance and direction between two points on Earth, given the locations' latitudes and longitudes. We validate the results by comparing them to those obtained from online calculators. This example invites a discussion of the benefits of teaching spherical trigonometry (the usual way of solving such problems) at the high-school level versus teaching how to use Geometric Algebra for the same purpose. "Given the latitudes and longitudes of the starting point A and the destination B, calculate the distance between these points, and the bearing of B from A." 1
As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to de... more As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors. "Find the equation, in the form z = z 0 +λû, of the line of intersection between the planes P 1 : (x − a 1) ∧ ˆ B 1 and P 2 : (x − a 2) ∧ ˆ B 2 ." 1 Figure 1: Plane P 1 consists of the endpoints of all those vectors x that satisfy the condition expressed by the equation (x − a 1) ∧ ˆ B 1 = 0. Plane P 2 consists of the endpoints of all those vectors x that satisfy the condition expressed by the equation (x − a 2) ∧ ˆ B 2 = 0.
As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concept... more As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two vectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors de ned for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Because the shortage of worked-out examples at introductory levels is an obstacle to widespread a... more Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting
We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve t... more We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
Published times of the Earth's perihelions do not refer to the perihelions of the orbit that the ... more 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. " At the equinoxes, the Earth's axis of rotation lies within the plane that is perpendicular to the ecliptic and to the line connecting the centers of the Earth and Sun. " 1
We show how to calculate the projection of a vector, from an arbitrary direction, upon a given pl... more 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.
We show how to express the representations of single, composite, and ``rotated" rotations in GA t... more 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.
We solve Problem 7.2.3 from Alan Macdonald's excellent textbook, "Linear and Geometric Algebra" t... more We solve Problem 7.2.3 from Alan Macdonald's excellent textbook, "Linear and Geometric Algebra" to develop a formula for the "rotation of a rotation".
We show how to express the representation of a composite rotation in terms that allow the rotatio... more We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.
As an example for newcomers to GA who may have difficulty applying its identities to real problem... more As an example for newcomers to GA who may have difficulty applying its identities to real problems, we use those identities to prove the equivalence of two expressions for rotations of a vector. Rather than simply present the proof, we first review the relevant GA identities, then formulate and explore reasonable conjectures that lead, promptly, to a solution.
The story behind this article is instructive, and even a bit troubling. I wrote it in 1991 as a c... more The story behind this article is instructive, and even a bit troubling. I wrote it in 1991 as a continuation of part of my Doctoral thesis, which I’d completed a few years earlier. During that research, I’d found that scientists who’d done very fine laboratory work on Ostwald ripening during the 1960s had made a curious error in simple mass balances when deriving a rate equation for Ostwald ripening starting from the minimum-entropy-production-rate (MEPR) principle. That error led the 1960s scientists to reject (with commendable honesty) their hypothesis that the MEPR principle is applicable to Ostwald ripening. Like all the rest of us metallurgists back then, I didn’t catch that error, until I examined the derivation of the MEPR-based rate equation in detail during my thesis work. However, I didn’t manage to re-derive the rate equation fully until I took up the subject again in the early 1990s. The scientists who did such fine lab work in the 1960s would no doubt have been pleased to learn that their empirical results agreed quite well with predictions made by the corrected equation. Thus, those scientists were correct in their hypothesis about the MEPR principle’s applicability. I continue to wonder how we metallurgists overlooked, for more than two decades, the simple error that led those scientists to conclude, mistakenly but honestly, that they’d been wrong. I never did manage to publish this article, but the same derivations and analyses were published by other researchers within a few years. Some of the reviewers’ comments on the article are addressed in the second article in this document, “Comments on ‘Ostwald Ripening Growth Rate for Nonideal Systems with Significant Mutual Solubility’”.
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Papers by James Smith
multiple-tangency problem. The solution is obtained in two ways, the
easiest of which transforms the relevant refl ections into a single rotation.
The solution is validated via a GeoGebra worksheet.
also the related outer product, to derive equations for the location and
radius of a triangle’s incircle.
That error led the 1960s scientists to reject (with commendable honesty) their hypothesis that the MEPR principle is applicable to Ostwald ripening. Like all the rest of us metallurgists back then, I didn’t catch that error, until I examined the derivation of the MEPR-based rate equation in detail during my thesis work. However, I didn’t manage to re-derive the rate equation fully until I took up the subject again in the early 1990s. The scientists who did such fine lab work in the 1960s would no doubt have been pleased to learn that their empirical results agreed quite well with predictions made by the corrected equation. Thus, those scientists were correct in their hypothesis about the MEPR principle’s applicability.
I continue to wonder how we metallurgists overlooked, for more than two decades, the simple error that led those scientists to conclude, mistakenly but honestly, that they’d been wrong.
I never did manage to publish this article, but the same derivations and analyses were published by other researchers within a few years. Some of the reviewers’ comments on the article are addressed in the second article in this document, “Comments on ‘Ostwald Ripening Growth Rate for Nonideal Systems with Significant Mutual Solubility’”.