Geometric algebra (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Geometric algebra" in English language version.

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ams.org

mathscinet.ams.org

Artin 1988. Artin, Emil (1988) [1957], Geometric algebra, Wiley Classics Library, Wiley, doi:10.1002/9781118164518, ISBN 978-0-471-60839-4, MR 1009557

archives-ouvertes.fr

hal.archives-ouvertes.fr

Breuils et al. 2019. Breuils, Stéphane; Fuchs, Laurent; Hitzer, Eckhard; Nozick, Vincent; Sugimoto, Akihiro (2019), "Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment" (PDF), Advances in Applied Clifford Algebras, 29 (3), doi:10.1007/s00006-019-0974-z, S2CID 253597480

arxiv.org

Lundholm & Svensson 2009, pp. 58 et seq. Lundholm, Douglas; Svensson, Lars (2009), "Clifford Algebra, Geometric Algebra and Applications", arXiv:0907.5356v1 [math-ph]
Lundholm & Svensson 2009, p. 58. Lundholm, Douglas; Svensson, Lars (2009), "Clifford Algebra, Geometric Algebra and Applications", arXiv:0907.5356v1 [math-ph]
Francis & Kosowsky 2008. Francis, Matthew R.; Kosowsky, Arthur (2008), "The Construction of Spinors in Geometric Algebra", Annals of Physics, 317 (2): 383–409, arXiv:math-ph/0403040v2, Bibcode:2005AnPhy.317..383F, doi:10.1016/j.aop.2004.11.008, S2CID 119632876
Wu 2022. Wu, Bofeng (2022), "A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession", Scientific Reports, 12 (1): 3981, arXiv:2111.07353, Bibcode:2022NatSR..12.3981W, doi:10.1038/s41598-022-06895-0, PMC 8901677, PMID 35256628
Sokolov 2013. Sokolov, Andrey (2013), Clifford algebra and the projective model of Minkowski (Pseudo-Euclidean) spaces, arXiv:1307.4179
Lasenby 2004. Lasenby, Anthony (2004), "Conformal Models of de Sitter Space, Initial Conditions for Inflation and the CMB", AIP Conference Proceedings, vol. 736, pp. 53–70, arXiv:astro-ph/0411579, doi:10.1063/1.1835174, S2CID 18034896

av8n.com

"Electromagnetism using Geometric Algebra versus Components". Retrieved 2013-03-19.

books.google.com

Li 2008, p. 411. Li, Hongbo (2008), Invariant Algebras and Geometric Reasoning, World Scientific, ISBN 9789812770110. Chapter 1 as PDF
Grassmann 1844. Grassmann, Hermann (1844), Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, Leipzig: O. Wigand, OCLC 20521674

cam.ac.uk

geometry.mrao.cam.ac.uk

Lasenby, Lasenby & Doran 2000. Lasenby, Joan; Lasenby, Anthony N.; Doran, Chris J. L. (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19

repository.cam.ac.uk

Hadfield & Lasenby 2020. Hadfield, Hugo; Lasenby, Joan (2020), "Constrained Dynamics in Conformal and Projective Geometric Algebra", Advances in Computer Graphics, Lecture Notes in Computer Science, vol. 12221, pp. 459–471, doi:10.1007/978-3-030-61864-3_39, ISBN 978-3-030-61863-6, S2CID 224820480

cambridge.org

assets.cambridge.org

Doran & Lasenby 2003, p. 102. Doran, Chris J. L.; Lasenby, Anthony N. (2003), Geometric Algebra for Physicists (PDF), Cambridge University Press, ISBN 978-0-521-71595-9, archived (PDF) from the original on 2009-01-06

davidhestenes.net

Hestenes 2003. Hestenes, David (2003), "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics" (PDF), Am. J. Phys., 71 (2): 104–121, Bibcode:2003AmJPh..71..104H, CiteSeerX 10.1.1.649.7506, doi:10.1119/1.1522700{{citation}}: CS1 maint: url-status (link)
Hestenes 2005. Hestenes, David (2005), Introduction to Primer for Geometric Algebra

doi.org

This definition follows Dorst, Fontijne & Mann (2007) and Perwass (2009) – the left contraction used by Dorst replaces the ("fat dot") inner product that Perwass uses, consistent with Perwass's constraint that grade of ⁠ $A$ ⁠ may not exceed that of ⁠ $B$ ⁠. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969 Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Dorst appears to merely assume $B^{+}$ such that ⁠ $B\;\rfloor \;B^{+}=1$ ⁠, whereas Perwass (2009) defines ⁠ $B^{+}=B^{\dagger }/(B\;\rfloor \;B^{\dagger })$ ⁠, where $B^{\dagger }$ is the conjugate of ⁠ $B$ ⁠, equivalent to the reverse of $B$ up to a sign. Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Hestenes 1986, p. 6. Hestenes, David (1986), "A Unified Language for Mathematics and Physics", in J.S.R. Chisholm; A.K. Commons (eds.), Clifford Algebras and Their Applications in Mathematical Physics, NATO ASI Series (Series C), vol. 183, Springer, pp. 1–23, doi:10.1007/978-94-009-4728-3_1, ISBN 978-94-009-4728-3
Hestenes 2003. Hestenes, David (2003), "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics" (PDF), Am. J. Phys., 71 (2): 104–121, Bibcode:2003AmJPh..71..104H, CiteSeerX 10.1.1.649.7506, doi:10.1119/1.1522700{{citation}}: CS1 maint: url-status (link)
Doran 1994. Doran, Chris J. L. (1994), Geometric Algebra and its Application to Mathematical Physics (PhD thesis), University of Cambridge, doi:10.17863/CAM.16148, hdl:1810/251691, OCLC 53604228
Lasenby, Lasenby & Doran 2000. Lasenby, Joan; Lasenby, Anthony N.; Doran, Chris J. L. (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19
Hildenbrand et al. 2004. Hildenbrand, Dietmar; Fontijne, Daniel; Perwass, Christian; Dorst, Leo (2004), "Geometric Algebra and its Application to Computer Graphics" (PDF), Proceedings of Eurographics 2004, doi:10.2312/egt.20041032, archived (PDF) from the original on 2015-09-06
Aragón, Aragón & Rodríguez 1997, p. 101. Aragón, G.; Aragón, J.L.; Rodríguez, M.A. (1997), "Clifford Algebras and Geometric Algebra", Advances in Applied Clifford Algebras, 7 (2): 91–102, doi:10.1007/BF03041220, S2CID 120860757
Wilmot 1988b, p. 2346. Wilmot, G.P. (1988b), "Clifford algebra and the Pfaffian expansion", Journal of Mathematical Physics, 29: 2346–2350, doi:10.1063/1.528118
Francis & Kosowsky 2008. Francis, Matthew R.; Kosowsky, Arthur (2008), "The Construction of Spinors in Geometric Algebra", Annals of Physics, 317 (2): 383–409, arXiv:math-ph/0403040v2, Bibcode:2005AnPhy.317..383F, doi:10.1016/j.aop.2004.11.008, S2CID 119632876
Selig 2005. Selig, J.M. (2005). Geometric Fundamentals of Robotics. Monographs in Computer Science. New York, NY: Springer New York. doi:10.1007/b138859. ISBN 978-0-387-20874-9.
Hadfield & Lasenby 2020. Hadfield, Hugo; Lasenby, Joan (2020), "Constrained Dynamics in Conformal and Projective Geometric Algebra", Advances in Computer Graphics, Lecture Notes in Computer Science, vol. 12221, pp. 459–471, doi:10.1007/978-3-030-61864-3_39, ISBN 978-3-030-61863-6, S2CID 224820480
Hrdina, Návrat & Vašík 2018. Hrdina, Jaroslav; Návrat, Aleš; Vašík, Petr (2018), "Geometric Algebra for Conics", Advances in Applied Clifford Algebras, 28 (3), doi:10.1007/s00006-018-0879-2, S2CID 125649450
Wu 2022. Wu, Bofeng (2022), "A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession", Scientific Reports, 12 (1): 3981, arXiv:2111.07353, Bibcode:2022NatSR..12.3981W, doi:10.1038/s41598-022-06895-0, PMC 8901677, PMID 35256628
Lasenby 2004. Lasenby, Anthony (2004), "Conformal Models of de Sitter Space, Initial Conditions for Inflation and the CMB", AIP Conference Proceedings, vol. 736, pp. 53–70, arXiv:astro-ph/0411579, doi:10.1063/1.1835174, S2CID 18034896
Dorst 2016. Dorst, Leo (2016), "3D Oriented Projective Geometry Through Versors of ⁠ $\mathbb {R} ^{3,3}$ ⁠", Advances in Applied Clifford Algebras, 26 (4): 1137–1172, doi:10.1007/s00006-015-0625-y
Perwass 2009. Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Breuils et al. 2019. Breuils, Stéphane; Fuchs, Laurent; Hitzer, Eckhard; Nozick, Vincent; Sugimoto, Akihiro (2019), "Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment" (PDF), Advances in Applied Clifford Algebras, 29 (3), doi:10.1007/s00006-019-0974-z, S2CID 253597480
Easter & Hitzer 2017. Easter, Robert Benjamin; Hitzer, Eckhard (2017), "Double Conformal Geometric Algebra", Advances in Applied Clifford Algebras, 27 (3): 2175–2199, doi:10.1007/s00006-017-0784-0, S2CID 253600526
Perwass 2009, §3.2.10.2 p. 83. Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Artin 1988. Artin, Emil (1988) [1957], Geometric algebra, Wiley Classics Library, Wiley, doi:10.1002/9781118164518, ISBN 978-0-471-60839-4, MR 1009557

gaalop.de

Hildenbrand et al. 2004. Hildenbrand, Dietmar; Fontijne, Daniel; Perwass, Christian; Dorst, Leo (2004), "Geometric Algebra and its Application to Computer Graphics" (PDF), Proceedings of Eurographics 2004, doi:10.2312/egt.20041032, archived (PDF) from the original on 2015-09-06

geometricalgebra.net

This should not be confused with Hestenes's irregular generalization ⁠ $\textstyle C\bullet _{\text{H}}D:=\sum _{r\neq 0,s\neq 0}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{\vert s-r\vert }$ ⁠, where the distinguishing notation is from Dorst, Fontijne & Mann (2007), p. 590, §B.1, which makes the point that scalar components must be handled separately with this product. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
This definition follows Dorst, Fontijne & Mann (2007) and Perwass (2009) – the left contraction used by Dorst replaces the ("fat dot") inner product that Perwass uses, consistent with Perwass's constraint that grade of ⁠ $A$ ⁠ may not exceed that of ⁠ $B$ ⁠. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969 Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Dorst, Fontijne & Mann 2007, p. 204. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
Dorst, Fontijne & Mann 2007, pp. 177–182. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
Dorst, Fontijne & Mann 2007, §3.6 p. 85. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969

github.com

Wilmot 2023. Wilmot, G.P. (2023). "The Algebra Of Geometry". GitHub.

handle.net

hdl.handle.net

Doran 1994. Doran, Chris J. L. (1994), Geometric Algebra and its Application to Mathematical Physics (PhD thesis), University of Cambridge, doi:10.17863/CAM.16148, hdl:1810/251691, OCLC 53604228

harvard.edu

ui.adsabs.harvard.edu

This definition follows Dorst, Fontijne & Mann (2007) and Perwass (2009) – the left contraction used by Dorst replaces the ("fat dot") inner product that Perwass uses, consistent with Perwass's constraint that grade of ⁠ $A$ ⁠ may not exceed that of ⁠ $B$ ⁠. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969 Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Dorst appears to merely assume $B^{+}$ such that ⁠ $B\;\rfloor \;B^{+}=1$ ⁠, whereas Perwass (2009) defines ⁠ $B^{+}=B^{\dagger }/(B\;\rfloor \;B^{\dagger })$ ⁠, where $B^{\dagger }$ is the conjugate of ⁠ $B$ ⁠, equivalent to the reverse of $B$ up to a sign. Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Hestenes 2003. Hestenes, David (2003), "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics" (PDF), Am. J. Phys., 71 (2): 104–121, Bibcode:2003AmJPh..71..104H, CiteSeerX 10.1.1.649.7506, doi:10.1119/1.1522700{{citation}}: CS1 maint: url-status (link)
Lasenby, Lasenby & Doran 2000. Lasenby, Joan; Lasenby, Anthony N.; Doran, Chris J. L. (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19
Vaz & da Rocha 2016, p. 58, Theorem 3.1. Vaz, Jayme; da Rocha, Roldão (2016), An Introduction to Clifford Algebras and Spinors, Oxford University Press, Bibcode:2016icas.book.....V, ISBN 978-0-19-878292-6
Francis & Kosowsky 2008. Francis, Matthew R.; Kosowsky, Arthur (2008), "The Construction of Spinors in Geometric Algebra", Annals of Physics, 317 (2): 383–409, arXiv:math-ph/0403040v2, Bibcode:2005AnPhy.317..383F, doi:10.1016/j.aop.2004.11.008, S2CID 119632876
Vaz & da Rocha 2016, §2.8. Vaz, Jayme; da Rocha, Roldão (2016), An Introduction to Clifford Algebras and Spinors, Oxford University Press, Bibcode:2016icas.book.....V, ISBN 978-0-19-878292-6
Wu 2022. Wu, Bofeng (2022), "A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession", Scientific Reports, 12 (1): 3981, arXiv:2111.07353, Bibcode:2022NatSR..12.3981W, doi:10.1038/s41598-022-06895-0, PMC 8901677, PMID 35256628
Perwass 2009. Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Perwass 2009, §3.2.10.2 p. 83. Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6

nih.gov

ncbi.nlm.nih.gov

Wu 2022. Wu, Bofeng (2022), "A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession", Scientific Reports, 12 (1): 3981, arXiv:2111.07353, Bibcode:2022NatSR..12.3981W, doi:10.1038/s41598-022-06895-0, PMC 8901677, PMID 35256628

pubmed.ncbi.nlm.nih.gov

Wu 2022. Wu, Bofeng (2022), "A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession", Scientific Reports, 12 (1): 3981, arXiv:2111.07353, Bibcode:2022NatSR..12.3981W, doi:10.1038/s41598-022-06895-0, PMC 8901677, PMID 35256628

projectivegeometricalgebra.org

"Projective Geometric Algebra". projectivegeometricalgebra.org. Retrieved 2023-10-03.

psu.edu

citeseerx.ist.psu.edu

Hestenes 2003. Hestenes, David (2003), "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics" (PDF), Am. J. Phys., 71 (2): 104–121, Bibcode:2003AmJPh..71..104H, CiteSeerX 10.1.1.649.7506, doi:10.1119/1.1522700{{citation}}: CS1 maint: url-status (link)

semanticscholar.org

api.semanticscholar.org

Lasenby, Lasenby & Doran 2000. Lasenby, Joan; Lasenby, Anthony N.; Doran, Chris J. L. (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19
Aragón, Aragón & Rodríguez 1997, p. 101. Aragón, G.; Aragón, J.L.; Rodríguez, M.A. (1997), "Clifford Algebras and Geometric Algebra", Advances in Applied Clifford Algebras, 7 (2): 91–102, doi:10.1007/BF03041220, S2CID 120860757
Francis & Kosowsky 2008. Francis, Matthew R.; Kosowsky, Arthur (2008), "The Construction of Spinors in Geometric Algebra", Annals of Physics, 317 (2): 383–409, arXiv:math-ph/0403040v2, Bibcode:2005AnPhy.317..383F, doi:10.1016/j.aop.2004.11.008, S2CID 119632876
Hadfield & Lasenby 2020. Hadfield, Hugo; Lasenby, Joan (2020), "Constrained Dynamics in Conformal and Projective Geometric Algebra", Advances in Computer Graphics, Lecture Notes in Computer Science, vol. 12221, pp. 459–471, doi:10.1007/978-3-030-61864-3_39, ISBN 978-3-030-61863-6, S2CID 224820480
Hrdina, Návrat & Vašík 2018. Hrdina, Jaroslav; Návrat, Aleš; Vašík, Petr (2018), "Geometric Algebra for Conics", Advances in Applied Clifford Algebras, 28 (3), doi:10.1007/s00006-018-0879-2, S2CID 125649450
Lasenby 2004. Lasenby, Anthony (2004), "Conformal Models of de Sitter Space, Initial Conditions for Inflation and the CMB", AIP Conference Proceedings, vol. 736, pp. 53–70, arXiv:astro-ph/0411579, doi:10.1063/1.1835174, S2CID 18034896
Breuils et al. 2019. Breuils, Stéphane; Fuchs, Laurent; Hitzer, Eckhard; Nozick, Vincent; Sugimoto, Akihiro (2019), "Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment" (PDF), Advances in Applied Clifford Algebras, 29 (3), doi:10.1007/s00006-019-0974-z, S2CID 253597480
Easter & Hitzer 2017. Easter, Robert Benjamin; Hitzer, Eckhard (2017), "Double Conformal Geometric Algebra", Advances in Applied Clifford Algebras, 27 (3): 2175–2199, doi:10.1007/s00006-017-0784-0, S2CID 253600526

springer.com

link.springer.com

Selig 2005. Selig, J.M. (2005). Geometric Fundamentals of Robotics. Monographs in Computer Science. New York, NY: Springer New York. doi:10.1007/b138859. ISBN 978-0-387-20874-9.

web.archive.org

Li 2008, p. 411. Li, Hongbo (2008), Invariant Algebras and Geometric Reasoning, World Scientific, ISBN 9789812770110. Chapter 1 as PDF
Lasenby, Lasenby & Doran 2000. Lasenby, Joan; Lasenby, Anthony N.; Doran, Chris J. L. (2000), "A Unified Mathematical Language for Physics and Engineering in the 21st Century" (PDF), Philosophical Transactions of the Royal Society A, 358 (1765): 21–39, Bibcode:2000RSPTA.358...21L, doi:10.1098/rsta.2000.0517, S2CID 91884543, archived (PDF) from the original on 2015-03-19
Hildenbrand et al. 2004. Hildenbrand, Dietmar; Fontijne, Daniel; Perwass, Christian; Dorst, Leo (2004), "Geometric Algebra and its Application to Computer Graphics" (PDF), Proceedings of Eurographics 2004, doi:10.2312/egt.20041032, archived (PDF) from the original on 2015-09-06
Doran & Lasenby 2003, p. 102. Doran, Chris J. L.; Lasenby, Anthony N. (2003), Geometric Algebra for Physicists (PDF), Cambridge University Press, ISBN 978-0-521-71595-9, archived (PDF) from the original on 2009-01-06

worldcat.org

search.worldcat.org

This should not be confused with Hestenes's irregular generalization ⁠ $\textstyle C\bullet _{\text{H}}D:=\sum _{r\neq 0,s\neq 0}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{\vert s-r\vert }$ ⁠, where the distinguishing notation is from Dorst, Fontijne & Mann (2007), p. 590, §B.1, which makes the point that scalar components must be handled separately with this product. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
This definition follows Dorst, Fontijne & Mann (2007) and Perwass (2009) – the left contraction used by Dorst replaces the ("fat dot") inner product that Perwass uses, consistent with Perwass's constraint that grade of ⁠ $A$ ⁠ may not exceed that of ⁠ $B$ ⁠. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969 Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Bibcode:2009gaae.book.....P, doi:10.1007/978-3-540-89068-3, ISBN 978-3-540-89067-6
Hestenes 1966. Hestenes, David (1966), Space–time Algebra, Gordon and Breach, ISBN 978-0-677-01390-9, OCLC 996371
Doran 1994. Doran, Chris J. L. (1994), Geometric Algebra and its Application to Mathematical Physics (PhD thesis), University of Cambridge, doi:10.17863/CAM.16148, hdl:1810/251691, OCLC 53604228
Dorst, Fontijne & Mann 2007, p. 204. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
Dorst, Fontijne & Mann 2007, pp. 177–182. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
Dorst, Fontijne & Mann 2007, §3.6 p. 85. Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-12-369465-2, OCLC 132691969
Grassmann 1844. Grassmann, Hermann (1844), Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, Leipzig: O. Wigand, OCLC 20521674