3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$ R 3 , 3

Dorst, Leo

doi:10.1007/s00006-015-0625-y

Cited by 20 publications

(14 citation statements)

References 12 publications

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“…There are, however, other models that use Clifford algebras but are based on different ideas, and therefore on different ways of representing geometric objects and their transformations. One such model was provided by Dorst [7] and is based on the study of oriented projective transformations of lines. In [7] an interesting comparison is made between the models of Dorst, GM and Klawitter [18].…”

Section: Discussionmentioning

confidence: 99%

“…One such model was provided by Dorst [7] and is based on the study of oriented projective transformations of lines. In [7] an interesting comparison is made between the models of Dorst, GM and Klawitter [18]. Because of the relationship between our model and GM one, the comparison made in [7] also serves as a comparison between the model presented here and the ones of Dorst and of Klawitter, although the criteria used in [7] are more relevant to projective geometry than they are to computer graphics.…”

Section: Discussionmentioning

confidence: 99%

“…In [7] an interesting comparison is made between the models of Dorst, GM and Klawitter [18]. Because of the relationship between our model and GM one, the comparison made in [7] also serves as a comparison between the model presented here and the ones of Dorst and of Klawitter, although the criteria used in [7] are more relevant to projective geometry than they are to computer graphics. There is also the PGA model developed by Gunn [19] based on a dual projectivized Clifford algebra.…”

Section: Discussionmentioning

confidence: 99%

“…In fact, quaternions [1,2] were shown to be an efficient tool for the interpolation of sequences of orientations in computer graphics, and other studies [3] involving applications of other algebraic systems in computer graphics have appeared in the literature, together with studies trying to identify geometric spaces that may be more suitable as an ambient space for computer graphics [4,5]. Models based on Clifford algebras [6,7,8] have been proposed as an alternative algebraic framework for computer graphics.…”

Section: Introductionmentioning

confidence: 99%

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On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

Vaz¹,

Mann²

2019

Preprint

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We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra Cℓ 3,3 of the quadratic space R 3,3 . We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspective projection can be written in this Clifford algebra as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation. An important point is that the expression for the operations of reflection and rotation in Cℓ 3,3 preserve the subspaces that can be associated with the algebras Cℓ 3,0 and Cℓ 0,3 , so that reflection and rotation can be expressed in terms of Cℓ 3,0 or Cℓ 0,3 , as well-known. However, all other operations mix those subspaces in such a way that they need to be expressed in terms of the full Clifford algebra Cℓ 3,3 . An essential aspect of our formulation is the representation of points in terms of objects called paravectors. Paravectors have been used previously to represents points in terms of an algebra closely related to the Clifford algebra Cℓ 3,3 . We compare these different approaches.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

Vaz¹,

Mann²

2019

Preprint

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“…One significant difference is that their model uses an extra dimension in the vector space, resulting in a Clifford algebra four times the size of our model. Dorst [15] develops a model for the study of oriented projective transformations of lines; our representation of lines is similar to that of Dorst but ours is a model of affine space rather than focused on lines, and our model can be generalized to arbitrary dimensions. A deeper analysis of both approaches should be done after we formulate a version of our work using the same algebra used in [15].…”

Section: Discussionmentioning

confidence: 99%

Paravectors and the Geometry of 3D Euclidean Space

Vaz

Mann

2018

Adv. Appl. Clifford Algebras

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We introduce the concept of paravectors to describe the geometry of points in a three dimensional space. After defining a suitable product of paravectors, we introduce the concepts of biparavectors and triparavectors to describe line segments and plane fragments in this space. A key point in this product of paravectors is the notion of the orientation of a point, in such a way that biparavectors representing line segments are the result of the product of points with opposite orientations. Incidence relations can also be formulated in terms of the product of paravectors. To study the transformations of points, lines, and planes, we introduce an algebra of transformations that is analogous to the algebra of creation and annihilation operators in quantum theory. The paravectors, biparavectors and triparavectors are mapped into this algebra and their transformations are studied; we show that this formalism describes in an unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale transformation. Using the concept of Hodge duality, we define a new operation called cotranslation, and show that the operation of perspective projection can be written as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation.Mathematics Subject Classification: 15A66, 15A75, 68U05.It can be proven [10] that this definition is equivalent tofor A k ∈ k (R 3 ). It follows then that ⋆ 1 = e 1 ∧ e 2 ∧ e 3 , ⋆ e 1 = e 2 ∧ e 3 , ⋆ e 2 = e 3 ∧ e 1 , ⋆ e 3 = e 1 ∧ e 2 , ⋆ e 1 ∧ e 2 = e 3 , ⋆ e 3 ∧ e 1 = e 2 , ⋆ e 2 ∧ e 3 = e 1 , ⋆ e 1 ∧ e 2 ∧ e 3 = 1. ParavectorsLet us fix the notation −1 (R 3 ) = ∅ and 4 (R 3 ) = ∅. We define k (V ) as k (R 3 ) = k−1 (R 3 ) ⊕ k (R 3 ), k = 0, 1, 2, 3, 4.

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Projective duality encodes complementary orientations in geometric algebras

Dorst

2023

Math Methods in App Sciences

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Oriented elements are part of geometry, and they come in two complementary types: intrinsic and extrinsic. Those different orientation types manifest themselves by behaving differently under reflection. Projective dualization in geometric algebras can encode them, or conversely, orientation types inform the interpretation of dualization. Oriented elements may be combined using the meet operation, and the dual‐join (which is here introduced for that purpose). We discuss algebra‐based implementation techniques on how to double the representational power of a geometric algebra, without explicitly doubling the algebra.

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3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$ R 3 , 3

Cited by 20 publications

References 12 publications

On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

Paravectors and the Geometry of 3D Euclidean Space

Projective duality encodes complementary orientations in geometric algebras

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