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- 22:10, 19 June 2022 Dual flector (hist | edit) [127 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''dual flector'' is an operator that performs an improper isometry in dual Euclidean space. == See Also == * Dual motor")
- 22:10, 19 June 2022 Dual motor (hist | edit) [124 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''dual motor'' is an operator that performs a proper isometry in dual Euclidean space. == See Also == * Dual flector")
- 22:09, 19 June 2022 Dual reflection (hist | edit) [130 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''dual reflection'' is an improper isometry of dual Euclidean space. == See Also == * Dual rotation * Dual translation")
- 22:08, 19 June 2022 Dual rotation (hist | edit) [127 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''dual rotation'' is a proper isometry of dual Euclidean space. == See Also == * Dual translation * Dual reflection")
- 06:39, 17 June 2022 Dual translation (hist | edit) [1,831 bytes] Eric Lengyel (talk | contribs) (Created page with "An ''antitranslation'' is a proper isometry of dual Euclidean space. The specific kind of antimotor :$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + 1$$ performs a perspective projection. The exact antitranslation calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Antitranslation |- | style="padding: 12px;" | Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \math...") originally created as "Antitranslation"
- 04:13, 17 June 2022 Duality (hist | edit) [3,272 bytes] Eric Lengyel (talk | contribs) (Created page with "400px|thumb|right|'''Figure 1.''' The coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as the one-dimensional span of a single vector representing a homogeneous point or as the $$(n - 1)$$-dimensional span of all orthogonal vectors representing a homogeneous plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other. The concept of duality can be understood geometric...")
- 00:42, 17 June 2022 Transformation groups (hist | edit) [6,417 bytes] Eric Lengyel (talk | contribs) (Created page with "In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, every Euclidean isometry of 3D space can be represented by a motor $$\mathbf Q$$ of the form :$$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ or by a flector $$\mathbf G$$ of the form :$$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \m...")
- 03:32, 18 May 2022 Quaternion (hist | edit) [3,091 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as :$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ , where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$. == Quaternions in 3D Geometric Algebra == The set of quaternions corresponds to a part of the three-dimensional geometric algebra. Specifically, the quaternio...")
- 03:30, 18 May 2022 Scalar (hist | edit) [491 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''scalar'' in a geometric algebra is an element having grade 0. Scalars are just ordinary real numbers, and they do not involve any basis vectors. The basis element representing the unit scalar is denoted by $$\mathbf 1$$, a boldface number one. The unit scalar $$\mathbf 1$$ is the multiplicative identity of the geometric product. For a general element $$\mathbf a$$, the notation $$a_{\mathbf 1}$$ means the scalar component of $$\mathbf a$$. == See Also ==...")
- 03:30, 18 May 2022 Antiscalar (hist | edit) [574 bytes] Eric Lengyel (talk | contribs) (Created page with "An ''antiscalar'' in a geometric algebra is an element having antigrade 0. Antiscalars are multiples of the volume element given by the wedge product of all basis vectors. The basis element representing the unit antiscalar is denoted by $$\large\unicode{x1D7D9}$$, a double-struck number one. The unit antiscalar $$\large\unicode{x1D7D9}$$ is the multiplicative identity of the geometric antiproduct. For a general element $$\mathbf a$$, the notation $$a_{\larg...")
- 03:28, 18 May 2022 Antivector (hist | edit) [261 bytes] Eric Lengyel (talk | contribs) (Created page with "An ''antivector'' in a geometric algebra is an element composed entirely of components having antigrade 1. In an ''n''-dimensional geometric algebra, these are the elements having grade $$n - 1$$. == See Also == * Vector * Bivector * Trivector")
- 03:28, 18 May 2022 Trivector (hist | edit) [166 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''trivector'' in a geometric algebra is an element composed entirely of components having grade 3. == See Also == * Vector * Bivector * Antivector")
- 03:27, 18 May 2022 Bivector (hist | edit) [166 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''bivector'' in a geometric algebra is an element composed entirely of components having grade 2. == See Also == * Vector * Trivector * Antivector")
- 03:27, 18 May 2022 Vector (hist | edit) [166 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''vector'' in a geometric algebra is an element composed entirely of components having grade 1. == See Also == * Bivector * Trivector * Antivector")
- 03:20, 18 May 2022 Rigid Geometric Algebra for 2D Space (hist | edit) [17,201 bytes] Eric Lengyel (talk | contribs) (Created page with "== Introduction == thumb|right|400px|'''Table 1.''' The 8 basis elements of the 3D rigid geometric algebra. In the three-dimensional rigid geometric algebra, there are 8 graded basis elements. These are listed in Table 1. There is a single ''scalar'' basis element $$\mathbf 1$$, and its multiples correspond to the real numbers, which are values that have no dimensions. There are three ''vector'' basis elements named $$\mathbf e_1$$, $$\mathbf e_...")
- 03:12, 18 May 2022 Magnitude (hist | edit) [1,046 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows: :$$\mathbf z = x + y {\large\unicode{x1d7d9}}$$ Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a bulk and a weight, and it is unitized by making the magnitude of its weight one. === Examples...")
- 03:10, 18 May 2022 Euclidean distance (hist | edit) [3,340 bytes] Eric Lengyel (talk | contribs) (Created page with "The Euclidean distance between geometric objects can be measured by using commutators to calculate homogeneous magnitudes. The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. The points, lines, an...")
- 01:59, 18 May 2022 Commutators (hist | edit) [1,113 bytes] Eric Lengyel (talk | contribs) (Created page with "In geometric algebra, there are four ''commutator'' products defined as follows. :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode...")
- 01:57, 18 May 2022 Projections (hist | edit) [7,117 bytes] Eric Lengyel (talk | contribs) (Created page with "Projections and antiprojections of one geometric object onto another can be accomplished using interior products as described below. The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. == Projection == The geometric projection of an object $$\mathbf a$$ onto an object $$\mathbf b$$ is given by the general formula $$(\mathbf b_\unicode{x25CB} \mathbin{\unicode{...")
- 01:55, 18 May 2022 Join and meet (hist | edit) [5,434 bytes] Eric Lengyel (talk | contribs) (Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The points, lines, and planes appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ :$$\mathbf...")
- 01:54, 18 May 2022 Dot products (hist | edit) [1,779 bytes] Eric Lengyel (talk | contribs) (Created page with "The ''dot product'' is the inner product in geometric algebra, and it makes up the scalar part of the geometric product. There are two products with symmetric properties called the dot product and antidot product. The dot product and antidot product are important for the calculation of norms. == Dot Product == The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$ and r...")
- 01:53, 18 May 2022 Interior products (hist | edit) [3,406 bytes] Eric Lengyel (talk | contribs) (Created page with "The left and right ''interior products'' are special products in geometric algebra that are useful for performing projections. These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there are several possible interior products. We define the interior products in terms of the left and right complements. Interior products are also known as contraction products. == Left and Right Interior Prod...")
- 01:48, 18 May 2022 Geometric products (hist | edit) [5,062 bytes] Eric Lengyel (talk | contribs) (Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct...")
- 01:47, 18 May 2022 Exterior products (hist | edit) [6,603 bytes] Eric Lengyel (talk | contribs) (Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. The exterior product between two elements $$\mathbf a$$ and $$\mathbf b$$ generally combines their spatial extents, and the magnitude of the result indicates how close they are to being orthogonal. If the spatial extents of $$\m...")
- 01:43, 18 May 2022 Reverses (hist | edit) [2,041 bytes] Eric Lengyel (talk | contribs) (Created page with "''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations. Every basis element $$\mathbf a$$ has a ''reverse'', which we denote by $$\mathbf{\tilde a}$$, that satisfies the equation :$$\mathbf a \wedge \mathbf{\tilde a} = \mathbf 1$$ . There is also an ''antireverse'', which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}$$, that satisfies the equation :$$\mathbf a \vee \smash{\mathbf{\underset{\Lar...")
- 01:42, 18 May 2022 Complements (hist | edit) [6,944 bytes] Eric Lengyel (talk | contribs) (Created page with "''Complements'' are unary operations in geometric algebra that perform a specific type of dualization. Every basis element $$\mathbf a$$ has a ''right complement'', which we denote by $$\overline{\mathbf a}$$, that satisfies the equation :$$\mathbf a \wedge \overline{\mathbf a} = {\large\unicode{x1D7D9}}$$ . There is also a ''left complement'', which we denote by $$\underline{\mathbf a}$$, that satisfies the equation :$$\underline{\mathbf a} \wedge \mathbf a = {\larg...")
- 01:42, 18 May 2022 Unitization (hist | edit) [2,463 bytes] Eric Lengyel (talk | contribs) (Created page with "''Unitization'' is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''. An element $$\mathbf a$$ is unitized by calculating :$$\mathbf{\hat a} = \dfrac{\mathbf a}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf a}{\sqrt{\mathbf a \mathbin{\unicode{x25CB}} \smash{\ma...")
- 01:41, 18 May 2022 Bulk and weight (hist | edit) [3,779 bytes] Eric Lengyel (talk | contribs) (Created page with "The components of an element of a rigid geometric algebra can be divided into two groups called the ''bulk'' and the ''weight'' of the element. The bulk of an element $$\mathbf a$$ is denoted by $$\mathbf a_\unicode{x25CF}$$, and it consists of the components of $$\mathbf a$$ that do not have the projective basis vector as a factor. The weight is denoted by $$\mathbf a_\unicode{x25CB}$$, and it consists of the components that do have the projective basis vector as a fact...")
- 01:39, 18 May 2022 Grade and antigrade (hist | edit) [823 bytes] Eric Lengyel (talk | contribs) (Created page with "The ''grade'' of a basis element in a geometric algebra is equal to the number of basis vectors present in its factorization. An arbitrary element whose components all have the same grade is also said to have that grade. The ''antigrade'' of a basis element is equal to the number of basis vectors absent from its factorization. The grade of an element $$a$$ is denoted by $$\operatorname{gr}(a)$$, and the antigrade is denoted by $$\operatorname{ag}(a)$$. In an ''n''-dime...")
- 01:38, 18 May 2022 Geometric norm (hist | edit) [10,130 bytes] Eric Lengyel (talk | contribs) (Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf a$$, d...")
- 01:37, 18 May 2022 Geometric property (hist | edit) [2,621 bytes] Eric Lengyel (talk | contribs) (Created page with "An element $$\mathbf a$$ of a geometric algebra possesses the ''geometric property'' if and only if the geometric product between $$\mathbf a$$ and its own reverse is a scalar, which is given by the dot product, and the geometric antiproduct between $$\mathbf a$$ and its own antireverse is an antiscalar, which is given by the antidot product. That is, :$$\mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde a} = \mathbf a \mathbin{\unicode{x25CF}} \mathbf{\...")
- 01:36, 18 May 2022 Transflection (hist | edit) [3,727 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''transflection'' is an improper isometry of Euclidean space consisting of a reflection through a plane and a translation parallel to the same plane. All combinations of a reflection and a translation, even if the original translation vector is not parallel to the original reflection plane, can be formulated as a transflection with respect to some plane. The specific kind of flector :$$\mathbf H = s_x \mathbf e_{1} + s_y \mathbf e_{2} + s_z \mathbf e_{3}...")
- 01:36, 18 May 2022 Inversion (hist | edit) [1,890 bytes] Eric Lengyel (talk | contribs) (Created page with "An ''inversion'' is an improper isometry of Euclidean space. When used as an operator in the sandwich product, a unitized point $$\mathbf q = q_x\mathbf e_1 + q_y\mathbf e_2 + q_z\mathbf e_3 + \mathbf e_4$$ is a specific kind of flector that performs an inversion through $$\mathbf q$$. The exact inversion calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Inversion |- | style="padding: 12px;" | Poin...")
- 01:36, 18 May 2022 Rotation (hist | edit) [3,465 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\mathbf L$$, the specific kind of motor :$$\mathbf R = \mathbf L\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ , performs a rotation by twice the angle $$\phi$$ about the line $$\mathbf L$$. This differs from a general motor only in that it is always the case that $$u_w = 0$$. The exact rotation calculations for points, lines, and planes are shown in the followin...")
- 01:35, 18 May 2022 Translation (hist | edit) [2,421 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ performs a translation by twice the displacement vector $$(t_x, t_y, t_z)$$ when used as an operator in the sandwich product. This can be interpreted as a rotation about the line at infinity perpendicular to the direction $$\mathbf t$$. The exact translation calculations f...")
- 01:35, 18 May 2022 Reflection (hist | edit) [2,833 bytes] Eric Lengyel (talk | contribs) (Created page with "A ''reflection'' is an improper isometry of Euclidean space. When used as an operator in the sandwich product, a unitized plane $$\mathbf g = g_x\mathbf e_{234} + g_y\mathbf e_{314} + g_z\mathbf e_{124} + g_w\mathbf e_{321}$$ is a specific kind of flector that performs a reflection through $$\mathbf g$$. The exact reflection calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Reflection |- | style="pad...")
- 01:34, 18 May 2022 Flector (hist | edit) [10,896 bytes] Eric Lengyel (talk | contribs) (Created page with "400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a reflection across a plane perpendicular to the same line. A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of reflections, inversions, transflections, and rotoreflectio...")
- 01:32, 18 May 2022 Motor (hist | edit) [19,359 bytes] Eric Lengyel (talk | contribs) (Created page with "400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a displacement along the same line. A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of rotations and translations. The name motor is a portmanteau of ''motion operator'' or ''moment vector...")
- 01:30, 18 May 2022 Plane (hist | edit) [1,415 bytes] Eric Lengyel (talk | contribs) (Created page with "400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf f$$ is a trivector having the general form :$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ . All planes possess the geometric property. The bulk of a plane is given by its $$w$$ coordinate, a...")
- 01:29, 18 May 2022 Line (hist | edit) [3,495 bytes] Eric Lengyel (talk | contribs) (Created page with "400px|thumb|right|'''Figure 1.''' A line is the intersection of a 4D bivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''line'' $$\mathbf L$$ is a bivector having the general form :$$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ . The components $$(v_x, v_y, v_z)$$ correspond to the line's direction, and...")
- 01:27, 18 May 2022 Point (hist | edit) [1,490 bytes] Eric Lengyel (talk | contribs) (Created page with "400px|thumb|right|'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ . All points possess the geometric property. The bulk of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and...")
