Difference between pages "Scalars and antiscalars" and "Origin and horizon"

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(Created page with "In rigid geometric algebra, there is a pair of special geometric objects called the ''origin'' and the ''horizon''. The origin has the same meaning here as it does in an ordinary Cartesian coordinate system. It is nothing more than the point having the 3D coordinates $$(0,0,0)$$. In 4D homogeneous coordinates, this becomes the point $$(0,0,0,1)$$, which is expressed on the vector basis as $$0\mathbf e_1 + 0\mathbf e_2 + 0\mathbf e_3 + 1\mathbf e_4$$. Thus, we ca...")
 
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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.
In rigid geometric algebra, there is a pair of special geometric objects called the ''origin'' and the ''horizon''.


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]].
The origin has the same meaning here as it does in an ordinary Cartesian coordinate system. It is nothing more than the [[point]] having the 3D coordinates $$(0,0,0)$$. In 4D homogeneous coordinates, this becomes the point $$(0,0,0,1)$$, which is expressed on the [[vector]] basis as $$0\mathbf e_1 + 0\mathbf e_2 + 0\mathbf e_3 + 1\mathbf e_4$$. Thus, we can simply equate the origin with the basis vector $$\mathbf e_4$$. The quantity $$\mathbf e_4$$ is the point at the origin.


For a general element $$\mathbf a$$, the notation $$a_{\mathbf 1}$$ means the scalar component of $$\mathbf a$$.
The horizon is the dual of the origin. In 3D space, points and planes are [[duals]] of each other, and they can be thought of as different interpretations of the same homogeneous coordinates on the [[vector]] basis and [[antivector]] basis. In the case of the horizon, we reinterpret the coordinates $$(0,0,0,1)$$ on the antivector basis to obtain $$0\mathbf e_{423} + 0\mathbf e_{431} + 0\mathbf e_{412} + 1\mathbf e_{321}$$. This is the plane containing all points at infinity, and it can be imagined as completely surrounding all of 3D space. The quantity $$\mathbf e_{321}$$ is the plane at the horizon.
 
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 $$a_{\large\unicode{x1D7D9}}$$ means the antiscalar component of $$\mathbf a$$.
 
== See Also ==
 
* [[Duality]]

Latest revision as of 06:06, 7 August 2022

In rigid geometric algebra, there is a pair of special geometric objects called the origin and the horizon.

The origin has the same meaning here as it does in an ordinary Cartesian coordinate system. It is nothing more than the point having the 3D coordinates $$(0,0,0)$$. In 4D homogeneous coordinates, this becomes the point $$(0,0,0,1)$$, which is expressed on the vector basis as $$0\mathbf e_1 + 0\mathbf e_2 + 0\mathbf e_3 + 1\mathbf e_4$$. Thus, we can simply equate the origin with the basis vector $$\mathbf e_4$$. The quantity $$\mathbf e_4$$ is the point at the origin.

The horizon is the dual of the origin. In 3D space, points and planes are duals of each other, and they can be thought of as different interpretations of the same homogeneous coordinates on the vector basis and antivector basis. In the case of the horizon, we reinterpret the coordinates $$(0,0,0,1)$$ on the antivector basis to obtain $$0\mathbf e_{423} + 0\mathbf e_{431} + 0\mathbf e_{412} + 1\mathbf e_{321}$$. This is the plane containing all points at infinity, and it can be imagined as completely surrounding all of 3D space. The quantity $$\mathbf e_{321}$$ is the plane at the horizon.

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