Published
Edited
Jul 12, 2022
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md`# What is dual euclidean space?`
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md`The question has arisen in a number of contexts regarding Euclidean PGA ${tex `P(R^*_{n,0,1})`}: why are 1-vectors planes and not points? Steven de Keninck's [Observable notebook](https://observablehq.com/@enkimute/plane-vs-point-based-pga) thoroughly answers this question by showing how using the geometric algebra ${tex `P(R_{n,0,1})`} (in which 1-vectors are points) gives incorrect answers for a catalog of calculations in euclidean space.

In this notebook I want to take a complementary approach and show how ${tex `P(R_{n,0,1})`} gives correct answers ... for a completely different metric space, so-called _dual euclidean space_, or counterspace for short. (I'm aware that counterspace is mostly thought of as what kitchens usually have too little of, I'm open to alternative naming suggestions:-)).

**Note:** In general, standard and dual geometric algebras can be *compatible*, that is, produce correct answers for the *same* metric space. For example ${tex `P(R_{3,0,0})`} and ${tex `P(R^*_{3,0,0})`} both model spherical trigonometry of the familiar 2-dimensional sphere. See [SIGGRAPH 2019 PGA course notes](https://arxiv.org/abs/2002.04509), Sec. 6.3. But this is here not the case.

The good news is that you know almost everything you need to know about counterspace: you only have to learn how to _dualize_ statements in projective geometry. That's as easy as going through the statement and replacing certain words and phrases with their _dual_ words and phrases. For example, *join* is dual to *meet*. Duality is symmetric, so *meet* is automatically dual to *join*. That is, the dual of the dual is the original word or phrase.

The importance of duality in projective geometry is this: a statement is true if and only if its dual statement is true!

Since this _dictionary of duality_ depends on the dimension, we'll start by restricting ourselves to the case of _n=2_, the euclidean plane ${tex `P(R^*_{2,0,1})`} -- and the dual euclidean plane ${tex `P(R_{2,0,1})`}.

From the course notes, you should be aware that the model of (dual) euclidean plane provided by ${tex `P(R^*_{2,0,1})`} (${tex `P(R_{2,0,1})`} ) is obtained by adding an inner product on lines (points) to the projective plane. We use this embedding of both spaces as dual partners in projective geometry to provide a full account of DES as the "dual" partner of ES.

Then our dictionary of duality looks like: (where I've included some pairs whose meaning will only become clear later)

X | Dual(X)
------------ | -------------
join | meet (or intersect)
pass through | lie on
point | line
segment | fan
triangle | trihedron
rotate | turn
translate | shift
${tex `P(R^*_{2,0,1})`} | ${tex `P(R_{2,0,1})`}
eucidean plane (EP) | dual euclidean plane (DEP)

**WARNING:** *There is no intrinsic significance for a given dual pair which partner is in the left column and which is in the right*

**Examples**:

Here is a fundamental property of plane projective geometry:

X | Dual(X)
------------ | -------------
Every two points have a unique joining line. | Every two lines have a unique meeting point.

Here we begin to consider how euclidean geometry sits inside of projective geometry:

X | Dual(X)
------------ | -------------
In ${tex `P(R^*_{2,0,1})`}, there is an ideal line | In DEP, two parallel points



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