## 8 Polarity for Rigidity
### Duality and Polarity for Projective Geometry  
The primary example we will explore in this section is polarity in dimension 3. However the broader applications of polarity will include whole sections of our companion paper Projective Geometry of Scene Analysis, Parallel Drawing and Reciprocal Drawings [27], where parallel drawing and liftings in scene analysis are explicitly explored as combinatorial duals, and geometric polars, of one another, in all dimensions.  
In plane projective geometry, there are axioms for points and lines, and they have an explicitly dual form: if you take theorems for points and lines, and swap the terms, replacing points by lines and lines by points, the entire theory is unchanged. For example, any two distinct points lie on a unique line, and any two distinct lines intersect in a unique point.  
In a general dimension \(d\), duality pairs subspaces of \(\mathbb{P}^{d}\) of projective dimension \(k\) and projective dimension \(d-k-1\), reversing inclusions and preserving incidences. More generally, such a map is also called a _correlation_. The correlation is invertible, and if this correlation is its own inverse (that is, if it is an involution) then it is called a _polarity_. If we write the projective points as \(x=(x_{1},x_{2},\ldots,x_{d},x_{d+1})\) and the hyperplanes in dual coordinates as \(u=(u_{1},u_{2},\ldots,u_{d},u_{d+1})\), then the incidence of the point \(x\) on the hyperplane \(u\) is given by the equation  
\[x_{1}u_{1}+x_{2}u_{2}+\ldots+x_{d}u_{d}+x_{d+1}u_{d+1}=0.\]  
In vector space terms this is sometimes named orthogonality but we prefer incidence, and the hyperplane can be identified with the space of points incident with the hyperplane.  
There are two polarities which are central to our vision. One is the 'natural' polarity in which we do not change any of the coordinates, but just switch our interpretation of the coordinates of the points as the coordinates of hyperplanes, and vice versa. This correlation is an abstract involution but does not immediately have a geometric representation.  
The second geometric construction (which can be visualized in \(\mathbb{P}^{d}\)) is named _polarity about a quadric_[116]. This already appeared for bipartite frameworks in Section 7.3. If we focus on the points which lie on their polar planes in the natural polarity, we see the equation \(x_{1}^{2}+x_{2}^{2}+\ldots+x_{d}^{2}+x_{d+1}^{2}=0\) is a quadric surface in \(\mathbb{P}^{d}\). For a given non-degenerate quadric surface in \(\mathbb{P}^{d}\), there is a geometric construction of a polarity. This takes points on the quadric to the tangent hyperplane through the point. For the plane, we saw a brief introduction in Figure 23.  
We also recall that going from a framework to the momenta of the vertices and centers of motions of the bars is also a duality, but is generally not a polarity, except for bipartite frameworks with vertices on a quadric (Proposition 1).  
The origins of the name 'polarity' become visible when we consider the 'natural' polarity on the sphere and the elliptical model of projective geometry with antipodal points identified. This is also referred to as duality on the sphere. Every hyperplane on the sphere (e.g., the equator) has two antipodal poles. When the pairs of antipodal points are collapsed to form the elliptic model of the projective space, there is a complete pairing of hyperplanes and points, so there is a duality which is an involution--a polarity. This a geometric image of the natural polarity above. This polarity on the sphere takes a distance constraint (bar) between two points to an angle constraint between the two hyperplanes. We will return to this in Section 8.4.