I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about the difference between Grassmann space and projective space.
Based on the books' definition, points have an associated weight in both spaces. However, in projective space all points on the same projective line are assumed to be equal. In other words, projective lines are equal to points. Can anyone explain an example application to illustrate this difference?
I also don't understand why projective space is used for defining conic sections (rational polynomials) instead of Grassmann space? Any help is highly appreciated.
The Grassmannian is the collection of all subspaces of a projective space having a fixed dimension. It can be represented using Plücker coordinates as an algebraic variety, which essentially lets you embed the Grassmannian into a projective space.
For example, if you have $V$ being a 4-dimensional vector space, then $P(V)$ is a 3-dimensional projective space. You can take the lines of $P(V)$ (these correspond to 2-dimensional vector subspaces) as the Grassmannian space $G(4,2)$; here the lines of $P(V)$ are represented as points, and two points are "collinear" in the Grassmannian if and only if the corresponding lines in $P(V)$ intersect nontrivially. Through the Klein correspondence, $G(4,2)$ can be embedded in a 5-dimensional projective space as the points of a hyperbolic quadric $\mathcal{Q}$.