GEOMETRY OF LINEAR TRANSFORMATIONS
... related to Section 2.2 of the book and also to the Geometry of linear transformations lecture notes. For the questions here, please use the following terminology. Suppose n is a fixed natural number greater than 1. For ease of geometric visualization, you can take n = 2 for the discussion. • A linea ...
... related to Section 2.2 of the book and also to the Geometry of linear transformations lecture notes. For the questions here, please use the following terminology. Suppose n is a fixed natural number greater than 1. For ease of geometric visualization, you can take n = 2 for the discussion. • A linea ...
Logic and Incidence Geometry
... Two models of incidence geometry are said to be isomorphic if there is a one-to-one correspondence P ↔ P 0 between the points of the models and a one-to-one correspondence ` ↔ `0 between the lines of the models such that P lies on ` if and only if P 0 lies on `0 . Such a correspondence is called an ...
... Two models of incidence geometry are said to be isomorphic if there is a one-to-one correspondence P ↔ P 0 between the points of the models and a one-to-one correspondence ` ↔ `0 between the lines of the models such that P lies on ` if and only if P 0 lies on `0 . Such a correspondence is called an ...
SYNTHETIC PROJECTIVE GEOMETRY
... not lie on L, and assume that M1 , · · · Mk are lines which pass through X and meet L in points Y1 , · · · Yk respectively. Prove that the points Y 1 , · · · Yk are distinct if and only if the lines M1 , · · · Mk are distinct. 5. Let (S, Π, d) be a regular incidence space of dimension ≥ 3, and assum ...
... not lie on L, and assume that M1 , · · · Mk are lines which pass through X and meet L in points Y1 , · · · Yk respectively. Prove that the points Y 1 , · · · Yk are distinct if and only if the lines M1 , · · · Mk are distinct. 5. Let (S, Π, d) be a regular incidence space of dimension ≥ 3, and assum ...
Log-rolling and kayaking: periodic dynamics of a nematic liquid
... Set up the dynamical equations as an ODE on the space V := {symmetric, traceless 3 × 3 matrices} ∼ = R5 . Think of Q ∈ V as the non-spherical part of the second moment of the probability that molecules will align in a given direction. Thus 0 ∈ V corresponds to the isotropic state: individual molecul ...
... Set up the dynamical equations as an ODE on the space V := {symmetric, traceless 3 × 3 matrices} ∼ = R5 . Think of Q ∈ V as the non-spherical part of the second moment of the probability that molecules will align in a given direction. Thus 0 ∈ V corresponds to the isotropic state: individual molecul ...
TWO CAMERAS 2009
... The investigation of image formation and object modeling is widely developed in the last thirty years. The formation of planar images of our three dimensional world plays an important role in our century of communications through computers. The geometry of multiple images provides us the description ...
... The investigation of image formation and object modeling is widely developed in the last thirty years. The formation of planar images of our three dimensional world plays an important role in our century of communications through computers. The geometry of multiple images provides us the description ...
4. Topic
... Every projective theorem has a translation to a Euclidean version, although the Euclidean result is often messier to state and prove. Euclidean pictures can be thought of as figures from projective geometry for a model of very large radius. (Projective plane is ‘locally Euclidean’.) ...
... Every projective theorem has a translation to a Euclidean version, although the Euclidean result is often messier to state and prove. Euclidean pictures can be thought of as figures from projective geometry for a model of very large radius. (Projective plane is ‘locally Euclidean’.) ...
Natural Homogeneous Coordinates
... • If a point ( x, y, 1) is on the line ax + by + cz=0, so is point (px, py, p) for any p. Note: apx + bpy + cpz = p * (ax + by + cz)=0 • Multiple projective coordinate points correspond to the same Euclidean coordinate. • To obtain Euclidean coordinates from non-ideal points represented as projectiv ...
... • If a point ( x, y, 1) is on the line ax + by + cz=0, so is point (px, py, p) for any p. Note: apx + bpy + cpz = p * (ax + by + cz)=0 • Multiple projective coordinate points correspond to the same Euclidean coordinate. • To obtain Euclidean coordinates from non-ideal points represented as projectiv ...
Projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient groupPGL(V) = GL(V)/Z(V)where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation ""Z"" reflects that the scalar transformations form the center of the general linear group.The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly:PSL(V) = SL(V)/SZ(V)where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant. Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in K (where n is the dimension of V and K is the base field).PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, projective transformation or homography. If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used.Note that PGL(n, F) and PSL(n, F) are equal if and only if every element of F has an nth root in F. As an example, note that PGL(2, C) = PSL(2, C), but PGL(2, R) > PSL(2, R); this corresponds to the real projective line being orientable, and the projective special linear group only being the orientation-preserving transformations.PGL and PSL can also be defined over a ring, with an important example being the modular group, PSL(2, Z).