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Category Archives: projective geometry
The geometry of a second order differential equation, part 2
We start over with a general second order ordinary differential equation and this time we won’t impose any restrictions on the right-hand side. We express and which gives that The variables and denote velocity and are tangent space coordinates. The … Continue reading
The geometry of a second order differential equation, part 1
We start with a general second order ordinary differential equation and express and which gives that The variables and denote velocity and are tangent space coordinates. The variables and denote acceleration and are jet space (of curves) coordinates. Since the … Continue reading
The geometry of y”+y=0
In the differential equation we rewrite and get and with polarisation we get We read off the Christoffel symbols to be The connection one-form becomes and the curvature two-form is where is the covariant exterior derivative and means the ordinary … Continue reading
The Schwarzian derivative
For a diffeomorphism between two Euclidean lines, the derivative is an invariant in the sense that to calculate the derivative we fix Euclidean coordinate systems on the domain and on the range and the calculated value will not depend on the … Continue reading
The cross ratio
The purpose of this post is to show how the cross ratio can be understood as a generalisation of the corresponding invariants in Euclidean and affine geometry and how it is closely related to the concept of a point frame. … Continue reading
Posted in projective geometry
Tagged cross ratio, harmonic, projective frame, projective line
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