Table of Contents
What is complex projective line?
Complex projective line: the Riemann sphere Adding a point at infinity to the complex plane results in a space that is topologically a sphere. It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface.
Who contributed to projective geometry?
Projective geometry has its origins in the early Italian Renaissance, particularly in the architectural drawings of Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72), who invented the method of perspective drawing.
What is a Riemann sphere in physics?
The Riemann sphere is a way of interpreting the complex numbers plus one point at infinity as the points on a sphere. The connection is done by the stereographic projection. Place a sphere on the complex plane. The point of tangency, call it the south pole, corresponds to the complex number 0.
How do you find the Riemann sphere from a stereographic projection?
The Riemann sphere can be visualized as the unit sphere x2 + y2 + z2 = 1 in the three-dimensional real space R3. To this end, consider the stereographic projection from the unit sphere minus the point (0, 0, 1) onto the plane z = 0, which we identify with the complex plane by ζ = x + iy.
What is the extended plane of Riemann model?
This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point “∞” is near to very large numbers, just as the point “0” is near to very small numbers. well-behaved.
What is the Riemann model of Infinity?
With the Riemann model, the point “∞” is near to very large numbers, just as the point “0” is near to very small numbers. well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity.