Explore the projective geometry approach to conics and quadrics in this differential geometry lecture. Delve into Mobius and Plucker's perspective, viewing the projective plane as one-dimensional subspaces of a three-dimensional vector space. Learn about homogeneous coordinates [X:Y:Z] and their advantages in representing points at infinity. Examine how curves like the parabola y=x^2 are expressed in homogeneous equations, providing a uniform view of conics similar to Apollonius' cone slices. Discover how homogeneous coordinates serve as a powerful tool for studying conics, algebraic curves, quadrics, and higher algebraic surfaces in space.