We visualized the two dimensional geometry by embedding it in a three dimensional setting. This is not the best way to proceed for our purposes. We want an intrinsic description of the geometry. One which does not go outside of the geometry to use an embedding dimension to supply the description. We will use differential and integral calculus to describe our geometry.
Straight lines are paths of shortest distance.
The angle subtended by an arc is the ratio of the arc length to the distance to centre of the arc. Areas and volumes will be computed by doing multiple integrals of area elements and volume elements , which are also defined in terms of the basic local distance function mentioned above. End Lecture 1. Back to top of Lecture. On to Lecture 2.