how can i create a hyperbolic plan?

Answer 1

M.C. Escher has already done a number of drawings based on the hyperbolic plane, based on Henri Poincaire's method of representing the entire hyperbolic plane. Check this out:

http://www.sciencenews.org/articles/20030830/mathtrek.asp

Answer 2

A plane (not a plan) is defined as a set of objects called points and and a set of objects called lines with a notion of "incidence" which is to say a notion that the point lies on the line or the line contains the point. These sets may be finite or infinite. See the article http://en.wikipedia.org/wiki/Projective_plane for a "plane" with seven points and seven lines and each line contains three points and each point has three lines passing through it. In projective geometry, there are no parallel lines; all lines have a common point of intersection.

In hyperbolic geometry (in extreme contradistinction to Euclidean geometry), for every line and point not on the line, there are an infinite number of lines passing through the point and parallel to the given line. In Euclidean geometry, there is a unique line parallel to the given line. This is equivalent to Euclid's famous parallel postulate.

Two standard models for the hyperbolic plane are used, the disc and the upper half plane. There is a map from one to another using complex numbers.

Here is your hyperbolic plane...

In the upper half plane model, we consider points on the plane {(x,y)} with y>0. Lines are either: 1) horizontal lines, say, y=5, 2) vertical half rays, say x=3, y>0, or 3) half circles that meet the x-axis perpendicularly, say x^2+y^2=1 and y>0.

Exercise: Consider the line y=2 and the point (0,1). Draw the half circles which are perpendicular to the x-axis, going through the point that don't meet the line.