Euclidean geometry and complex plane

In summary, Euclidean Geometry is connected to the complex plane through angle preservation, distance, Mobius Transformations, and isometries. Similarly, hyperbolic geometry can be described using complex numbers, with the unit disc in the complex plane serving as a model. Mobius transformations in this model act as rigid motions in the hyperbolic plane. Another equivalent model is the upper-half plane model. Reading "Visual Complex Analysis" by Needham is recommended for a better understanding of these connections.
  • #1
GcSanchez05
17
0
Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice.

Also, how can hyperbolic geometry be described with complex numbers?
 
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  • #2
Perhaps you should read a good complex analysis book. Try to read "visual complex analysis" by Needham. A very easy book, but with a lot of intuition and connections to geometry.
 
  • #3
Multiplying one complex number by another scales it and rotates it. Adding complex numbers give you translation. So, complex numbers can give you any dilation. So, if two figures are the similar, you can get from one to the other by multiplying and adding complex numbers to it.

There's a model for the hyperbolic plane which is the unit disc in the complex plane. Mobius transformations that map the unit disc to itself act as the rigid motions of the hyperbolic plane.
 
  • #4
representing the hyperbolic plane by complex numbers z with real part > 0, one obtains that maps defined by [az+b]/[cz+d/ with a,b,c,d real and ad-bc >0, are exactly the hyperbolic isometries. I think. oops that's what the previous post said.
 
  • #5
representing the hyperbolic plane by complex numbers z with real part > 0, one obtains that maps defined by [az+b]/[cz+d/ with a,b,c,d real and ad-bc >0, are exactly the hyperbolic isometries. I think. oops that's what the previous post said.

I referred to the complex numbers of modulus less than 1. That's the disk model.

Mathwonk is referring to the upper-half plane model, which is an equivalent way to think about it.
 

1. What is Euclidean geometry?

Euclidean geometry is a branch of mathematics that focuses on the study of geometric figures and shapes in a two-dimensional space. It is based on the principles and postulates established by the Greek mathematician Euclid, and it is often referred to as "flat" or "classical" geometry.

2. What is the complex plane?

The complex plane, also known as the Argand plane, is a geometric representation of the complex numbers. It consists of a horizontal x-axis and a vertical y-axis, with the origin (0,0) located at their intersection. The complex numbers are represented by points on the plane, with the real part of the number corresponding to the x-coordinate and the imaginary part corresponding to the y-coordinate.

3. How are Euclidean geometry and the complex plane related?

The complex plane is a powerful tool for representing and analyzing geometric problems in Euclidean geometry. It allows for the use of complex numbers to solve geometric equations and problems, and it provides a visual representation of geometric transformations such as rotations, translations, and reflections.

4. What is the difference between Euclidean geometry and non-Euclidean geometry?

Euclidean geometry is based on the five postulates established by Euclid, which include the concept of a straight line being the shortest distance between two points and the parallel postulate. Non-Euclidean geometry, on the other hand, is based on different postulates, which lead to different geometric principles and theorems. One example of non-Euclidean geometry is hyperbolic geometry, which allows for multiple parallel lines through a given point.

5. How is the Pythagorean theorem used in Euclidean geometry and the complex plane?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is fundamental in Euclidean geometry, as it is used to determine the distance between two points and to prove many geometric theorems. In the complex plane, the Pythagorean theorem can be extended to the Pythagorean identity, which relates the square of a complex number to its real and imaginary parts.

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