Non-Euclidean/Hyperbolic Geometry

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In summary, hyperbolic geometry involves measuring angles and distances on a curved surface. The angle between two lines can be found by taking the inner product of the two lines, and the distance between two points can be calculated using the hyperbolic metric equation. To find the original distances, the Pythagorean theorem can be applied. Additional information and resources on hyperbolic geometry can be found on various websites, including Wikipedia, MathWorld, and MathPages.
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Storm Butler
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Hey I am reading "The Road to Reality" and I'm trying to get a grasp on hyperbolic Geometry. One thing that i wasn't really sure about was how do you measure the curved angles (i.e. when there is a triangle that's formed by the hyperbolic "circular arc" lines how do you find the angle between those two "lines" would you just take the tangent and use that slope to determine it as if it were a straight line)? Also in the book they say that the distance between two points is the log (QA*PB)/(QB*PA), where does this come from? and if that is the distance between two points then how would you find the original distances (QA, PB ect) to plug in for the formula?

Also any good websites on the subject would help.
 
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To measure the angle between two lines, you would need to take the inner product of the two lines. This is a measure of how much they are pointing in the same direction, and can be calculated as the dot product of two vectors representing the lines. The formula for the distance between two points comes from the definition of the hyperbolic metric, which is defined by the equation ds^2 = dx^2 + dy^2 - dz^2. The logarithm of the ratio of two distances (QA*PB)/(QB*PA) is a measure of the distance along the hyperbolic surface between two points. To find the original distances, you can use the Pythagorean theorem in the hyperbolic plane. For example, if the coordinates of your points are (x1,y1,z1) and (x2,y2,z2), then the distance between the points is given by: d = sqrt((x2-x1)^2 + (y2-y1)^2 - (z2-z1)^2). For more information on hyperbolic geometry, you can check out these websites: https://en.wikipedia.org/wiki/Hyper.../www.mathpages.com/home/kmath458/kmath458.htm
 

1. What is Non-Euclidean/Hyperbolic Geometry?

Non-Euclidean/Hyperbolic Geometry is a type of geometry that explores the properties and relationships of shapes and objects in a curved space, rather than the flat space of traditional Euclidean Geometry. It is based on the work of mathematicians such as Gauss, Lobachevsky, and Bolyai.

2. How does Non-Euclidean/Hyperbolic Geometry differ from Euclidean Geometry?

Non-Euclidean/Hyperbolic Geometry differs from Euclidean Geometry in that it allows for the existence of parallel lines that intersect at a point, and the sum of the angles of a triangle can be less than 180 degrees. In Euclidean Geometry, parallel lines never intersect and the sum of the angles of a triangle is always 180 degrees.

3. What are the real-world applications of Non-Euclidean/Hyperbolic Geometry?

Non-Euclidean/Hyperbolic Geometry has applications in fields such as physics, engineering, and architecture. It is used to study the curvature of spacetime in Einstein's theory of general relativity, and to design curved structures such as bridges and domes.

4. Is Non-Euclidean/Hyperbolic Geometry still relevant today?

Yes, Non-Euclidean/Hyperbolic Geometry is still relevant and widely studied today. It has led to new discoveries and advancements in fields such as physics and mathematics, and its applications continue to be explored and utilized in various industries.

5. Are there any practical implications of Non-Euclidean/Hyperbolic Geometry?

Yes, Non-Euclidean/Hyperbolic Geometry has several practical implications, particularly in the fields of physics and engineering. It has also been used to develop new technologies, such as GPS systems, that require precise measurements in non-Euclidean spaces.

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