Understanding the Formulas for Area of Spherical Triangles

In summary, the conversation discusses the two formulas for calculating the area of spherical triangles, Lambert's and Harriot's, and whether the radius of the sphere must be the same for each segment for the formulas to work. It is explained that in spherical geometry, the radius used is the radius of the sphere itself and that all sides of a spherical triangle are arcs of great circles with the same radius. The concept of relative measurement in spherical geometry is also discussed, using units such as radians and steradians. It is concluded that the area of a spherical triangle can be calculated using either formula, depending on the perspective being used.
  • #1
denni89627
71
0
I've been reading Penrose's Road to Reality where he presents two formulas for area of shperical triangles. the first is Lamberts which is
pi-(A+B+C)=area (where A,B,C are angles of triangle)

the other is Harriot's which is
Area=R^2(A+B+C-Pi)

What I'm trying to figure out is if the radius must be the same for each segment for these formulas to work. In other words, if you extrapolated each curved side of the triangle into a complete circle, do they all have to be the same size circles? I'm guessing they do, and increasing R would just change the scaling. But Lambert's formula does not require a radius so would that formula work for a triangle created from cirlces of different radius?

Hope I worded this in an understandable way.

dennis
 
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  • #2
The radius being used is the radius of the sphere. Also the sides of a spherical triangle (by definition) are all arcs of great circles, which all have the same radius as the sphere.
 
  • #3
Incidentally, this is spherical geometry, not hyperbolic geometry.

Area, like any other measurement, is a relative quantity... the area of something depends on what you define to be a unit area!


Spherical geometry has some intrinsic measures (e.g. the length of any "line", or the total area of the entire "plane"), so it is natural to define your unit of measure relative to those. For example, I might decide to degree that the length of a line is 2 pi, and the area of the plane is 4 pi.

In that case, the area of a triangle is A + B + C - pi.


On the other hand, we might be might really be interested in the geometry of a Euclidean sphere. (which is a model of spherical geometry) In that case, we might want to speak about Euclidean lengths and areas, rather than the natural units of spherical geometry.

In that case, the area of a triangle is R^2 (A + B + C - pi).


There's a nice compromise; use a different word for the natural units of spherical geometry. Measure lengths in "radians", and areas in "steradians". Then, whatever perspective we are using, there are

2 pi radians in a spherical line or a Euclidean great circle.
4 pi steradians in a spherical plane or a Euclidean sphere.

and to convert into Euclidean lengths and areas, we simply multiply by R or R^2 as appropriate.


(note: when doing spherical geometry, people probably use the word "sphere" instead of "plane" -- I was using the latter to emphasize the fact I was talking about spherical geometry, and not just the Euclidean geometry of a Euclidean sphere)
 

Related to Understanding the Formulas for Area of Spherical Triangles

1. What is hyperbolic geometry?

Hyperbolic geometry is a non-Euclidean geometry that describes the properties of space in a curved, saddle-shaped surface. It is based on the principles of Euclid's parallel postulate, but it does not hold true in hyperbolic space. This leads to unique and interesting properties, such as angles of a triangle adding up to less than 180 degrees.

2. How is hyperbolic geometry different from Euclidean geometry?

Hyperbolic geometry differs from Euclidean geometry in several ways. Euclidean geometry is based on the assumption that parallel lines never meet, while hyperbolic geometry allows for multiple parallel lines to intersect at a single point. Additionally, the angles of a triangle in Euclidean geometry add up to 180 degrees, but in hyperbolic geometry, they add up to less than 180 degrees.

3. What are some real-world applications of hyperbolic geometry?

Hyperbolic geometry has many real-world applications, particularly in the fields of architecture and art. It has been used to design structures such as the Guggenheim Museum in Bilbao, Spain and the Lotus Temple in New Delhi, India. It has also been used in computer graphics and animation to create visually interesting and complex shapes.

4. Who discovered hyperbolic geometry?

Hyperbolic geometry was first developed by mathematician Carl Friedrich Gauss in the early 19th century. However, it was further developed and popularized by mathematicians such as Nikolai Lobachevsky and János Bolyai in the mid-19th century.

5. How is hyperbolic geometry relevant to modern science?

Hyperbolic geometry has many applications in modern science, particularly in the fields of physics and cosmology. It has been used to describe the behavior of light in curved space-time and has been used in models of the universe to explain the curvature of space. It also has practical applications in fields such as computer graphics, robotics, and computer vision.

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