Geometry (great circles)

In summary, the conversation discusses the orientation of a composition of reflections on three great circles with respective normals. It is suggested to use a matrix with the normals to determine if the composition is orientation preserving or reversing. The notation used includes "R" for reflection and "C" for great circle.
  • #1
Pearce_09
74
0
consider the 3 great circles C1,C2,C3 with respective normals
(0,-1,1) (1,0,1) (1,1,0), and let f = Rc3Rc2Rc1

is f orientation preserving or orientation reversing.

can i make a matrix using the normals... then calculate the determinate of A. and if its negative then its reversing and if its positive then its preserving... or can i only calculate the matrix when its givin to me...
 
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  • #2
Pearce_09 said:
consider the 3 great circles C1,C2,C3 with respective normals
(0,-1,1) (1,0,1) (1,1,0), and let f = Rc3Rc2Rc1
is f orientation preserving or orientation reversing.
can i make a matrix using the normals... then calculate the determinate of A. and if its negative then its reversing and if its positive then its preserving... or can i only calculate the matrix when its givin to me...

I don't understand your notation. What does "Rc3Rc2Rc1" mean? I assume that the c3, c2, c1 should really be C3, C2, C1, but what is R?
 
  • #3
R means reflection, sorry i should have stated that
so reflection in C3 Reflection in C2 and Reflection in C1
 
  • #4
"Reflection in C3"? C1, C2, C3 are great circles on a sphere? And reflection is to the other hemisphere?
Reflection, generally, is orientation reversing. Two reflections would be orientation preserving and 3, as you have here, orientation reversing.
 

1. What is a great circle in geometry?

A great circle is a circle on the surface of a sphere that has the same center as the sphere and divides it into two equal halves. It is the largest circle that can be drawn on a sphere and is also known as an equator for the sphere.

2. How is a great circle different from a small circle?

A great circle is different from a small circle in terms of their sizes. A great circle has the same diameter as the sphere it is drawn on, whereas a small circle has a smaller diameter. Additionally, a great circle divides the sphere into two equal halves, while a small circle divides it into two unequal halves.

3. What is the significance of great circles in navigation?

Great circles are significant in navigation as they provide the shortest distance between two points on a spherical surface. This is because they are the largest circles that can be drawn on a sphere and, therefore, have the shortest distance between any two points on the circle. This makes them crucial in calculating the most efficient route for travel or navigation.

4. How are great circles used in map projections?

Great circles are used in map projections to create accurate maps of the Earth's surface. They are used as reference lines to accurately depict the curved surface of the Earth on a flat map. By using great circles as reference lines, map projections can preserve the shapes and sizes of landmasses, making them more accurate for navigation and measurement purposes.

5. Can great circles be used to measure distances on any curved surface?

Yes, great circles can be used to measure distances on any curved surface, not just the Earth's surface. This is because great circles are the largest circles that can be drawn on any spherical surface and, therefore, have the shortest distance between any two points on the circle. This makes them useful for measuring distances on other spherical objects, such as planets, moons, or even bubbles.

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