- #1
Reshma
- 749
- 6
Explain me why the sum of the interior angles of a triangle is not equal to 180 degrees on a spherical surface?
Interior angles of a triangle on a spherical surface
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SUMMARY
The sum of the interior angles of a triangle on a spherical surface exceeds 180 degrees due to the positive curvature of the sphere. Unlike planar geometry, where angles sum to exactly 180 degrees, spherical triangles exhibit greater angle sums because of the unique properties of curved surfaces. This phenomenon can be visualized using a model such as a tennis ball with rubber bands representing the triangle's sides, demonstrating that angles can exceed 90 degrees at vertices. The mathematical explanation is grounded in the definition of angles between coordinate curves on a positively curved surface.
PREREQUISITES- Understanding of basic geometric concepts, including triangles and angles.
- Familiarity with spherical geometry and curvature.
- Knowledge of coordinate systems and their applications in geometry.
- Basic visualization skills to comprehend geometric models like rubber bands on a sphere.
- Explore the properties of spherical geometry and its differences from Euclidean geometry.
- Study the concept of curvature, focusing on positive and negative curvature in geometry.
- Learn about the applications of spherical triangles in fields such as navigation and astronomy.
- Investigate visual aids and models, such as the use of rubber bands on spheres, to better understand geometric concepts.
Students of geometry, educators teaching advanced mathematics, and anyone interested in the applications of spherical geometry in real-world scenarios such as navigation and physics.
- #2
dextercioby
Science Advisor
- 13,404
- 4,173
In simple terms,because the surface (unlike the simple case of a plane) is not flat but curved and the definition of an angle between 2 coordinate curves is the same in all possible cases.Since the total curvature of the 2-sphere is positive (more,it is even constant),the sum of all angles in a spherical triangle is not 180° anymore,but more...If the surface had negative total curvature (the saddle,or mathematically rogurous:hyperbolic paraboloid),the angles in a triangle would add to less than 180°.
This part of geometry is really interesting...
Daniel.
This part of geometry is really interesting...
Daniel.
- #3
vsage
Here's my take on it:
First you have to consider what a triangle is. It's three straight lines L1, L2 and L3 that have the property that L1 and L2 intersect where L3 does not intersect them, L1 and L3 intersect where L2 does not intersect them, and L2 and L3 intersect where L1 does not intersect them.
Spherical surfaces have the interesting property that two lines can intersect even if they share a common line perpendicular to each (not possible in a plane). I think that's the most important part of understanding how this works. Angles on a spherical surface are also measured at the vertex, before curvature dictates a smaller angle, so think of the angle measurement as like a tangent line to this curve at a vertex.
I think a physical description of this is easiest to see on a tennis ball with 3 rubber bands to represent lines, so I'll try and describe what I mean using that as a model. First, construct two lines with rubber bands (they'll fly off if you don't put them around the fattest circumference of the sphere in the particular way you angle them, so might as well do that). The rubber bands should look something like longitudinal lines on a globe. You'll notice the lines intersect. All you have to do now is place a third rubber band perpendicular to one of the rubber bands, but if it's perpendicular to one you'll see it's perpendicular to both. In this fashion you've made two triangles (actually three but just disregard it), so pick one to study.
Look at each vertex straight on. The angle measurements of two of those vertices should be 90 degrees because you placed a third rubber band perpendicular to the other two. The third one has some angle measurement too but I think that suffices to prove the sum of the interior angles is greater than 180 degrees. If you're still a little shaky because I told you how to make a specific triangle, try moving the first two lines closer together: Sure one angle gets smaller but the other two stay the same at 90 degrees, so you're still > 180 degrees by some amount.
The above is just how I look at it and I'm sure some would disagree with me. mathworld has a nice picture to illustrate the point too: http://mathworld.wolfram.com/SphericalTriangle.html
First you have to consider what a triangle is. It's three straight lines L1, L2 and L3 that have the property that L1 and L2 intersect where L3 does not intersect them, L1 and L3 intersect where L2 does not intersect them, and L2 and L3 intersect where L1 does not intersect them.
Spherical surfaces have the interesting property that two lines can intersect even if they share a common line perpendicular to each (not possible in a plane). I think that's the most important part of understanding how this works. Angles on a spherical surface are also measured at the vertex, before curvature dictates a smaller angle, so think of the angle measurement as like a tangent line to this curve at a vertex.
I think a physical description of this is easiest to see on a tennis ball with 3 rubber bands to represent lines, so I'll try and describe what I mean using that as a model. First, construct two lines with rubber bands (they'll fly off if you don't put them around the fattest circumference of the sphere in the particular way you angle them, so might as well do that). The rubber bands should look something like longitudinal lines on a globe. You'll notice the lines intersect. All you have to do now is place a third rubber band perpendicular to one of the rubber bands, but if it's perpendicular to one you'll see it's perpendicular to both. In this fashion you've made two triangles (actually three but just disregard it), so pick one to study.
Look at each vertex straight on. The angle measurements of two of those vertices should be 90 degrees because you placed a third rubber band perpendicular to the other two. The third one has some angle measurement too but I think that suffices to prove the sum of the interior angles is greater than 180 degrees. If you're still a little shaky because I told you how to make a specific triangle, try moving the first two lines closer together: Sure one angle gets smaller but the other two stay the same at 90 degrees, so you're still > 180 degrees by some amount.
The above is just how I look at it and I'm sure some would disagree with me. mathworld has a nice picture to illustrate the point too: http://mathworld.wolfram.com/SphericalTriangle.html
- #4
Reshma
- 749
- 6
Thanks for the explanation!
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