There Are No Dumb Questions.

[hazelm]

In a saddle curve, a triangle always has fewer than 180 degrees.  In a hill curve, a triangle always has more than 180 degrees.  I can't see why they would not both have the same.  That is, it the two curves measure the same except in opposite directions, why would they not get the same triangular measurements?  Aren't they traveling the same curved arc but in opposite directions?

[GAHD]

One's a pinch, and one's a pull.

[hazelm]

One's a pinch, and one's a pull.

"I am not quite sure what my right honorable friend said, but we hold precisely the same view."  (British Prime Minister Margaret Thatcher)

 

Good night, GAHD.  I am sure you are right.  I just do not understand.

[ralfcis]

Think of a convex and concave magnifying glass. In a convex the lines forming the triangle bulge out and in extreme cases could bulge out to be a circle. A circle has 360 degrees which is twice the 180 degrees of a flat triangle. Trace a flat triangle over a concave lens and the marker will sink inwards pinching the triangle and  its angles.

Edited January 19, 2020 by ralfcis

[hazelm]

Think of a convex and concave magnifying glass. In a convex the lines forming the triangle bulge out and in extreme cases could bulge out to be a circle. A circle has 360 degrees which is twice the 180 degrees of a flat triangle. Trace a flat triangle over a concave lens and the marker will sink inwards pinching the triangle and  its angles.

Give me 12 hours.  OK?  Thanks.

[OceanBreeze]

In a saddle curve, a triangle always has fewer than 180 degrees.  In a hill curve, a triangle always has more than 180 degrees.  I can't see why they would not both have the same.  That is, it the two curves measure the same except in opposite directions, why would they not get the same triangular measurements?  Aren't they traveling the same curved arc but in opposite directions?

 

Nice question! (Definitely not a dumb one)

 

There is no difference in the curvature of the inside and the outside of a saddle curve because, as you say “they are travelling in the same curved arc”. So, both the inside and the outside have negative curvature and a triangle drawn on either side has less than 180 degrees.

 

Similarly, both the inside and the outside of a sphere have positive curvature and a triangle drawn on either side will have more than 180 degrees.

 

The point of confusion is to compare the saddle curve, which is hyperbolic, with the “hill” curve which is spherical. These two do not have the same curvature.

[hazelm]

Your last line is what I was trying to use in my question  cave or hill being the same size but in opposite directions.  Why can't they be the same size?  What makes a hill spherical?  Hills are all shapes and sizes.  And a cave could be spherical, seems to me.

 

I'll think on it.  I am missing something. Thanks

[ralfcis]

I'm missing something too. Let's say you made a triangle with an elastic band and then moved that triangle towards the surface of a larger ball. The sides of the elastic band will hit the ball first. If you tape the sides where they touch the ball and keep moving the points of the triangle down, the angles will sharpen. If you didn't tape the sides, they would have just spread out over the ball and the angles of the triangle would have widened.

 

If you cut the hollow ball in half and did the same thing with the triangle but from inside the hollow ball, the ends of the triangle would hit first. Tape those ends and keep moving the sides forward and the angles would sharpen. I don't even know how the sides would bulge out from this perspective. It should work the same from both perspectives.

[hazelm]

I'm missing something too. Let's say you made a triangle with an elastic band and then moved that triangle towards the surface of a larger ball. The sides of the elastic band will hit the ball first. If you tape the sides where they touch the ball and keep moving the points of the triangle down, the angles will sharpen. If you didn't tape the sides, they would have just spread out over the ball and the angles of the triangle would have widened.

 

If you cut the hollow ball in half and did the same thing with the triangle but from inside the hollow ball, the ends of the triangle would hit first. Tape those ends and keep moving the sides forward and the angles would sharpen. I don't even know how the sides would bulge out from this perspective. It should work the same from both perspectives.

But I am trying to keep the hill and the cave the same size and shape but pointing in opposire directions.  To me, if you spread either one wider or taller, its angles will spread - or shrink?