What is a triangle

[CraigD]

Now I'm confused; you said the figure had an area A = 0.5, when x = 1.543.

That’s correct, though [math]x = 1.543[/math] is an approximate value. The exact value is [math]x= \frac{e^2 +1}{2e}[/math]. The dot over the equal signs in my previous posts indicates that the values following them are approximate.

What's the value of y, when x = 1.543? I was sure it's 1.

Don’t be sure – use a calculator! ;)

 

The equation for a unit hyperbola is [math]y= \sqrt{x^2 -1}[/math].

 

If you plug in the value [math]x=1.543[/math], you get [math]y = \sqrt{1.543^2 -1} = \sqrt{1.380849} \dot= 1.17509532[/math], not [math]y=1[/math]. Plugging this [math]x[/math] value into the equation for area from post #14 gives [math]A = \frac{\ln(1.543 + \sqrt{1.543^2-1})}{2} \dot= 0.49996569[/math]. If you used its exact value, which can’t be written as a decimal number or entered into most calculators, you’d get [math]A= 0.5[/math] exactly.

 

If you plug in [math]x= \sqrt2[/math] (an exact value), you get [math]y = \sqrt{\left(\sqrt2\right)^2 -1} = \sqrt{2-1} =\sqrt{1} = 1[/math], but [math]A[/math] isn’t very close to 0.5 for this value.

 

PS: can you reproduce how I got [math]x= \frac{e^2 +1}{2e}[/math] by solving [math]\frac12 = A = \frac{\ln(x + \sqrt{x^2-1})}{2} [/math] in post #14? It’s not very difficult algebra, involving no calculus, but it’s as important as the calculus in understanding how I found this value.

[Boof-head]

Ok, I see I got the slope of the asymptote mixed with the slope of the intersect from (0,0) to the hyperbola at (x,1); x = 1.4142.

So the enclosed figure has an area < 0.5; so x is = the slope of the asymptote, y = x.

 

I drew a rough sketch and somehow thought the areas were equal (it looks like you can 'fit' the figure with a hyperbolic arc for a side, into the right triangle inside the unit square.

Oh well.

 

If you draw another line from (0.5, 0) to the same point on the curve at (x,1), what's the area of the triangle described by this line, assuming it divides the 'angle' on the curve?

 

Since you used logarithms does this not already describe the cosh, sinh trig relations?

 

I've done calculus and I know how to derive the curve for a hanging cable with trig.

Can you see an analytic soln. by dividing the x-axis (as I suggest)? if you keep dividing the remainder by 2 I mean. Next step after finding the area of the above triangle, is drawing a line from (0.75,0) to the curve at (x,1), and so on. Does this work?

 

(It's been awhile since I did any basic calculus; after learning the basics you sort of push them into the background,, so I want to review my grasp of it.)

[Qfwfq]

By all standard terminology, each side is a geodesic of whatever manifold the figure is in. In this sense the maximal circles of a spherical surface are "straight" and therefore so are the arcs which form sides of a triagle in spherical geometry.

 

A polygonal is a case of a curve which is regular by arcs, specifically wherein each arc is straight; a polygon (including a triangle) is a closed polygonal. Thus by definition the 3 sides of a triangle are straight (i. e. geodesics).

[Boof-head]

This exercise and the analysis of the unit square and circle, or 'squaring the circle', plus another conic, with a general form ab = 1, encapsulate Kepler's Law of equal areas, since a hyperbolic triangle and section are related to 1/2 and 1/4 the area of a unit square.

 

Geodesics on a sphere and hyperbolic curves allowed Islamic astronomers to predict phases of the moon; Newton to predict planetary and cometary motion; and modern astronomy to predict nutations and gyrations in the planet's motion with GPS and bright celestial objects at an essentially fixed inertial distance (which is Newtonian).

 

This is all related to the way a cable hangs between two poles. The cable is an astronomical observatory, a kind of mathematical telescope.

[Qfwfq]

Al Khawrizmi!!!!!! :evil:

 

It's also the name from which the word algorithm is derived. The terminology "hyperbolic triangle" has more than one meaning so it would br good if you clarified which you mean when mentioning Kepler's laws.

[Boof-head]

When is a hyperbolic section equal to a hyperbolic triangle? Is it a) never :) sometimes, depending on who you ask c) neither a or b?

[CraigD]

When is a hyperbolic section equal to a hyperbolic triangle? Is it a) never …

The answer to this question depends on what you mean by “hyperbolic section”. “hyperbolic section” isn’t a commonly recognized mathematical term. If you mean “conic section that is a hyperbola”, the answer is “never – no conic section is a triangle”, because this graph isn’t a closed figure and has not vertexes, and by any conventional definition, a triangle must be a closed figure with 3 sides and 3 vertexes.

… :) sometimes, depending on who you ask c) neither a or b?

This choice makes the question not a mathematical one, but an epistemological one.

 

Usually in math, one assumes an objective ontology, widely known as realism. In this case, well formed propositions are either true or false, independent of who asserts their truth or falsehood. So if two people answer assert a different true/false value for the same proposition, either one person is wrong, or miscommunication has occurred, in which case the proposition is not well formed.