Some geometry trivia curiosity

Discussion in 'Physics & Math' started by curioucity, Aug 27, 2005.

  1. curioucity Unbelievable and odd Registered Senior Member

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    Hi genius ^_^

    Well, I just thought of some things about 2D geometry, that is, about deciding if a certain shape can be determined if some parts of it (points in Cartessian coordinates, more specifically) are known.

    Like this. I know that if (and only if, curiously enough) just 3 distinct points of a circle is known (AND provided the three points are not in one single line), we already can define where the circle is located and what its radius is. So...

    How many points (either at least or exactly) in Cartessian coordinates do we need to determine how these shapes are defined:
    1) Ellipse
    2) Parabole
    3) Hyperbole
    Or is it impossible to determine those shapes by points only?

    Okay, I could have tried working it out myself, but I'm both a bit too lazy, and I can't vizualize it immediately withOUT paper.... so, please?


    BTW, here's some of my other guesses:
    It is impossible to determine a square if only 3 of its points are known, provided that these points are NOT necessarily their corners.

    Any light discussions are welcome, thanks ^_^

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    PS: forgot about some thing... corrected about the circle.
     
    Last edited: Aug 29, 2005
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  3. Fraggle Rocker Staff Member

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    Let's see. A hyperbola is just an ellipse with one of the coefficients turned negative. A circle is a special case of an ellipse with both foci on the same point. A parabola is a special case of either an ellipse or a hyperbola, with one of the coefficients stretched to infinity.

    So, it requires two more variables to define an ellipse than a circle, the two coordinates of the second focus. Therefore it should take five points to define it. The same should be true of a hyperbola since it behaves like an ellipse squashed through imaginary space: five points to define it.

    That leaves the parabola. Since it's a special case of the ellipse/hyperbola, it should need at least one less point to define it. It's an ellipse or a hyperbola with the second focus at infinity. But we have to know in which direction the focus lies at an infinite distance. We have the length of that vector, as it were, but we need the angle. Therefore it should take four points to define a parabola.

    This makes sense because a parabola is intuitively more complex than a circle. Circles have no orientation, parabolas do. So it's reasonable that defining a parabola takes one more point than a circle.

    And to test the other shapes against intuition, it's also reasonable that it would take even one more point to define an ellipse/hyperbola. Its eccentricity is variable so you need to specify it. All parabolas have the same eccentricity (infinity) as do all circles (zero).

    This is the math paradigm we worked with 45 years ago. I hope it makes sense to you younger people who grew up with "new math."

    It's possible to define any quadratic function by being given the right number of points because quadratic equations can be solved. The shapes are merely the graphs of quadratic equations and the points are merely pairs of x,y values. Plug the values into the equations and you can solve for the coefficients.

    I don't know if it's possible to solve higher-order equations this way. We couldn't do it, but I can't recall if it had been proven impossible or if just nobody had figured it out yet. One of the younger math majors here should be able to answer that one.
     
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  5. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    If you inscribe an irregular hexagon in a circle (6 corners on the circle) without any pair of opposite sides being parallel and then extend each of the three pairs of opposite sides until they meet at points A, B and C you will find that points A, B and C always are on the same straight line! If I knew how to post drawings I would show a few examples. (Perhaps some one else will.) Gauss proved this at age 12. Perhaps you would like to try.
     
    Last edited by a moderator: Aug 28, 2005
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  7. curioucity Unbelievable and odd Registered Senior Member

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    Wow, Fraggle, those are great explanations, thanks. To tell you the truth, I usually describe things visually, so these word-explanations are quite a good info to try. I've never been told to relate ellipse with hyperbole and parabole (really.... my maths teacher merely related parabole with a point, its focus, and a line 'over the tip of the parabole', and that by definition, a parabole is a group of points of which distance to the focus is the same as their distance to the line)

    And Billy, that's nice. I'll try that some time ^_^

    BTW, I'm still trying to think about if a square in a plane can be determined if only its points (not necessarily corners) are known..... Like I suggested, 3 is uncertain, and currently, 4 means it may or may not be one........ but, with what you said about ellipse, Fraggle, and from my square findings:

    Just like how 3 points are not enough to define one set square (it can make multiple), 5 random points are not enough to define a single rectangle....... in fact, the 5 points may even not make one.

    Guess that settles it... looks like we can't take random points and make a certain polygon which is made by straight lines...... Well, well, lemme try thinking of any other random things....

    PS: Just thought about this: If a segment of a straight line happens to be a side of a square (of which side length is unknown at current), and two random points located away from the line are known, PLUS that the straight line between the two points does not intersect with the main straight line, a square can be defined from them.

    Same things can't always be done with rectangles though, I think....
     
    Last edited: Aug 29, 2005
  8. Fraggle Rocker Staff Member

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    I'm not visual at all. I think in words or music. Geometry was a wee bit of a challenge for me.
     
  9. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    They are not called "conic sections" for no reason. All are just the sections of a cone cut by a plane, but remember that the geometic "cone" is like two co-axial icecream cones joined that the tip. The plane cuts thru both, parallel to the cone axis, but not thru the "tip point" to get the two curves of the hyperbole.
     
    Last edited by a moderator: Sep 2, 2005
  10. curioucity Unbelievable and odd Registered Senior Member

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    OKay, so I guess I missed some things on those......

    Anyway, I just tried to think about this same problem, but with 3D objects (hollow 3D shapes, by the way).
    Starting from sphere. It took 3 points to define a circle, and it takes just an extra point to determine a sphere, as long as all 4 points are NOT on one same 2D plane (erased the no-3-on-a-line part, since they'll simply imply the four points are in one plane already). Here's my thought on that:

    First, take the 3 points and find the circle. Next, find the circle's center-point, and from that point, make a line which is perpendicular to the circle. This line will be the sphere's axis, and thus the center-point of the sphere will be in that axis (possible thinking, since we can imagine a circle (and in different cases later, all 3D objects) as a stack of thin circles). Voila, just try our best then to actually find the sphere with the aid of the fourth point ^_^

    I have a question about 3D shapes: Other than spheres, are there any other orderly 3D hollow shapes which are, by definition, a collection of points which are related in distance to certain points (thus I guess polyhedrals are out) ?


    PS: Oh, and by the way, since the geometrical cone is actually double-cone (like Billy T said), is there actually any formula that defines such shape?
     
    Last edited: Sep 6, 2005
  11. Physics Monkey Snow Monkey and Physicist Registered Senior Member

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    curioucity,

    A double cone can be described as all the point that satisfy the equation a^2 z^2 = x^2 + y^2 . a is constant that determines how quickly the cone "opens". Check mathworld for details.

    Also, it seems to me that a sphere in 3d can be determined by two points. One point gives the center of the sphere while the distance between the two points gives the radius. The equations for a sphere is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, and so the point (a,b,c) is the center, the radius is r, and everything is determined by two points.
     
  12. curioucity Unbelievable and odd Registered Senior Member

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    I see, thanks.

    About the sphere...... surething, though I just felt like finding out if we can define certain 3D shapes by knowing points which are located only on the surface, not the inside.


    PS: Just thought of adding to what Froggle suggested about how many points are needed to determine the conical sections in 2D Cartessian coordinate:
    For those points to be usable to define the said shapes, some conditions have to be satisfied:
    When any 3 random points are connected from point to point, they must make a triangle, and inside the triangle, there must be no point.
     
    Last edited: Sep 6, 2005
  13. Physics Monkey Snow Monkey and Physicist Registered Senior Member

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    My mistake, curioucity, I didn't realize we were trying to stick with points on the surface. Perhaps I should have read the thread more closely

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    .
     
  14. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    Sure:
    tan (A) = {sqrt(x^2 +y^2)}/z where "A" is half the cone's apex angle and (x,y,z) is any point on the surface of the cone.
     
  15. curioucity Unbelievable and odd Registered Senior Member

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    No problem, Physics Monkey

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    And looks like the formulas you and Billy T give are similar, except that you use a polynomial variable, while Billy T uses a trigonometrical one... so I get it that the formula is as such ^_^ thankies
     

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