Quod Erat Demonstrandum


sine-like curve

Filed under: Fun,mathematics — johnmayhk @ 10:41 下午




9 則迴響 »

  1. Will It be a one-to-many function from right to left?

    迴響 由 Simon YAU YAU — 2013/10/02 @ 8:10 上午 | 回應

    • Yes, there are many choices other than the ones shown on the left.

      迴響 由 johnmayhk — 2013/10/02 @ 9:16 上午 | 回應

  2. Can we obtain the equation on the right to generate the equation on the left? For example, for the first one, it is a sine function, with which we obtain the so-called circular function so as to have a circle. Can the same reasoning be applied on other figures?

    迴響 由 Simon YAU YAU — 2013/10/02 @ 9:23 上午 | 回應

    • Note that if the method works, we will obtain all possible equations those generate figures on the left by studying the curves on the right.

      迴響 由 Simon YAU YAU — 2013/10/02 @ 9:25 上午 | 回應

    • If we have a method to parametrize x and y of the equations of the graphs on the left, i.e.


      the figures on the right are just the graphs of


      Now, given an equation of a periodic function y=y(\theta) on the right, it is quite arbitrary to take different equation


      to obtain different graphs on the left.

      迴響 由 johnmayhk — 2013/10/02 @ 10:08 上午 | 回應

      • Can you prove it mathematically?

        迴響 由 Simon YAU YAU — 2013/10/03 @ 1:41 下午

      • Your point is NOT correct. For the circle, y=sint or y=cost, without loss of generality, let y=sint but x must be cost but NOT any arbitrary equation, e.g. x cannot tant or sect etc.

        迴響 由 Simon YAU YAU — 2013/10/03 @ 2:13 下午

  3. In my post, I said “給出右圖來估左圖", the word “估" means “guess", not “determine".

    In fact, when we are given the equation on the right


    only without any extra condition, it is impossible to “determine" the equation of graph on the left of course.

    Take arbitrary function


    is just a ‘guess’,

    may be it is playful to take different functions of x=x(\theta) to obtain many different choices of left graph for having some fun, that’s why I said “可能幾好玩" in the post.

    迴響 由 johnmayhk — 2013/10/03 @ 3:56 下午 | 回應

    • I think x=f(t) and y=g(t) will give the same shapes when t varies, but with different “X" ( I do not know the word, I hope you can get what I want to point out. ). Of course, this is an assumption and has to be proven or disproven.

      迴響 由 Simon YAU YAU — 2013/10/03 @ 6:20 下午 | 回應

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