# Quod Erat Demonstrandum

## 2013/08/30

### sine-like curve

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

source: ROBERT T. GONZALEZMATHS

## 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.

$x=x(\theta)$
$y=y(\theta)$

the figures on the right are just the graphs of

$y=y(\theta)$

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

$x=x(\theta)$

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

$y=y(\theta)$

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

Take arbitrary function

$x=x(\theta)$

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