# Parametric Equations For A Sine Wave With Changing Angle Relative To Growth / Decrease Factor

I've been working to create parametric equations for a sine wave that corresponds to the expressions beneath (Please Read My Description After The Expressions, or my post may not make sense!):

If, $$0≤t≤ 360$$

then:

$$x(t) = exp\left(t\frac{ln(1.4)}{360(1.2)}\right)sin(t)$$

$$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360}\right)$$

If, $$360≤t≤ 360(2)$$

then:

$$x(t) = exp\left(t\frac{ln(1.4)}{360}\right)sin(t)$$

$$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360(\frac{1}{1.2})}\right)$$

If, $$360(2)≤t≤ 360(3)$$

then:

$$x(t) = exp\left(t\frac{ln(1.4)}{360(\frac{1}{1.2})}\right)sin(t)$$

$$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360(\frac{1}{1.2})^2}\right)$$

If, $$360(3)≤t≤ 360(4)$$

then:

$$x(t) = exp\left(t\frac{ln(1.4)}{360(\frac{1}{1.2})^2}\right)sin(t)$$

$$y(t) = exp\left(t\frac{ln(\frac{1}{1.4})}{360(\frac{1}{1.2})^3}\right)$$

And so on....

What I want to do can be viewed as combining the above sine waves at the given values. I can't seem to find a way of doing this myself. But, I've reached the conclusion that there are two general forms the answer could take:

1. A sine wave where the angle shifts seamlessly, reaching exactly my values for the angle at the input values given, and never changing other than at multiples of 360 i.e., full revolutions.

2. A sine wave where my angle values are reached but only at the given inputs, that is, one where they do not remain constant in-between full revolutions; a wave where they gradually decrease to my values.

Ideally, I would like to find both solutions, or even another way of combining these waves which I haven't thought of but which still corresponds to the above.

Sorry for a super long post, but I really would be very interested in the answer as I can't seem to discover it myself nor does it appear to be in any of my books or online!

Thank you guys so much for your time and talent.

• Please ask if you want to know / see where I've gotten, or rather, failed to get (lol), with the problem thus far. – Jinny Ecckle Oct 4 '19 at 20:36
• What do you mean by $t=360, 360(2), 360(3)$?... Do you mean "if $0\leq t \leq 360$ then ..." ,"if $360 \leq t \leq 2 \times 360$ then ...", etc... – Jean Marie Oct 4 '19 at 22:15
• @JeanMarie You got it PERFECTLY, the computer was expressing the code oddly. Thank you! – Jinny Ecckle Oct 4 '19 at 22:21
• @JeanMarie I changed it in the post, that is, indeed, better expressed than how I had it. – Jinny Ecckle Oct 4 '19 at 22:33

These are all of the form: $$x(t) = \exp(\alpha(t) t)\sin(t), y(t) = \exp(\beta(t) t)\sin(t)$$ where $$\alpha(t)$$ and $$\beta(t)$$ are constant within each subinterval, but jump discontinuously at the subinterval boundaries. This time dependence can be modeled using step functions. For example, we can define a function that is 1 between $$t=0$$ and $$t=1$$ and 0 outside that interval: $$\theta(t) = \begin{cases} 1, & 0\le t \lt 1 \\ 0 & \text{otherwise} \end{cases}$$ Then if you have a piecewise constant function that changes values at $$t = 0, 1, 2, 3$$ and is 0 if $$t < 0$$ or $$t >= 4$$ say, you could write: $$f(t) = f_0 \theta(t) + f_1 \theta(t-1) + f_2 \theta(t-2) + f_3 \theta(t-3) +\ ...$$ where $$f_0$$ is the constant value in the $$0\le t \lt 1$$ subinterval and similarly for the others. You can change the scale of t in order to accommodate your subintervals, so since you want your subinternvals to be e.g $$0\le t \lt 360$$, you could write the $$\alpha(t)$$ function above: $$\alpha(t) = \alpha_0 \theta(t/360) + \alpha_1 \theta(t/360 - 1) + \alpha_2 \theta(t/360-2) + \alpha_3 \theta(t/360-3) +\ ...$$ where $$\alpha_0 = \frac{\ln(1.4)}{1.2\cdot 360}\\ \alpha_n = ({1.2})^n \alpha_0$$ and similarly for $$\beta(t)$$.

• Great work! Would you mind showing me an example of what this might look like for my (or a similar) parametric equations, up to say, 4 revolutions? – Jinny Ecckle Oct 5 '19 at 17:34
• This issue for me here is the definition of $$θ$$ – Jinny Ecckle Oct 5 '19 at 17:35
• By o, do you mean θ? – Jinny Ecckle Oct 5 '19 at 18:50
• "By o, do you meant $\theta$?" - I don't understand your question: what "o" are you talking about? And what issue do you have with the definition of $\theta(t)$? – NickD Oct 5 '19 at 19:08
• t=0,1,2,3 and is "o" if t<0 or t> – Jinny Ecckle Oct 5 '19 at 19:20

Maybe something like this,

$$x(t) = exp\left(t\frac{ln(1.4)}{360\left(\frac{1}{1.2}\right)^\frac{t}{360}}\right)sin(t)$$

but I'm really unsure though

• Thanks for taking the time; but I've tried things like this, and as far as I can tell, they don't appear to achieve my goal of combining the different sine waves accurately. But Thank you! – Jinny Ecckle Oct 4 '19 at 23:45
• sorry, that's all I could bring to mind. if I think of anything better i'll post it.... – user711404 Oct 4 '19 at 23:47