• Page
• Discussion
• View source
• History

From Odwiki

Contents 1 Cycloids 2 Ordinary Cycloid 2.1 Trochoids 3 Epicycloid 3.1 Epitrochoids 3.2 Cardioid 4 Hypocycloid 4.1 Astroid 4.2 Hypotrochoids 5 Lissajous-Functions 5.1 Circle 5.2 Figure 1 5.3 Figure 2

Cycloids

Cycloids are curves defined by a fixed point on a wheel as it rolls along a curve or straight line. Several mathematicans in the 15-16th century studied them in detail. Although they are quite intresting from the math point of view, they are for most of you even more intresting to create nice swirling particle effects and alike.
Usually we find these function in the following form like this (implicit):

x = a arccos ((a-y)/a) - sqrt(y(2a-y))

For Animationpaths this is usually of little usage unfortunally, therefore we need to 'seperate' this function, so we have no dependencies of x and y and can calculate each value seperatly. This involves some math trickery, i.e. I'll skip it and directly give the result. For the function above I'd write:

x = a (t - sin t)

y = a (1 - cos t)

For free paremeters I'll use the variables a,b,c and you can replace them with any number ∈ R unless noted otherwise.

• Usage: Simply write write a expression for the translate channels of the object you intend to animate. For the function above we could do the following:
  

We choose a = 2 and place according to the formula the following expressions:

TranslateY = 2 * ($F - sin($F))

TranslateZ = 2 * ($1 - cos($F))

Looks intresting, doesn't it?

Ordinary Cycloid

The simplest cycloid that is around.

• x = a arccos ((a-y)/a) - sqrt(y(2a-y)) with a > 0

• x = a (t - sin t)
  

y = a (1 - cos t)

Trochoids

Trochoids are elongated or shortend cycloids

• x = at - c sin t

y = a - c cos t

elongated cycloid: c > a

shortend cycloid: c < a

Epicycloid

• x² + y² = (a+² + b² - 2b(a+ cos(a² t / b²)

• x = (a+ cos t - b cos((a+t/b)

y = (a+ sin t - b sin((a+t/b)

Epitrochoids

Similar to Trochoids, Exitrochoids are the elongated or shortend version of epicycloids

• x = (a+ cos t - c cos((a+t/b)

y = (a+ sin t - c sin((a+t/b)

elongated epicycloid: c > b

shortend epicycloid: c < b

Cardioid

The ratio m of a/b gives the number of cusps in thte epicycloids. With one cusp the resulting curves resembles a heartshape and thus the name, i.e. m = 1.

Hypocycloid

• x = (a- cos t + b cos((a-t/b)

y = (a- sin t - b sin((a-t/b)

with a > b m = a/b

Astroid

for m = 4 we can rewrite the equation like this:

• x = a cos³(t)

y = a sin³(t)

Hypotrochoids

• x = (a- cos t + c cos((a-t/b)

y = (a- sin t - c sin((a-t/b)

elongated Hypotrochoid: c > b

shortend Hypotrochoid: c < b

Lissajous-Functions

Though no cycloids, Lissajous figures / functions are a superposition of harmonic functions and are cyclic.

Circle

• x = a sin (wt)

y = a sin(wt - pi/2) = cos (wt)

Figure 1

• x = a sin (wt)

y = a sin (2wt - pi/4)

Figure 2

• x = a sin (2wt)

y = a sin (3wt)