Exotic spirals in excel
Hypo-cycloid
In geometry, a hypo-cycloid is a special plane curve generated
by the trace of a fixed point on a small circle that rolls within a
larger circle. It is comparable to the cycloid but instead of the
circle rolling along a line, it rolls within a circle. [4]
Figure 1 generating a hypocycloid
These are the parametric equation of the hypocycloid the angle being
the parameter (if the rolling circle rotates with constant angular
velocity, will be proportional to the elapsed time since the motion
began). The general shape of the curve depends on the ratio R= r. If
this ratio is a fraction m=n in lowest terms, the curve will have m
cusps (corners), and it will be completely traced after moving the
wheel n times around the inner rim. If R=r is irrational, the curve
will never close, although going around the rim many times will
nearly close it. [1]
A hypocycloid with three cusps is known as a deltoid. A hypocycloid
curve with four cusps is known as an asteroid.[2] The evolutes of a
hypocycloid are an enlarged version of the hypocycloid itself, while
the involute of a hypocycloid is a reduced copy of itself. The pedal
of a hypocycloid with pole at the center of the hypocycloid is a rose
curve. The isoptic of a hypocycloid is a hypocycloid.[3]
Step1: Enter the formula =RAND()*1000 in cells A2 , B2 and C2.
Step2: Enter the values 1.00 , 1.00 and 0 in cells A5, B5 and C5
respectively.
Step3: Enter the formula =A5+a_inc in cell A6 and fill and handle to
cell A381.
Step4: Enter the formula =B5+b_inc in cell B6 and fill and handle
to cell B381.
Step5: Enter the formula =C5+c_inc in cell C6 and fill and handle to
cell C381.
Step6: Enter the formula =(A5-B5)*COS(C5)+B5*COS((A5/B5-1)*C5) in
cell D5 and
fill and handle to cell D381.
Step6: Enter the formula =(A5-B5)*SIN(C5)-B5*SIN((A5/B5-1)*C5) in
cell E5 fill and handle to cell E381.
Step7: Select the cells from D5:E381 and use
Insert->Chart->XY(Scatter) and select the
3rd graph type as shown in figure 2 and click next till
you reach Finish.
Step8: Press Random button for changing the random values.
Sit and enjoy the different spirals generated by excel.
Fig 3
Exotic spiral in excel
Here we
can save our favorite spirals using the button Add to Favorites,
internally it copy the corresponding value of random number in to
separate location. Then after anyone want favorite just select the
option show favorites it shows all stored favorite.
Draw a
dependancy diagram
References
for further reading
- H. Martyn Cundy and A. P. Rollett, Mathematical Models (London: Oxford University Press, 1961), chapters 2 and 5.
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