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Stefan Maehlmann August 10, 2000 11:26

Superellipses
 
Hello,

does anyone know something about Superellipses or so called "modified Superellipses" of Lin, maybe formulas or literature references?

Thanx,

Stefan Maehlmann

IL Kon Choi August 10, 2000 18:17

Re: Superellipses
 
The cross section of the transition duct is described usually by the equation of a superellipse.

Your 'Modified Superellipses' may need some cumbersome algebra with special functions such as 'gamma function' in these equations.

'Anything about Superellipse' can be found in a typical mathematical table or handbook.

IL Kon Choi


Mehdi BEN ELHADJ August 11, 2000 06:26

Re: Superellipses
 
Hi,

you can write your question at "Math Forum" on http://forum.swarthmore.edu/.

Best Regards


John C. Chien August 11, 2000 09:36

Re: Superellipses
 
(1). A circle is: (x*x) + (y*y) = a*a (2). Formula for ellipse is: (x*x)/(a*a) + (y*y)/(b*b) =1 (3). A superellipse is something like (not quite sure of it): (x/a) to the power of (n) + (y/b) to the power of (n)=1. (4). If you set n=2, then you get formula for ellipse. Further setting b=a, you get back the equation for a circle. (5). You can plot the curves to find out the shapes. By changing the power of (n), the shape will approach a rectangular shape, form the basic shape of a circle or ellipse.

John C. Chien August 11, 2000 09:58

Re: Superellipses, errors
 
(1). Item-5, should read By ..., the shape..., from the basic shape of a circle or ellipse.

Steve Amphlett August 11, 2000 10:00

Re: Superellipses
 
What if n is odd though?

John C. Chien August 11, 2000 11:33

Re: Superellipses
 
(1). I don't have the math handbook with me, but I think, the general form is something like that. (2). It may not be the exact form, but it is in the right direction. I don't recall any limit on the value of (n). If (n) is greater than one, the curve will be convex. If (n) is less than one, the curve will be concave.

Mehdi BEN ELHADJ August 11, 2000 11:50

Re: Superellipses
 
In general case we take the absolute values of (x/a) and (y/b) that's why no limitation on the value of (n).

John C. Chien August 11, 2000 14:20

Re: Superellipses
 
(1). The complete shape is symmetric wrt x- and y-axis, therefore, only positive x and y are needed to define the one quarter of the complete shape. (2). The rest of the missing curves can be easily obtained by mirror reflection to get a complete shape.

Mehdi BEN ELHADJ August 11, 2000 16:50

Re: Superellipses
 
We must take (n) different for 0 if not we have 1+1=1?

we must take (n) different for "infty" if not we have 0+0=1? because abs(x)<a and abs(y)<b

Mehdi BEN ELHADJ August 11, 2000 16:53

Re: Superellipses; Incomplete
 
We must take (n) different for 0 if not we have 1+1=1?

we must take (n) different for "infty" if not we have 0+0=1? because abs(x)<a and abs(y)<b

Mehdi BEN ELHADJ August 11, 2000 16:54

Re: Superellipses; Incomplete
 
We must take (n) different for 0 if not we have 1+1=1?

we must take (n) different for "infty" if not we have 0+0=1? because abs(x)is less than (a) and abs(y)is less than b

clifford bradford August 11, 2000 17:12

Re: Superellipses
 
stefan, the eqtn for a superellipse looks just like the one for an ellipse except the power can vary from 1 to infinity. when the power is one you get a kite and when it tends to infinity you get a rectangle. super-ellipses are good for some kind of arbitrary fillet (I assume that's what you're using them for)

John C. Chien August 11, 2000 23:54

Re: Superellipses; Incomplete
 
(1). (y/b)to the power of (n)=1-(x/a)to the power of (n). (2). For (x/a)=0, (x/a)to the power of (n) is zero. (3). So, at (x/a)=0, (y/b)=(1)to the power of (1/n). So, (y/b)=1. (4). For (x/a)greater than zero, (x/a) to the power of (n) is one, if (n) is zero. Therefore, (y/b)=(1-1)to the power of (1/n), which is zero. (5). Hence, for (n) goes to zero, (y/b)=1 at (x/a)=0. and (y/b)=0, for (x/a) greater than zero. (6). So, the profile is (y/b)=0 except at (x/a)=0,where (y/b)=1. (7). For (x/a)less than one, (x/a)to the power of (n) is zero, if (n) is infinity. Thus, for (n) goes to infinity, (y/b)to the power of (n)=1. Therefore (y/b)=1 for (x/a)less than one, when (n) goes to infinity. (8). At (x/a)=1, (x/a)to the power of (n) is one. Therefore, (y/b)to the power of (n)=(1-1)=0. (9). So, For (n) goes to infinity, (y/b)=1,except at (x/a)=1, where (y/b)=0. (10). Therefore, the formula is good for (n) from 0 to infinity. (11). Sorry for the lengthy answer.

Mehdi BEN ELHADJ August 12, 2000 05:04

Re: Superellipses;
 
case if n=0

** For (x/a)=0, (x/a)to the power of (n) is zero only if n is different of zero. So we can't say (y/b)=1 at (x/a)=0

** For abs(x/a) greater than zero and less than 1, we should have (abs(y/b))to the power of (n=0)=1 so y must be different for zero and we don't have (y/b)=0, for (x/a) greater than zero and less than one. If abs(x/a)=1,we have (abs(y/b))to the power of (n=0)=0 so it's impossible.

So the case of n=0 is not true.


Mehdi BEN ELHADJ August 12, 2000 05:09

Re: Superellipses;errors
 
case if n=0

** For (x/a)=0, (x/a)to the power of (n) is zero only if n is different of zero. So we can't say (y/b)=1 at (x/a)=0

** For abs(x/a) greater than zero, we should have (abs(y/b))to the power of (n=0)=0 so it's impossible and we don't have (y/b)=0, for (x/a) greater than zero .

So the case of n=0 is not true.

Mehdi BEN ELHADJ August 12, 2000 05:19

Re: Superellipses
 
when (n) tends to infinity you get a rectangle private of the points (a,-b); (a,b); (-a,b) and (-a,-b)

John C. Chien August 12, 2000 09:27

Re: Superellipses;?
 
(1). I think we both have a long way to go. (2). I have been using this Japanese Sharp palm scientific computer EL-5500III for well over 12 years now. (3). So, either the computer is too old, or the Sharp engineers were not smart, or the Japanese math is different. (4). As I said at the begining of the message, I don't have a math handbook with me. It would be nice to have an English version of a German math handbook. (5). It seems to me that in this digital age, we still need a lot of help from Mr. Newton. Or perhaps we should call him Sir Newton.

Mehdi BEN ELHADJ August 12, 2000 10:41

Re: Superellipses;?
 
I use only a pen and a paper!

John C. Chien August 12, 2000 13:21

Re: Superellipses;#
 
(1). A company dies because it refuses to change. (2). A person dies because it refuses to learn. It is hard to learn by hanging to the old pen . (3). Last year, to study the rollerball pen , I tried Waterman, Parker, Mont Blanc, Cross, uni-ball, pilot G2, Optima,unigel,Panther,Reflection. (I own these pens) (4). The best smooth writing roller ball pen is Cross gold pen with 0.7mm gel type refill made in Japan, which does not use internal sponge material to retain the ink.(which is available only recently) (5). The superellipse equation has been used in aerospace vehicle design for a long time in the CAD program. There is nothing new there.


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