topic Re: Ellipse (again) in Libraries & objects

Ellipse (again)

Anonymous — Wed, 12 Jul 2006 00:59:51 GMT

Hello,
Today I was informed by Tom Waltz that using the MUL2 command
in a 2D script on a command that draws a circle such as CIRCLE2
does not make a true ellipse but an approximation.
I did a search on GDL TALK using the key word "ellipse"
and found a post by Alex Schamenek the text of which follows.

-------------------------------------------------------------------------------------
"Okay, now I really need to create an ellipse. I know you can fake one by using
MUL2 with a circle, but when you explode a stretched object, anything round gets
faceted. So I can't just use a MUL2 statement to convert a circle into an
ellipse.

So PLEASE Graphisoft!!! Provide a way for us script an ellipse, without MUL
transformations!!!!"
-------------------------------------------------------------------------------------

I don't know how many scripts by many different people
I have seen that use either the 3D command MUL or
the 2D command MUL2 to change a circular object or figure into an elliptical shape.
So I am not the ownly one that thought that using a MUL command
on a circular shape produced an elliptical shape.

The Cartesian formula for an ellipse is
"x" squared over "a" squared plus "y" squared over "b" squared equals one

I have used this formula to find the the x and y coordinates
for points along elliptical curves in several GDL scripts.

I am not mathematical enough to write a formula that
describes the resultant shape of a circle whose "Y" coordinate values
for all points has been multiplied by some factor and then
compare this formula with the Cartesian formula to see if
they are equivalent.

Does any one have more information on these issues ?

If what happens when a circular shape is distorted
using a Mul command does not produce an ellipse then
what does it produce and what is the difference between it
and a true ellipse ?

Or, said another way, does anyone know how to write a formula that
describes the resultant shape of a circle whose "Y" coordinate values
for all points have been multiplied by some factor ?

Thank you,
Peter Devlin

Re: Ellipse (again)

LiHigh — Wed, 12 Jul 2006 04:18:45 GMT

Peter, the formula you have is the answer you're looking for!

the scripts shall look like this:

--------------------------------------------------
a0=a/2
b0=b/2
c=SQR(a0^2-b0^2)

DIM x_pos[],y_pos[]

n=20
delta_x=a/n

FOR u=1 TO 2
FOR i=1 TO n+1

x_pos=a0-delta_x*(i-1)

y_pos=SQR((1-(x_pos^2)/a0^2)*b0^2)

hotspot2 x_pos,y_pos

NEXT i
MUL2 1,-1
NEXT u
---------------------------------------------------------------------------

Re: Ellipse (again)

Anonymous — Wed, 12 Jul 2006 04:35:14 GMT

Hello LiHigh,
Thank you for posting back.
It is going to take me a while to understand your script
because, as I said, I am not very mathematical.

I'm going to copy your script into the 2D script of a new object
and see if I can figure it out.

Do you know whether a circular shape operated on
by a MUL command produces a true ellipse or some other shape ?

Thank you,
Peter Devlin

Re: Ellipse (again)

Anonymous — Wed, 12 Jul 2006 05:02:37 GMT

Hello LiHigh,
I think I understand your script now.
I have written something quite similar
using the ellipse formula except I did not use an array
but used put and get and drew the ellipse as lines segments.
I used a much larger "n" value to make the arc look smoother.

Using the ellipse formula to draw an ellipse does not answer
the question about whether an ellipse drawn by MULing a circle
is the same shape (meaning a true ellipse).

Thank you,
Peter Devlin

Re: Ellipse (again)

Anonymous — Wed, 12 Jul 2006 05:57:54 GMT

Peter,

I don't have a proof for it, but intuitively it seems to me that the mul statement should produce a true ellipse. Since an ellipse is just a circle viewed obliquely it makes sense to me that this would be equivalent to a uniform distortion along one axis. There may even be a simple geometric proof for this but I am too tired to think about it right now.

Re: Ellipse (again)

LiHigh — Wed, 12 Jul 2006 06:16:21 GMT

Peter wrote:
Hello LiHigh,
I think I understand your script now.
I have written something quite similar
using the ellipse formula except I did not use an array
but used put and get and drew the ellipse as lines segments.
I used a much larger "n" value to make the arc look smoother.

Using the ellipse formula to draw an ellipse does not answer
the question about whether an ellipse drawn by MULing a circle
is the same shape (meaning a true ellipse).

Thank you,
Peter Devlin

Oh! sorry, I tought you're talking about the formula.

I think ellipse drawn by MULing a circle
is infact a true ellipse.

You can double check by adding a equal-size stretched circle to the above script.

Re: Ellipse (again)

Frank Beister — Wed, 12 Jul 2006 06:21:57 GMT

You can proof, if ArchiCAD aproximates an ellipse (if there is a need to approxiamte), by drawing it with the mul2 command and overlay it by hotspots. As you can see with e.g. theEllipse object it is the same.

If you use openGL for the 2D-views: I have mentioned some aberrations if you zoom in a very large scale.
If you explode an ellipse object the resulting segment lines (!) can't be a true ellipse, but if you draw the ellipse inside the script not by mul2 but by the formula it wouldn't be even correct as proper object.

There was a long thread about the ellipse issue on german GDL Talk. It was for creating true elliptical shapes with prism_ without the mul command. We tried to appriximate with arcs. This works not bad. Not really exact, but very less points than the iterative way with polylines. (For this the EllipsePro object was written for. A case study.)

My advice: Use MUL and MUL2. 😉
MHO: Tom fails here. 😉

Re: Ellipse (again)

James Murray — Wed, 12 Jul 2006 12:25:32 GMT

MUL is the way.

Link. "If the major and minor axes are equal, the ellipse is a circle. Shall we define an ellipse as a distorted circle, then, or the circle as a special case of the ellipse? Either one would be reasonable and useful."

Re: Ellipse (again)

Anonymous — Wed, 12 Jul 2006 20:34:07 GMT

Hello to you all and thank you for your interest.

There seems to be a consensus here that
using the MUL commands on a circular element
results in a true ellipse.

I have done the test of placing an instance of an object
that draws an ellipse by MULing a circle and then making
an ellipse with the circle tool with the same dimensions
as the object and it appears that the two exactly coincide.

That they appear to be the same, to me, does not prove
that they are indeed the same.

I wish I could figure out a way of writing a formula/equation
that mimics what happens when distance values along
one axis have a multiplication factor applied to them.
Or, more satisfying to me, Matthew's suggestion
of a geometric proof.

I wish I could read German so I could read the
discussion Frank referred to. It sounds fascinating.

It appears that, for the moment, we have a consensus
but not a proof.

I thank you all for your interest and commentary.
Peter Devlin

Re: Ellipse (again)

James Murray — Wed, 12 Jul 2006 23:40:37 GMT

This guy points out, when you cut a cone with a plane you always get an ellipse, and if you use a plane perpendicular to the axis you get a circle, therefore a circle is an ellipse.

This looks like a proof. "In other words, an ellipse is the projection of a circle." Step by step from ellipse equation to circle equation.

Interesting.

Re: Ellipse (again)

Anonymous — Thu, 13 Jul 2006 00:39:40 GMT

Hello James,
Very interesting indeed.
In all the math sites I have gone to it is always stated
that a circle is a special case of an ellipse where the
the foci are coincident at 0,0 or the center of the ellipse.
I would suggest that one could say that if the major axis
and minor axis are equal then the ellipse is a circle.

On the subject of a mathematical proof that the mul command
operating on a circle results in a true ellipse, a man
posted to GDL TALK the following code for drawing
an ellipse.

gosub 10
rot2 180
gosub 10
end

10:
for i=-1 to 1 step .05
hotspot2 i, b*sqr(1-(i*i/a*a))
next i
return

I don't understand this code yet
but it appears to be in the form
where the "Y" values are calculated
by multiplying by a factor.

I have asked this person to show the
steps in the derivation of this code.

Thanks,
Peter Devlin

Re: Ellipse (again)

LiHigh — Thu, 13 Jul 2006 02:09:00 GMT

y=b*sqr(1-(i*i/a*a))

Peter, this equation is equivalent to what we have here.

Again, I believe strectched-circle is an ellipse.

In order to proof it, we must first define what is an ellipse. The foolowing definition is quoted fron Answer.com:

A plane curve, especially:
1. A conic section whose plane is not parallel to the axis, base, or generatrix of the intersected cone.
2. The locus of points for which the sum of the distances from each point to two fixed points is equal.

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Re: Ellipse (again)

Anonymous — Thu, 13 Jul 2006 02:44:45 GMT

Hello LiHigh,
Then you agree, this man's code is in the form
where y values are multiplied by a factor just like Mul .
Then we have a proof that a mul operation performed
on the values along one axis of a circle
indeed generates a true ellipse.

To be complete, all that needs to be done is transform
this equation y=b*sqr(1-(i*i/a*a))
into the Cartesian equation for an ellipse.

Intuitively, it seemed that it had to be true
but I had not satisfied myself that it could be proved.

I, for one, am relieved and pleased.

Thank you LiHigh,
Peter Devlin

Re: Ellipse (again)

Fabrizio Diodati — Thu, 13 Jul 2006 08:32:03 GMT

Hi Peter,

try the attached one, is a "dirty" experiment made in some minutes... it can be improved but it could be a could start for you.

Friendly
Fabrizio

Re: Ellipse (again)

Anonymous — Thu, 13 Jul 2006 18:04:13 GMT

Hello Fabrizio,
Thank you for the object.
Nice touch with the extra hotspots where the curve gets tight.
The object is drawn using the Cartesian equation for an ellipse
I have several of them now including a few I have made.

The man who wrote the code that used the equation
y=b*sqr(1-(i*i/a*a))
posted back with the derivation. He said:

Hello peter

thse script is derived from x^2/a^2 + y^2/b^2 = 1

x^2/a^2 + y^2/b^2 = 1
y^2/b^2=1-x^2/a^2
y^2=(1-x^2/a^2)/b^2
y=sqr((1-x^2/a^2)*b^2)
y=sqr((1-x^2/a^2))*b
y=b*sqr((1-x^2/a^2))

replace y and x by i
and the result is

i, b*sqr(1-(i^2/a^2))

Regards...
Eric Wilk
Webmaster FC-CadLink.com

So the proof is complete.

Peter Devlin