## Re: [EMHL] Greek Columns

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• Thank you Patrick. The first formulation of Problem is correct: Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k.
Message 1 of 7 , Nov 30, 2009
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Thank you Patrick. The first formulation of Problem is correct:

Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k. The two tangents from A to circle w intersect m at two points B and C. Prove that the length of BC is lesser if point A is closer to circle w.

Note that the illusion (that Greek columns which are closer to observer look as smaller) which I mentioned in my 1st message can also experimented with equal sized balls (spheres). See also the links:

http://www.istockphoto.com/file_thumbview_approve/6779165/2/
http://lifehackery.com/qimages/5/used-tennis-balls.jpg

Note that the columns/balls in these pictures which are closer to observer look as smaller with respect to columns/balls in the right and left sides of the pictures.
Yakub

--- In Hyacinthos@yahoogroups.com, Patrick Morton <patrickmorton289@...> wrote:
>
> Yakub,
>
> I think there's a problem with the way you pose your question: Don't you mean the length of BC is "greater" if point A is closer to circle w?
>
> Patrick
>
>
>
> ________________________________
> From: yakub.aliyev <yakub.aliyev@...>
> To: Hyacinthos@yahoogroups.com
> Sent: Thu, November 26, 2009 4:33:54 AM
> Subject: [EMHL] Greek Columns
>
>
> Dear friends. The well known controversial visual illusion effect that in pictures the farthest column seems to be greater and closer column seems to be smaller is related to the following geometrical problem:
>
> Problem:
Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k. The two tangents from A to circle w intersect m at two points B and C. Prove that the length of BC is lesser if point A is closer to circle w.

>
> Who knows the solution of this problem? Inform us.
> Yakub.
>
>
>
>
>
>
>
> [Non-text portions of this message have been removed]
>
• Dear friends, we received a solution from Angel Montesdeoca using calculus. See the link:
Message 2 of 7 , Dec 3, 2009
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Dear friends, we received a solution from Angel Montesdeoca using calculus.

It is also possible to present completely synthetic solution for the problem. Patrick Morton proposed the following correction:

Problem: Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k. The two tangents from A to circle w intersect m at two points B and C. As the point A gets closer to w along the fixed line k, the distance BC decreases, the minimum value of BC being reached when A is on the perpendicular to k through the center of w.

Thanks to all.
Yakub.

--- In Hyacinthos@yahoogroups.com, "yakub.aliyev" <yakub.aliyev@...> wrote:
>
> Dear friends. The well known controversial visual illusion effect that in pictures the farthest column seems to be greater and closer column seems to be smaller is related to the following geometrical problem:
>
> Problem: Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k. The two tangents from A to circle w intersect m at two points B and C. Prove that the length of BC is lesser if point A is closer to circle w.
>
> Who knows the solution of this problem? Inform us.
> Yakub.
>
• Dear friends, ... To discuss what is perceived, I prefer to use an observer centered frame. Let A(0,0) be the observer, and K (c,-t) the centers of the columns
Message 3 of 7 , Dec 3, 2009
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Dear friends,
--- In Hyacinthos@yahoogroups.com, "yakub.aliyev" <yakub.aliyev@...> wrote:
> Dear friends. The well known controversial visual illusion effect that in pictures the farthest column seems to be greater and closer column seems to be smaller is related to the following geometrical problem...

To discuss what is perceived, I prefer to use an observer centered frame. Let A(0,0) be the observer, and K (c,-t) the centers of the columns (c fixed, t changing).
Then BC = 2*r*c*W/(c^2-r^2) where W^2=c^2-r^2+t^2
as given by Angel Montesdeoca.

If I, the observer, was the sun, BC would be the shadow of the column on some wall behind the column line. But this quantity cannot be used to describe what is the human perception of the columns. When t tends to infinity, the shadow becomes infinite too, but the human perception of the column becomes "too small to be observed".

We can try to consider the length of the chord that joins the contact points. We obtain ST= 2*r*sqrt(d^2-r^2)/d where d^2=c^2+t^2, d being the distance from observer to the center of the column. If radius of columns is 1 meter, and you are standing 7 meters away from the column line, ST increases of 1% when t goes from 0 (the column in front of you) to infinity (the last column in the line).

But perception of size is not obtained by measuring ST. What is measured is angle (AS,AT) and from **binocular** vision, an estimation of the distance. Does our embedded telemeter focus on midpoint of ST (inside the column) or on the proximal point of the column ? In the second case, factor (d+r)/d must be applied and we obtain 2r*sqrt((d-r)/(d+r)) as estimated size of the column, instead of the obvious 2r. Using again r=1, c=7, then "estimated" ST ranges from 1.73 when t=0 to 2 when t=inf. This time, effect is huge.

Therefore, the main question is : what are the laws of perception ?

Best regards.
• Dear Pierre, ... The illusion which I mentioned in my previous messages considers the size of image of the Greek column in picture (line m) but not its image
Message 4 of 7 , Dec 8, 2009
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Dear Pierre,
>Original Problem: Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k. The two tangents from A to circle w intersect m at two points B and C. As the point A gets closer to w along the fixed line k, the distance BC decreases, the minimum value of BC being reached when A is on the perpendicular to k through the center of w.

The illusion which I mentioned in my previous messages considers the size of image of the Greek column in picture (line m) but not its image on human eyes. The line m in the original problem is this picture. I think the following problem modelling human perception is quite natural.
Problem: Let k and w be fixed line and circle, respectively. The point A is taken on the circle v with variable center O on k and fixed radius, such that OA perpendicular to k. Let tangents from A to w intersect circle v at the points B and C. Prove that as the point O gets closer to w along the fixed line k, the length of arc (segment) BC increases.

The solution is immediate from the facts which you mentioned. Angle BAC is getting greater as O is getting closer to w.
The remaining question is which is the human perception of column: segment BC or arc BC?
Yakub

--- In Hyacinthos@yahoogroups.com, "Pierre" <pldx1@...> wrote:
>
>
>
>
> Dear friends,
> --- In Hyacinthos@yahoogroups.com, "yakub.aliyev" <yakub.aliyev@> wrote:
> > Dear friends. The well known controversial visual illusion effect that in pictures the farthest column seems to be greater and closer column seems to be smaller is related to the following geometrical problem...
>
> To discuss what is perceived, I prefer to use an observer centered frame. Let A(0,0) be the observer, and K (c,-t) the centers of the columns (c fixed, t changing).
> Then BC = 2*r*c*W/(c^2-r^2) where W^2=c^2-r^2+t^2
> as given by Angel Montesdeoca.
>
> If I, the observer, was the sun, BC would be the shadow of the column on some wall behind the column line. But this quantity cannot be used to describe what is the human perception of the columns. When t tends to infinity, the shadow becomes infinite too, but the human perception of the column becomes "too small to be observed".
>
> We can try to consider the length of the chord that joins the contact points. We obtain ST= 2*r*sqrt(d^2-r^2)/d where d^2=c^2+t^2, d being the distance from observer to the center of the column. If radius of columns is 1 meter, and you are standing 7 meters away from the column line, ST increases of 1% when t goes from 0 (the column in front of you) to infinity (the last column in the line).
>
> But perception of size is not obtained by measuring ST. What is measured is angle (AS,AT) and from **binocular** vision, an estimation of the distance. Does our embedded telemeter focus on midpoint of ST (inside the column) or on the proximal point of the column ? In the second case, factor (d+r)/d must be applied and we obtain 2r*sqrt((d-r)/(d+r)) as estimated size of the column, instead of the obvious 2r. Using again r=1, c=7, then "estimated" ST ranges from 1.73 when t=0 to 2 when t=inf. This time, effect is huge.
>
> Therefore, the main question is : what are the laws of perception ?
>
> Best regards.
>
• Dear friends, ... I must also note that in this new problem modeling human perception: 1) the circle v is observers eye 2) point A is the entering point for
Message 5 of 7 , Dec 8, 2009
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Dear friends,
> "New" Problem: Let k and w be fixed line and circle, respectively. The point A is taken on the circle v with variable center O on k and fixed radius, such that OA perpendicular to k. Let tangents from A to w intersect circle v at the points B and C. Prove that as the point O gets closer to w along the fixed line k, the length of arc (segment) BC increases.

I must also note that in this new problem modeling human perception: 1) the circle v is observers eye 2) point A is the entering point for light rays to eye 3) BC is image of circle w on back side of eye. Unfortunately, I don't know scientific names of these parts of eye.
Yakub

--- In Hyacinthos@yahoogroups.com, "yakub.aliyev" <yakub.aliyev@...> wrote:
>
> Dear Pierre,
> >Original Problem: Let k and m be parallel lines and w be a circle not intersecting with k. Let A be a point on k. The two tangents from A to circle w intersect m at two points B and C. As the point A gets closer to w along the fixed line k, the distance BC decreases, the minimum value of BC being reached when A is on the perpendicular to k through the center of w.
>
> The illusion which I mentioned in my previous messages considers the size of image of the Greek column in picture (line m) but not its image on human eyes. The line m in the original problem is this picture. I think the following problem modelling human perception is quite natural.
> New Problem: Let k and w be fixed line and circle, respectively. The point A is taken on the circle v with variable center O on k and fixed radius, such that OA perpendicular to k. Let tangents from A to w intersect circle v at the points B and C. Prove that as the point O gets closer to w along the fixed line k, the length of arc (segment) BC increases.
>
> The solution is immediate from the facts which you mentioned. Angle BAC is getting greater as O is getting closer to w.
> The remaining question is which is the human perception of column: segment BC or arc BC?
> Yakub
>
> --- In Hyacinthos@yahoogroups.com, "Pierre" <pldx1@> wrote:
> >
> >
> >
> >
> > Dear friends,
> > --- In Hyacinthos@yahoogroups.com, "yakub.aliyev" <yakub.aliyev@> wrote:
> > > Dear friends. The well known controversial visual illusion effect that in pictures the farthest column seems to be greater and closer column seems to be smaller is related to the following geometrical problem...
> >
> > To discuss what is perceived, I prefer to use an observer centered frame. Let A(0,0) be the observer, and K (c,-t) the centers of the columns (c fixed, t changing).
> > Then BC = 2*r*c*W/(c^2-r^2) where W^2=c^2-r^2+t^2
> > as given by Angel Montesdeoca.
> >
> > If I, the observer, was the sun, BC would be the shadow of the column on some wall behind the column line. But this quantity cannot be used to describe what is the human perception of the columns. When t tends to infinity, the shadow becomes infinite too, but the human perception of the column becomes "too small to be observed".
> >
> > We can try to consider the length of the chord that joins the contact points. We obtain ST= 2*r*sqrt(d^2-r^2)/d where d^2=c^2+t^2, d being the distance from observer to the center of the column. If radius of columns is 1 meter, and you are standing 7 meters away from the column line, ST increases of 1% when t goes from 0 (the column in front of you) to infinity (the last column in the line).
> >
> > But perception of size is not obtained by measuring ST. What is measured is angle (AS,AT) and from **binocular** vision, an estimation of the distance. Does our embedded telemeter focus on midpoint of ST (inside the column) or on the proximal point of the column ? In the second case, factor (d+r)/d must be applied and we obtain 2r*sqrt((d-r)/(d+r)) as estimated size of the column, instead of the obvious 2r. Using again r=1, c=7, then "estimated" ST ranges from 1.73 when t=0 to 2 when t=inf. This time, effect is huge.
> >
> > Therefore, the main question is : what are the laws of perception ?
> >
> > Best regards.
> >
>
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