Circles in Perspective 3

[dalgoma]

CIRCLES IN PERSPECTIVE 3

 

When you are not looking at them!

 

 

When we view the rim of a cup on a table directly in front of us, the major axis of the elliptical image we see is always horizontal from whatever angle we view it.

 

However, it does tilt if the cup is positioned to left or right of our centre of vision.

 

Here is a way to establish that tilt angle.

 

On essentially a picture plane -

 

a. Draw a horizontal line AB representing eye level and the vertical centre line of vision CD intersecting at O.

 

b. Describe a "boundary" circle radius d centred on O intersecting CD at X and Y. "d" is equivalent to the distance of the viewpoint in front of this picture plane and is that self same circle as the "bubble" circle used in CiP2 to generate those truly circular images.

 

c. Now pick a point P that is the centre of an elliptical image.

 

d. Join PX (or PY) and construct its perpendicular bisector to meet AB at N.

 

e. Describe a circle centre N and radius PN intersecting AB at E and F.

 

f. Join PE and PF. These are the axes of an elliptical image centred on P viewed from distance d.

 

 

Note that for a series of locations for point P on a horizontal line, the tilt goes from zero at CoV, to a maximum at the boundary circle, gradually returning to zero again at infinity!

[TonyMcC]

I think you will find that drawing a circular surface tilted away from you leaves you drawing a distorted ellipse due to perspective. This effect is usually ignored in engineering drawing.

[michel123456]

It is not because cups are positioned left or right of our centre of vision, it is because our axis of vision is not parallel to the plane of the circles.

Edited November 19, 2010 by michel123456

[TonyMcC]

I think you misunderstand me. Any circle looked at from an angle will not look like a true ellipse. An artist would need to be aware of this. Engineering drawings are not drawn in perspective and I'm sure your method could be helpful to engineering draughtsmen or draughtwomen. Go to http://www.drawingcoach.com/1-point-perspective.html for an explanation.

[michel123456]

My post was addressed to dalgoma. You made your point, Tony.

[TonyMcC]

Sorry Michell - I hadn't noticed that yours was a reply to Dalgoma. Bit sloppy of me. My comment was also really for Dalgoma.

[dalgoma]

For Michel - Your construction of the central axes of the cups, rightly going to a vanishing point, confirms that they are not at right angles to the major axes of the elliptical images of their rims. I contend that my contruction provides a more accurate result for the tilt.

 

for Tony - The link shows the traditional rough & ready method of constructing the image. How does the circle know that it is surrounded by a square? My post (CiP) I think, supplies the answer without recourse to circumscriptions.

 

Thank you all for your interest.

[TonyMcC]

dalgoma - Do we agree that a circle viewed from an angle is seen as a distorted ellipse due to perspective?

[dalgoma]

no we don't!

 

see for example http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm#circle

[michel123456]

For Michel - Your construction of the central axes of the cups, rightly going to a vanishing point, confirms that they are not at right angles to the major axes of the elliptical images of their rims. I contend that my contruction provides a more accurate result for the tilt.

(..)

 

I have to admit I don't not fully understand your construction, better say where does it come from? and how do you insert it in a perspective view?

 

Thank you all for your interest.

You're welcome.

 

Wonderful link. Thank you.

[TonyMcC]

dalgoma - I guess it's learn a little every day! What a strange thing drawing a circle at an angle is. Both a parallel projection and a perspective drawing, I now believe, produce ellipses. However if you consider a disc in front of you divided into two equal areas by a diameter and each half painted in seperate colours and tilt it away from you keeping the dividing line horizontal you will see the following :-

The half further away will appear smaller with less depth whereas the half nearer will appear larger with greater depth. I would think it quite natural to assume this would distort the ellipse. However taking the disc as a whole the widest point of what you see is not the circles diameter, but a horizontal line forward of the diameter. This line forward of the diameter becomes a new major axis. I hope I now understand the situation - anyway I have enjoyed thinking about it. Thanks.

[michel123456]

dalgoma - I guess it's learn a little every day! What a strange thing drawing a circle at an angle is. Both a parallel projection and a perspective drawing, I now believe, produce ellipses. However if you consider a disc in front of you divided into two equal areas by a diameter and each half painted in seperate colours and tilt it away from you keeping the dividing line horizontal you will see the following :-

The half further away will appear smaller with less depth whereas the half nearer will appear larger with greater depth. I would think it quite natural to assume this would distort the ellipse. However taking the disc as a whole the widest point of what you see is not the circles diameter, but a horizontal line forward of the diameter. This line forward of the diameter becomes a new major axis. I hope I now understand the situation - anyway I have enjoyed thinking about it. Thanks.

 

Well, the maths are very persuasive.

From the link

 

"The conic sections in the plane are given as the locus, that is the set of all points (u,v) in E2 which satisfy a quadratic equation of the form (...)

 

D = a c - b2 is negative, then the conic is a hyperbola, if D=0 the conic is a parabola and if D is positive the conic is an ellipse."

 

The entire demonstration begins with the premise that we are examining a conic section. Correct. It is well known that there are 3 kinds of conic sections: hyperbola, parabola, ellipse. But here we have an unusual oblique cone, and I can currently not find any backup showing that all known equations of sections are still valid for an oblique cone. An oblique cone is a cone "in which the axis does not pass perpendicularly through the centre of the base" _see wiki

 

Or to say it otherways, since only the ellipse is a closed curve (opposite to hyperbola & parabola), the supposition of a conic intersection points directly to the ellipse, and the whole demonstration is worthless.

Edited November 20, 2010 by michel123456

[michel123456]

I spent some time searching about sections of oblique cone, nothing.

Then I made an Autocad drawing, which is very accurate : the graphic result is indeed an ellipse.

[dalgoma]

For Tony. Absolutely right!

 

For Michel- The construction is my own and seems to me simple, have a certain mathematical elegance and satisfies observatioin.

 

However, I cannot at the moment relate it to the projection of a particular circle at a particular location. What it does do is predict the tilt angle of an elliptical image whose centre P is in that position in relation to O whatever its size.

 

Also I cannot establish major an minor axes.

 

For instance, if we plot the image of concentric circles, They would not share the same common centre. Hence, they would have their own tilt angles if the were not on CoV.

 

Can I suggest that you view CiP 2. This produces circular images! and they are generated by projection from plan and section in the traditional way. I would be interested in your comments.