Circles in perspective

[dalgoma]

Have you noticed that when you look at a clock on the wall above you, the axis of the perceived elliptical image is at a "funny angle" and it moves when you move, yet the major axis of the rim of a cup on a table in front of you is always horizontal - and you cannot move to a position where it is not so. Why?

[Sisyphus]

It's because you are standing upright, so that the "down" of your field of vision is always perpendicular to the plane of the rim of the cup. With a clock on the wall, your down is at some other angle unless you're standing directly in front of it. Tilt your head, and the situation changes.

[dalgoma]

 

Here is a way to establish what that "funny angle" is.

 

Naturally it all depends on the location of the viewpoint.

 

I took the picture of The London Eye from the north end of Westminster Bridge, the camera was inclined at about 8 degrees.

 

I estimate that the distance to the centre of the wheel to be 400 metres and the angle to the

 

plane of it approx. 45 degrees. I know that The London eye is 62.5 metres radius.

 

Hence d (for distance) = 400

 

A (for Azimuth) = 45 degrees

 

E (for Elevation) = 82 degrees (90-8)

 

r (for radius) = 62.5

 

 

Referring to Figure 1

 

1 Draw verticals WX and YZ any distance apart, together with horizontal baseline intersecting the verticals at O and V

 

2 Draw line at angle E from O to meet YZ at N

 

3 Describe arc centred on V and radius VN

 

4 Draw line at angle A from V to meet arc at K

 

5 Join OK and label point P where it intersects with YZ, and label angle KOV, E1

 

6 Draw horizontal from P to meet arc at S

 

7 Join SV. THIS IS THE REQUIRED TILT ANGLE T

 

Distance d and radius r have no bearing on the tilt angle, as long as d is not zero.

 

Angle E1 is required to establish the ratio of major to minor axis and thence show that the approach is a correct one.

 

That is if anybody is interested.

[dalgoma]

"Heed thy ellipse, note two axes" misheard in the Oval Office.

 

Major and minor axes.

 

The one time we can be sure that we are creating the right image of a circle in perspective is when the major axis is horizontal and we are looking directly at its centre as in the case of the cup on the table in front of us.

 

Draw the side view of the setup viewing the disc edge on from the viewpoint V, at angle E and distance d.

 

The picture plane, where the image is projected, can be at any anywhere but we know it is always at right angles to the line of sight.

 

Eye level is established by drawing a line from viewpoint V, parallel to the disc to meet the picture plane.

 

We can now draw sightlines from V to the extremities of the disc and extend to the picture plane as necessary.

 

Horizontals from these two points will determine the top and bottom of the image – the MINOR AXIS.

 

Looking directly down on the disc we know that it will appear to be foreshortened to an Ellipse. The major axis of it will be its diameter 2*r. The minor axis will be cos(E1)*2*r

 

Place the plan at right angles and again draw sightlines from V, this time to the tangents of the ellipse.

 

Again extend to the picture plane and project vertically. The distance apart of these lines will be the MAJOR AXIS of the perceived image. The elliptical image can now be drawn.

 

See Figure 2

 

But there is an easier way (figure 3)

 

The side elevation is as figure 2 but for the plan, we draw a circle of radius r centred on O. And compensate by moving V to V1 a distance d1 = d/cos(E)

 

Sightlines are projected in the same way and the resultant image is exactly the same

 

It makes the maths a lot easier!

 

See Figure 3

 

Now we can return to the image of the London Eye.

 

We proceed as figure 3, but instead of angle E, we use angle E1 (figure 1) for the inclination.

 

It only remains to transfer angle T (figure 1) to the image passing through O and rotate to obtain the required tilt and add the centre of vision through O.

 

See Figure 4

[dalgoma]

 

"The puck stops here". Misread in the Oval Office.

 

Eye Level

 

Figure 5 shows the setup that created the ellipse in figure 4a. Using inclination E1 and distance on plan d1.

 

1 We just have to add the tilt angle T (the blue line) and extend it to meet eye level at VP1.

 

2 Use this to orientate a square on the plan that would generate this VP from viewpoint V1.

 

3 Hence position VP2 from the right angle to this line.

 

4 Similarly, locate VP3 by drawing the line at right angles to the inclination (E1) on elevation.

 

5 Now connect VP2 and VP3 with a straight line.

 

The result is shown at Figure 6

 

Now rotate the whole so that the blue tilt line is vertical. (Figure 7)

 

It will be seen that the tilt line has become the centre of vision and the line joining VP1 and VP2 has become the eye level!

 

Finally superimpose the construction on the view from upon Westminster Bridge ...

 

"Earth has not anything to show more fair,

 

Dull would he be of soul who could pass by,

 

A sight so touching in its majesty"

 

[dalgoma]

Here is a way of visualising what is going on.

 

Imagine one of those flat circular lollipops and hold it directly in front of you, then lean it away from you at say 8 degrees.

 

Now spin the stick in your fingers so that the lollipop rotates 360 degrees. The effect will be seen by clicking on the attachment which has been constructed by animating 18 frames with a rotation interval of 10 degrees, using the method as previously described. OK I know it wobbles a bit.

 

The stick that is embedded in the lollipop (albeit foreshortened) remains the same length throughout the rotation, it also remains vertical and becomes the common chord to all the ellipses.

 

As I know the length of the axes of each I can locate their centres, calculate eccentricities and hence foci. That is what those dancing spots are...*

 

*See what I have done there? Those three dots are also called ellipses!