Circles in Perspective 2

[dalgoma]

Circular Circles in Perspective

 

 

 

We know that a circle in perspective is an ellipse.

 

 

Here is the traditional method of constructing such a view.

 

 

With a viewpoint a horizontal distance d from the centre of a target circle of radius r (Fig 1)

 

 

The eye (camera) is horizontal and viewing a target circle on a horizontal surface directly in front but a

 

 

vertical dimension h below eye level

 

 

The resultant image is – an ellipse.

 

 

If however the circle is further below, using the same construction, the image rotates 90 degrees. (Fig 2)

 

 

Could there be therefore, a location for the target circle where the image is a true circle?

 

 

Consider that your eye (camera) is located on the circumference of a circular "bubble" of diameter d.

 

 

The target circle of radius r settles in the bowl of the bubble. (Fig 3)

 

 

Distance h is therefore sq rt (d^2 – r^2) (Fig 4)

 

 

The resultant image is a true circle and its radius is dependant upon radius of the target circle r, viewing

 

 

distance d and viewing height h being Pythagorean.

 

 

It follows then that if you stand say 354 mm back from the edge of a circular pool of 3 metres radius and

 

 

take a photo of the opposite bank with camera horizontal, 1500 mm above the water. The image of the

 

 

opposite bank will be the arc of a true circle!

 

[michel123456]

Your construction looks O.k. but it is not clear how does it come from.

IMHO it is simpler to put the picture plane tangent to the back of the circle, and also choose a close distance so that the radius of the image becomes clearly larger than the original circle.

The most peculiar in this construction is that the image-center of the green image-circle is not in its center.

[dalgoma]

I put the picture plane at the centre of the circle to make the maths easier when evolving the formula.

 

This figure shows another circle as well as their surrounding squares - and yes indeed, the centres are at the intersection of the diagonals of the squares.