一点透視図法の正当性、なぜ遠くのものが小さく見えるのか Principle of Perspective (in my opinion) why objects are smaller in the distance
As I mentioned in the self-introduction, I am a huge admirer of Leonard da Vinci. One of his many achievements is his study on perspective.
The basic principle of perspective is that objects get smaller as seen by our naked eyes. Therefore they recede to a point in the distance. If you extend lines in Leonardo's paintings, you see that all those lines intersect at one point as seen in the painting below.
Then I thought, "In Euclidean geometry, parallel lines are not supposed to intersect, how come this technique describe nature of matters?"
I thought about it for a couple of minutes, and the answer was fairly simple. But I first need to explain how we see objects.
The light first passes through a tough protective sheet called the cornea and then moves into the lens. This adjustable structure bends the light, focusing it down to a point on the retina, at the back of the eye. (from Independent). The image created on the retina is processed in our brain, thereby we perceive objects.
Now we consider how the lens work. This is a revision of high school physics.
As you can see below, there is an equation that enables us to find the sides of the image on the retina. Since the ratio o : i = (size of the object):(size of image), we can find the size of the image in the following equation
Size of image = (size of object) times (i/o) ・・・・(1)
The distance between lines, in general, can be thought as the length of the line segment perpendicular to both lines. When the length is zero, two lines intersect. Therefore, if we regard the line segments as objects, these will be smaller as it goes far away because the larger distance makes o bigger, leaving the size of an image (line segment) smaller. If we take o as infinity, the length of line segment approaches zero, meaning lines intersect.
In the discussion above I hope it is clear why things look smaller in the distance. I also hope this can validate the use of intersecting lines in perspective.
The reason for my question was the structure of our eyes. There is even an area of mathematics called projective geometry that allows for two parallel lines to intersect in the distance in an attempt to describe the world as we see it. Our sensory organs are distorting the world but we don't know that.
On hindsight, I am now wondering how we can be sure that we can rely on our sensory experience?
