When do straight lines become curved when talking about projection?

Question

This in not a question directly pertaining to photography, but the theory of projection in general.

Imagine that I'm drawing my surroundings, say, in a city with straight lines everywhere. I start by drawing what is directly in front of me, and going further and further out. I intend to draw everything around me. When will I be forced to draw curved lines rather than straight ones? Or are straight lines actually unrealistic? I'm just having a hard time wrapping my head around how we can make "flat" images from a "spherical" field of view, I'm not even sure if I'm using the right words to describe the questions.

There is undoubtedly extensive research and papers made on the subject, I'd appreciate if someone can point out some of that for me to get learned on the subject.

Here follows a painting which inspired me to ask the question:

The Courtyard by Irvine Peacock

Answer 1

When do straight lines become curved when talking about projection?

When you decide what type of projection you wish to use.

A rectilinear projection will preserve straight lines (but not correct angles between straight lines) at the expense of the relative sizes of objects in the center of the frame versus the edge of the frame. Things near the edge will be distorted in size to preserve straight lines. This works best with photographic lenses when the angle of view is relatively narrow. The wider the field of view, the more distorted the angles between straight lines become and the more distorted the relative sizes of things in the center and things on the edges becomes.

A fisheye projection will preserve relative sizes at the expense of not preserving straight lines.

An orthographic projection will preserve both straight lines and the angles between them. To capture an orthographic projection with a camera requires a telecentric lens that is larger in diameter than the width/height of the scene being projected. Many architectural drawings ('side view', 'top view', etc.) use orthographic projection. Each point in the scene is drawn from a constant angle rather than from the perspective of a constant point of view. Scanners and copiers capture an orthographic view of flat documents by travelling across the face of the document as they image each part.

There are many other types of projections, but these are the most common seen within the context of photographic lenses.

In the photographic disciplines (which is the specifically expressed subject of this site) the term projection is used when discussing how a lens projects a 2D image of the 3D world onto a flat imaging plane. In the disciplines related to artistic drawing, the term perspective is used in much the same way. But in the photographic disciplines the term perspective refers not to how an image is projected by a lens, but rather the relative positions of the objects within the camera's angle of view and how the relative distances from the camera to each of those objects affects their relative shapes and sizes as well as what part of a more distant object may be hidden by a closer object. Changing projection will alter the shapes of things, but not what parts are hidden by closer objects. Only changing the camera position, that is the perspective, can change that.

For more, please see:
What is the difference between perspective distortion and barrel or pincushion distortion?
Does wide angle equivalent in crop sensor skew image?
How do i take a photo of an object perfectly vertical like a top view?
Why the faces in the corner tends to skew a bit in almost all smartphone cameras
How wide can a lens be before it begins to produce distortion?
What does it really mean that telephoto lenses "flatten" scenes?
What's the opposite of a fisheye lens?
How can I achieve an effect in still photography that's analogous to a cinematographic effect?

Answer 2

With "true" perspective (with in photographic terms corresponds to a perfect rectilinear lens), straight lines remain straight lines. As you imply in the question, extremely wide angles cannot avoid distortion: in practice very wide angle (short focal length) lenses are typically not rectilinear but fish-eye, and the straight lines which remain straight are those which pass through the centre of the image.

If you move beyond the realm of photography into the more general realm of projecting large portions of a sphere onto a flat surface then cartography is the application which motivated the subject, and you're probably better off asking on the GIS site.

Answer 3

Linear perspective is a model of reality. It is useful, but like all models it is wrong. The key is that you have to "look around" as you draw.

Human foveal vision is restricted to the an angle of about 5 degrees and the that five degrees gets sharper toward the center so the zone of high sharpness is even narrower. The human eye also contains a blind spot where the optical nerve connects in lieu of rods and cones. This means that our eyes scan around a scene and our brains form sharp cohesive mental images. [a] The sample image works when our eye scans around it looking at details. It crumbles when viewed as a gestalt.

The utility of a linear perspective rendering is that it models [b] part of that cohesive mental image reasonably well. And part of that modeling is that our eye scans around the perspective rendering using foveal sharpness to discern details much the same way as our scans around the real world using foveal sharpness to discern details.

The straight lines of linear perspective renderings are usually not a problem because the model is usually good enough. Except if you're an ancient Greek building temples for the Gods, then entasis comes into play...but you didn't have linear perspective anyway.

If it is really an issue, one can draw a curvilinear perspective.

[a]: Another example of this is that we discern color across the entire visual field despite color vision being largely limited to the fovea.

[b]: The station point of a perspective rendering is similar to the point of maximum cone density in the human eye.

Answer 4

Map makers have the same problem; “how to depict a spherical earth on flat paper”. They solved by projection. They place a lamp inside a transparent globe of the earth. Then they place a small sheet of paper tangent to the globe and trace. There are many different projections. One solution is to fold the paper into a cone and place this figure over the globe. Check out conic projections.