Index Making Pictures

Image Formation: Projection

Here is a basic explanation of how optical systems produce images.

Figure: Durer engraving of ray tracing
We can mimic image formation by tracing rays from the object towards a centre of projection. In the above, the centre of projection is the pulley on the wall (the pulley and weight are there only to keep the string tight). The diagram shows the person on the right measuring the coordinates of where the ray (the string) intersects the image plane (the open frame). These coordinates are then transferred to the paper (held by the person to the left) and a dot marked. Notice that all rays pass through the centre of projection; that is, the "lens" is effectively a point. This is a form of pinhole camera. In the figure, the image is drawn on a plane between the centre of projection and the object. In consequence, it is the right way up. Contrast this with a lens, where the centre of projection/lens is between the object and the image.

One feature of this projection is that the further away from the CoP that an object is, the smaller it will appear in the image. This is the basis of perspective: the projected size of an object is inversely proportional to how far away it is. You can use the above diagram to get an understanding of why this happens.

Figure: Orthographic projection: no perspective
When we calculate an image, we have a lot of control on how we project the model onto the image plane. This box has no perspective effect. This is an orthographic projection. It preserves lengths (unlike perspective). We achieve this by parallel projecting the 3D coordinates to the 2D plane (possibly scaling them to fit the required image size), but displacing them to one side in proportion to their distance away. There are no vanishing points in this projection: parallel lines remain parallel. You could also say that there are vanishing points but they are at infinity.

Figure: Orthographic projection: Roller Coaster Tycoon

This technique is used in many games, for example those requiring you to build a city, theme park etc, because it gives a good impression of an overview. In addition, all the graphic elements are the same size, no matter how far away they are, so they can be rapidly copied onto the screen to animate it. The above is one example. In this case, the projection is oblique. The faces of the buildings are at an angle, rather than parallel to the screen. In addition, the viewpoint is above the ground plane, to give an overview. However, parallel lines remain parallel.

Figure: One-point perspective
This image has a single vanishing point and shows one-point perspective as a result.

Figure: Two-point perspective
Here is a two-point perspective, with two vanishing points.

Figure: Two--point perspective lettering

Figure: Three-point perspective
Here is a three-point perspective, with dotted lines showing the three vanishing points. Perspective projections preserve straight lines but not angles or lengths.

Figure: Curved projection
This picture shows how some projections do not preserve straight lines. With this mirror-glass sphere, the reflection does not take place in a single plane. (Thin lenses perform refraction in approximately a plane, the lens equation can be used and straight lines are retained.) The image shows curves where we know edges to be straight. We might claim this image is "distorted" but that isn't really fair: you cannot map the 3D world onto a 2D plane and retain everything, as map-makers know. You simply have to decide what is important to you and make sure your chosen projection retains that.
Strictly an image is only correct when it is viewed from the correct distance, according to the centre of projection. In practice we can tolerate quite a wide variation, unless a wide-angle close-up image is made. Another kind of apparently "distorted" image then results because our eyes cannot focus at the very close distance needed to view it. Of course it is not distorted but, at normal viewing distances, the perspective is too exaggerated to be acceptable.