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: Onepoint perspective
This image has a single vanishing point and shows onepoint
perspective as a result.
Figure: Twopoint perspective
Here is a twopoint perspective, with two vanishing points.
Figure: Twopoint perspective lettering
Figure: Threepoint perspective
Here is a threepoint 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 mirrorglass 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 mapmakers 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 wideangle closeup 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.
