The situation is that every point in the aperture acts as an infinitesimal aperture in its own right.
The (observed) image is the sum of the infinitesimal aperture images.
However, these do NOT exactly overlap, and so the image is somewhat fuzzy (i.e., lacking perfect sharpness).
As the aperture gets bigger, the image approaches just being the shape of the aperture which is why a rectangular window with sunlight streaming in creates a rectangular spot of illumination.
The imperfectly overlapping aperture images are spread out by of order "a". The ratio of spreading to image size is of order
a
f = -------- ,
d*tan(θ)
the fuzziness factor.
If f << 1 sufficiently, the image will look sharp to the human eye.
If f >∼ 1, then the image is totally fuzzed.
Just put the projection screen farther away from the aperture to increase sharpness.
-
Nota bene:
as f→0, the image gets
sharper only as long as you are in the
the limit of
geometrical optics.
As the aperture
approaches the size scale of the
wavelength of
visible light (fiducial range 0.4--0.7 μm),
geometrical optics increasely
fails and
physical optics effects
(i.e., interference
and diffraction effects)
become increasely important.
In this case, the shape of the aperture becomes
increasingly important again since it helps determine the
diffraction pattern of
the image.
In geometrical optics,
there are, of course, NO
interference
and diffraction effects.
You may be able to observe sunspots with pinhole projection---but yours truly has always failed. It is probably marginal at best (see, e.g., Cloudy Nights Telescope Review).
Pinhole projection does work well for observing solar eclipses---especially when you are too lazy to do anything more elaborate.
In fact, when the Sun images become crescents ☽, there is a partial solar eclipse. The crescents are sufficiently striking that people do notice them when they never notice the round light spots. In fact, since partial solar eclipses dim the sky similarly to thin cloud cover, maybe the only casual way to notice partial solar eclipses is by being spooked by the crescents.