Locus of light falling on a plane surface

hale2bopp

When there is a lamp on a wall and the light from the lamp falls on the wall, we notice that the shape formed by the light is a hyperbola. I would like to know what the explanation for this is.
Also, when you have a convex lens and you kee it flat on a horizontal surface, and sunlight falls at an angle on to the lens, you can see a curve of light on the surface that looks to me very much like a hyperbola.
A hyperbola is the locus of a point moving such that the difference of its distances from two fixed points is constant. That makes sense when you're wondering why the locus of an interference pattern should be a hyperbola, because the path difference is constant. But I can't relate this to the phenomenon of light falling on a wall, or the lens thing.
Thanks in advance!

haruspex

I assume the light has a shade with a circular opening.
The light passing through the opening projects a cone.
The intersection of a cone with a plane produces various "conic sections" depending on the angle.
If the axis of the shade's opening is parallel to the wall then you will see a hyperbola.
If you tilt the shade down so as to direct the light more fully onto the wall, at a critical angle (i.e. when the light falling furthest from the wall is going straight down) it becomes a parabola.
Tilt a fraction more and you have an ellipse.

In general, conic sections are the curves that satisfy quadratic equations. The locus of intersection of a cone with a plane will also satisfy a quadratic equation. And the locus of a point that's always further from one fixed point than another by a constant amount is also a quadratic equation.

Lenses are ground with spherical surfaces - more quadratics. So it wouldn't surprise me if that also gives you hyperbolae, though I haven't checked it in detail.

Btw, have you noticed the pattern light from a point source forms at the bottom of a cylindrical cup? No conic this time.

hale2bopp

Thank you so much!! This explained things perfectly. I will try to check out the light source at the bottom of a spherical cup. :)

haruspex

Actually I said cylindrical, the usual shape for a mug, and the light source is outside the mug. It's the pattern produced at the bottom that's interesting.