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]

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

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.

 

[sardar]

The explanation for the shape formed by light falling on a wall from a lamp is due to the property of light called reflection. When a ray of light hits a surface, it reflects off the surface at an angle that is equal to the angle at which it hit the surface. In the case of the lamp on a wall, the light rays are hitting the wall at different angles and reflecting off at different angles, creating a shape that appears to be a hyperbola.

Similarly, when sunlight falls at an angle on a convex lens, the light rays are refracted (bent) as they pass through the lens. This creates a curved shape of light on the surface, which appears to be a hyperbola due to the way the light rays are being bent by the lens.

The phenomenon of light forming a hyperbola on a surface can be explained by the principle of superposition, which states that when two or more waves (in this case, light waves) intersect, the resulting wave is the sum of the individual waves. In the case of light falling on a surface, the light waves are intersecting and creating a pattern of constructive and destructive interference, resulting in the shape of a hyperbola.

Furthermore, the constant path difference between the two fixed points (in this case, the light source and the surface) also contributes to the formation of a hyperbola, as this constant difference creates a consistent pattern of interference.

In conclusion, the phenomenon of light falling on a wall or a convex lens and forming a hyperbola can be explained by the principles of reflection, refraction, and superposition of light waves. The constant path difference between the light source and the surface also plays a role in the formation of this shape.