L/(L+M) = n/N

Posted by isaac on November 21, 2001 at 12:16:51:

In Reply to: Projective Geometry - Math Question posted by JP on November 20, 2001 at 13:18:14:

it may help to draw this out...

let's say you have a point source, call that point A, on the x-axis.
to make things simpler, sya that the object is of height n, and is positioned so that it is perfectly normal to the x-axis, and sitting on it a distance L from the source. (thus, the light from A can go in a straight line along the axis, as if you had a point source on a table, and stood a pencil up normal to the table.) then, set up a screen a distance M from the object.

so, you have a right triangle from A to the top of the object (point L,N), and the bottom of the object (point L,0). then, the shadow on the screen makes a similar right triangle from A to (L+M) to the top of the shadow. say that the shadow has height of N. what you're looking for is a relationship between L, M, and N/n. since they're similar triangles...

L/(L+M) = n/N

is that what you were looking for?

like bart pointed out, it gets a little more hairy if it's not a point source, or if the screen isn't positioned roughly normal to the rays hitting it, as in how your shadow gets stretched when cast onto a the ground when the sun is behind you. the ground is at a steep angle to the light hitting it. otherwise, however, because L+M is roughly equal to L (what's a few feet in the presence of 93E6 miles?) n/N approaches 1. as far as objects go, every object can be looked at as a composite of normal surfaces, since the light only sees that anyway. (unless the size of the object is enough for gravity to have a significant effect, like when light passes by a large planet or star.) if it's not a point source, then you have to take into consideration the difference between the inner and outer shadow (the inner where ALL the light is blocked, and the outer where SOME of the light is blocked.)

isaac

Follow Ups:

• The question was already answered, Isaac. Bartholomew 14:58:29 11/21/01 (4)
  • testy, testy, bart. have you had your cheerios today? isaac 17:10:14 11/21/01 (3)
    • I'm a Wheat Chex man, myself Bartholomew 18:12:27 11/21/01 (0)
    • oops, correction... isaac 17:12:42 11/21/01 (1)
      • no, it was an incorrection... Bartholomew 18:14:56 11/21/01 (0)
• What my uncle says JP 14:02:04 11/21/01 (0)

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