What are the XY coordinates of a point P(x,y,z) projected through a pinhole on to a screen?

Question

Light from a point P(x,y,z) in 3D space is projected in a straight line POQ through a pinhole O(0,0,0) on to a flat screen, photographic plate or plane, a fixed distance d behind the pinhole, to a point Q(X,Y) in 2D space, where (0,0) in this XY plane, representing the screen, is defined by the projection of a beam of light along the z axis perpendicular to the screen passing through the hole (i.e., all points (0,0,z) a distance z from O are projected on to one point (0,0) in the XY plane); the X axis is parallel to but is in the opposite direction to the x axis; and the Y axis is parallel to but is in the opposite direction to the y axis. (The mapping of points in 3D space through an aperture on to a flat screen is the basic principle on which optics, perspective and photography rest.) Find the values of X and Y in terms of x, y, z and d.

Comments

Hint:

The pinhole is just the origin O of the orthogonal x, y and z axes in 3-dimensional space. The y coord of P is the height (positive) of P above O or the depth of P (negative) below O; the x coord is the distance of P to the left (positive) or right (negative) of O; and z is the distance of P in front of O. Now draw two diagrams, diagram 1 representing the view above as if looking down the y axis at the x-z plane, where the point P can be represented by (x,z); and diagram 2 representing the view looking along the x axis at the y-z plane, where P is (y,z). In each diagram the screen is a line parallel to the vertical axis: in diagram 1 the vertical axis is the line x=-d, in diagram 2 it's the line y=-d. The point O' is where the screen line meets the horizontal axis. On each diagram draw a line from P through O to a point Q on the screen line. The screen is the X-Y plane, a 2-dimensional set of axes. In diagram 1 The X coord of Q (the length of line O'Q) can be calculated from the x and z coords of P using the geometry of similar figures, because OO'=d. The same applies in diagram 2 to the Y coord. 

 

Answer 1

Because it helps you see where the point is. Like point A for example, the Y cordinate is at the bottom and the X cordinate is at the top. You use it as a elivator to go up or down.

Comments

The equations you need to use for the similar figures are x/z=X/d and y/z=Y/d, so X=xd/z, Y=yd/z. These are the equations that map 3-space (x,y,z) onto 2-space (X,Y). Yes, you can think of the line from 3-space passing through the pinhole as a rigid seesaw, where the pinhole is the pivot. As one end (P) in 3-space goes up or down the other end goes down or up in the opposite direction in 2-space marking a point (Q). Just like the elevator and counterweight.