Changes

== Data Recording ==

== Data Recording ==

In our lab experiments, we must measure and record a collimnated light beam. This is done by shining the beam into a camera (the Casio Exilim EX-F1) and photographing the results. In order to properly interpret the information, it is necessary to understand the effects of the lenses on the image. This is incredibly difficult because of a number of factors. Most importantly, we have no way to directly measure the camera and acquire most of the information that we need. The lens assembly on the Exilim EX-F1 cannot be removed, so we cannot replace it with a custom- built lens assembly of known dimensions. All we can measure are a limited number of lengths, and we know the ranges of F-numbers and focal lengths. In order to solve this problem, a number of significant approximations and generalizations must be made. While the actual lens assembly contains twelve disparate lenses, our approximation has only four, and unlike the actual assembly, our approximation contains only perfect, idealized spherical lenses. These approximations would certainly affect our final results, but most likely not enough to severely skew our results.

In our lab experiments, we must measure and record a collimated light beam. This is done by shining the beam into a camera (the Casio Exilim EX-F1) and photographing the results. In order to properly interpret the information, it is necessary to understand the effects of the lenses on the image. This is incredibly difficult because of a number of factors. Most importantly, we have no way to directly measure the camera and acquire most of the information that we need. The lens assembly on the Exilim EX-F1 cannot be removed, so we cannot replace it with a custom- built lens assembly of known dimensions. All we can measure are a limited number of lengths, and we know the ranges of F-numbers and focal lengths. In order to solve this problem, a number of significant approximations and generalizations must be made. While the actual lens assembly contains twelve disparate lenses, our approximation has only four, and unlike the actual assembly, our approximation contains only perfect, idealized spherical lenses. These approximations would certainly affect our final results, but most likely not enough to severely skew our results.

== Inside the Lens Assembly ==

== Inside the Lens Assembly ==

where <math> M _1 </math> is the size of the object and <math> M _2 </math> is the size of the image.

where <math> M _1 </math> is the size of the object and <math> M _2 </math> is the size of the image.

By repeating these calculations for all lenses in the lens assembly, a final image magnification could be calculated.

By repeating these calculations for all lenses in the lens assembly, a final image magnification could be calculated.

Unfortunately, there were problems with this approximation, namely that it uses the wrong type of light. This approximation assumes that the object is a physical thing, which would allow light rays to leave it from any angle, allowing the required light rays shown in the illustration to pass through the lens. However, the light used is collimnated laser light, meaning that all the light rays are parallel when they enter the lens assembly. Therefore, this approximation is invalid.

Unfortunately, there were problems with this approximation, namely that it uses the wrong type of light. This approximation assumes that the object is a physical thing, which would allow light rays to leave it from any angle, allowing the required light rays shown in the illustration to pass through the lens. However, the light used is collimated laser light, meaning that all the light rays are parallel when they enter the lens assembly. Therefore, this approximation is invalid.

== Second Iteration ==

== Second Iteration ==

[[Image:Thin_Lenses_005_(Single_Ray).png|thumb|Two light paths, from an object off the left side of the frame. The vertical lines are the lenses. Notice how the light rays cross through the focal point and continue through to the next lens.]]

[[Image:Thin_Lenses_005_(Single_Ray).png|thumb|Two light paths, from an object off the left side of the frame. The vertical lines are the lenses. Notice how the light rays cross through the focal point and continue through to the next lens.]]

If the light entering the lens assembly is colimnated and enters the , then the top and bottom beams (the only two beams needed to locate the image) will travel through the first lens parallel, and both will cross through the focal point. The size of the image will then be

If the light entering the lens assembly is collimated and enters the , then the top and bottom beams (the only two beams needed to locate the image) will travel through the first lens parallel, and both will cross through the focal point. The size of the image will then be

<math> \frac{M _1}{f} = \frac{M _2}{L-F}</math>

<math> \frac{M _1}{f} = \frac{M _2}{L-F}</math>

== Fifth Iteration ==

== Fifth Iteration ==

[Iteration complete and pending upload; the fifth iteration allows for a 3D lens and a 2D image, where the light can enter the lens assembly from any angle and any distance in three dimensions]

The fifth iteration must generate a proper two-dimensional image through a three-dimensional camera. This model must allow for the light to enter the camera at any angle and offset. To simplify this, the two-dimensional model is used for each axis. The user will input an entry angle in both the X and Y directions, and he will input an entry offset in the same fashion.

 

We know logically that the image will not be compressed or stretched by passing through the lenses, as this cannot be seen in photography. First, therefore, we will track the center of the image as it passes through the lenses. This is done by calculating the offsets using the equation

 

<math>O = F_p \tan{(\theta_p)}+{(S_p-F_p)}\tan{(\theta)}\,</math>

 

 

 

where <math>\theta</math> is the entry angle at a given lens, <math>O</math> is the directional offset at the given lens, F is the focal length of the given lens, S is the spacing between the given lens and the next lens, and <math>R_p</math> is the value of any value R at the previous lens.

 

[[Image:Lens_arrangement.png|thumb|The lenses, as viewed by the sensor. The sensor perceives only the light that passes through every lens. The lenses appear off-center because the light is entering at some offset and angle.]]

 

Given these equations, we can further calculate the magnification of the image by recalculating the position of its edges. Assuming the image is symmetrical on some X and Y axis, we can find its size by tracing the position of its edges, defined as the offset plus or minus half the size of the object. Given these, we can calculate the image's size and location on each of the lenses, as well as the sensor.

== Aperture ==

== Aperture ==

Math for the aperture has been tried. For the third iteration, an aperture can be calculated by finding the magnification of an image at the location of the aperture, and if it is larger than the aperture allows, the amount of image remaining can be calculated by simple ratio.

[[Image:Lens_side_trim.png|thumb|A side view of the lens assembly. Notice that because of the entry angle, one of the illustrated beams is cut off before reaching the second lens. The final image will be only the remaining half of the original.]]

An aperture would be more difficult to add for the fourth iteration, but it should be possible.

 

The above equations can generate images magnified or offset beyond what the camera can resolve, as the camera lens assembly has a radius of 3.75 cm. To compensate for this and prevent resolution of an image larger than the camera can allow, we need to superimpose over our final offset and magnified image the effects of the lens assembly. At any given lens, the image will be at either its maximum or minimum magnification counting from the previous lens. These lenses can be generated by the equation

 

 

where O is the O-coordinate for the given lens, <math>O_c</math> is the O-coordinate for the center of the image circle at this lens, and M is the magnification of the image at the given lens.

An aperture for the fourth iteration is being calculated and will be posted alongside the fourth iteration.

These coordinates are used to set the center of the lens superpositions. With this known, and with the image magnifications known, we can draw the lens superpositions and observe what of the image is cut off by the lenses.

== Future Improvements ==

== Future Improvements ==

* Add diagrams to this page

* Unknown