Vertical translationexamples

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To transform the reference parabola down, we make k negative. This can look like a subtraction. Let's work at x = 5 and perform a vertical translation downward of 3 units. So, k = -3. Here's the calculation for the y-coordinate:

Vertical translationequation

Hopefully, it is clear that on this transformed, vertically translated parabola at x = 3 we simply have the reference y-coordinate, 9, with 5 added to it, lifting the reference parabola up 5 units to land on the transformed parabola. A similar lift happens at each x-coordinate.

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Here's the graph for this translation. The reference parabola ( y = x2 ) is drawn in transparent light gray, and the transformed parabola which is vertically translated 5 units ( y = x2 + 5 ) is drawn in black:

More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

Vertical translation for the parabola is changed by the value of a variable, k, that is added into the calculation for y after x is squared. So, our starting or reference parabola formula looks like this:

Vertical translationmath

Since the squaring of x occurs before the addition of k, the y-coordinates of the transformed, vertically translated parabola are simply the corresponding y-coordinates of the reference parabola with k added to them. If k is positive, this lifts the y-coordinates up. If k is negative, this moves them down.

Vertical translationrule

The size of the spot is related to your resolution. Resolution is being able to tell the difference between two closely positioned bright objects, and one big object. If two objects are closer together than your resolution, then they blur together in the microscope image and it's impossible to tell that they are two points (except maybe the combined image is twice as bright as one object: but still, you can't measure their separation). The best resolution for an optical microscope is about 0.2 microns = 200 nm. The good news is, there's a difference between resolution and "ability to locate the position". If you have one tiny and isolated fluorescent object, you can often locate the position of that object to better than your resolution. The image of the object will show up as an extended blob, and you can find the "center of mass" of that blob-shaped image. If the blob is N pixels wide and each pixel is M microns across, you can estimate the center of the blob to about M/N accuracy, which often beats the optical resolution. This is a useful trick, but not solving the same problem as resolution. In some cases you can do various tricks to make the spot size bigger (increase N) so that you can locate the center even better. Various experiments I've heard of have claimed to be able to locate the centers of spots to within 10-30 nm using this sort of method. You may be interested in some software available for identifying particle positions, which implements this center-of-mass method. The magnification is something different altogether. There's a technical definition which compares the apparent angular size of the image, to the actual angular size of the object as it would appear if it were 25 cm away from your eye. This is a somewhat arbitary definition and in my opinion is mainly relevant for devising problems when I teach optics in my introductory physics classes. In real life, one often takes pictures using a CCD camera on a microscope, and projects them on a monitor. Using a larger monitor certainly can magnify the image further. But, it will still be just as blurry or sharp as the resolution. Fortunately, in general higher magnification lenses also have better resolution. In our lab a 10x objective has a resolution of 0.7 microns and a 100x objective has a resolution of 0.2 microns. One other tradeoff to consider: higher magnification lenses look at smaller fields of view, in proportion to their magnification. A 100x objective that sees a field of view of 100 x 100 micron^2 can be contrasted with a 10x objective looking at a 1000 x 1000 micron^2 field of view. So, when worrying about how good a microscope is, the most important question is what the resolution is. And in some science applications (such as my work) you care a lot about how well you can locate the centers of objects, and hopefully you can beat the resolution. Magnification is a much less useful specification (in my opinion). Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

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Vertical translationgraph

The magnification is something different altogether. There's a technical definition which compares the apparent angular size of the image, to the actual angular size of the object as it would appear if it were 25 cm away from your eye. This is a somewhat arbitary definition and in my opinion is mainly relevant for devising problems when I teach optics in my introductory physics classes. In real life, one often takes pictures using a CCD camera on a microscope, and projects them on a monitor. Using a larger monitor certainly can magnify the image further. But, it will still be just as blurry or sharp as the resolution. Fortunately, in general higher magnification lenses also have better resolution. In our lab a 10x objective has a resolution of 0.7 microns and a 100x objective has a resolution of 0.2 microns. One other tradeoff to consider: higher magnification lenses look at smaller fields of view, in proportion to their magnification. A 100x objective that sees a field of view of 100 x 100 micron^2 can be contrasted with a 10x objective looking at a 1000 x 1000 micron^2 field of view. So, when worrying about how good a microscope is, the most important question is what the resolution is. And in some science applications (such as my work) you care a lot about how well you can locate the centers of objects, and hopefully you can beat the resolution. Magnification is a much less useful specification (in my opinion). Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

You can change the value for k using the upper left input boxes. Press the 'Draw graph' button after you change k, and you will see how your change effects the graph. For more information about EZ Graph click the following link:

Vertical translationdown

Here's the calculation for the y-coordinate on the transformed, vertically translated parabola at x = 3. Let's imagine the the translation to be 5 units:

So, when worrying about how good a microscope is, the most important question is what the resolution is. And in some science applications (such as my work) you care a lot about how well you can locate the centers of objects, and hopefully you can beat the resolution. Magnification is a much less useful specification (in my opinion). Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

Vertical translationand horizontaltranslation

Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

What follows is an animation that presents many vertical translations for our reference parabola. Note that the value for k is added into the calculation for y after x is squared. See that positive values for k translate the parabola to the up, negative values for k translate the parabola to the down.  When k = 0, the reference and transformed parabola would be the same.

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Vertical translationequation example

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Fortunately, in general higher magnification lenses also have better resolution. In our lab a 10x objective has a resolution of 0.7 microns and a 100x objective has a resolution of 0.2 microns. One other tradeoff to consider: higher magnification lenses look at smaller fields of view, in proportion to their magnification. A 100x objective that sees a field of view of 100 x 100 micron^2 can be contrasted with a 10x objective looking at a 1000 x 1000 micron^2 field of view. So, when worrying about how good a microscope is, the most important question is what the resolution is. And in some science applications (such as my work) you care a lot about how well you can locate the centers of objects, and hopefully you can beat the resolution. Magnification is a much less useful specification (in my opinion). Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

The good news is, there's a difference between resolution and "ability to locate the position". If you have one tiny and isolated fluorescent object, you can often locate the position of that object to better than your resolution. The image of the object will show up as an extended blob, and you can find the "center of mass" of that blob-shaped image. If the blob is N pixels wide and each pixel is M microns across, you can estimate the center of the blob to about M/N accuracy, which often beats the optical resolution. This is a useful trick, but not solving the same problem as resolution. In some cases you can do various tricks to make the spot size bigger (increase N) so that you can locate the center even better. Various experiments I've heard of have claimed to be able to locate the centers of spots to within 10-30 nm using this sort of method. You may be interested in some software available for identifying particle positions, which implements this center-of-mass method. The magnification is something different altogether. There's a technical definition which compares the apparent angular size of the image, to the actual angular size of the object as it would appear if it were 25 cm away from your eye. This is a somewhat arbitary definition and in my opinion is mainly relevant for devising problems when I teach optics in my introductory physics classes. In real life, one often takes pictures using a CCD camera on a microscope, and projects them on a monitor. Using a larger monitor certainly can magnify the image further. But, it will still be just as blurry or sharp as the resolution. Fortunately, in general higher magnification lenses also have better resolution. In our lab a 10x objective has a resolution of 0.7 microns and a 100x objective has a resolution of 0.2 microns. One other tradeoff to consider: higher magnification lenses look at smaller fields of view, in proportion to their magnification. A 100x objective that sees a field of view of 100 x 100 micron^2 can be contrasted with a 10x objective looking at a 1000 x 1000 micron^2 field of view. So, when worrying about how good a microscope is, the most important question is what the resolution is. And in some science applications (such as my work) you care a lot about how well you can locate the centers of objects, and hopefully you can beat the resolution. Magnification is a much less useful specification (in my opinion). Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

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So, if k = 5, we say that the reference parabola is vertically translated by 5 units, and our equation for this would appear:

If you have one tiny and isolated fluorescent object, you can often locate the position of that object to better than your resolution. The image of the object will show up as an extended blob, and you can find the "center of mass" of that blob-shaped image. If the blob is N pixels wide and each pixel is M microns across, you can estimate the center of the blob to about M/N accuracy, which often beats the optical resolution. This is a useful trick, but not solving the same problem as resolution. In some cases you can do various tricks to make the spot size bigger (increase N) so that you can locate the center even better. Various experiments I've heard of have claimed to be able to locate the centers of spots to within 10-30 nm using this sort of method. You may be interested in some software available for identifying particle positions, which implements this center-of-mass method. The magnification is something different altogether. There's a technical definition which compares the apparent angular size of the image, to the actual angular size of the object as it would appear if it were 25 cm away from your eye. This is a somewhat arbitary definition and in my opinion is mainly relevant for devising problems when I teach optics in my introductory physics classes. In real life, one often takes pictures using a CCD camera on a microscope, and projects them on a monitor. Using a larger monitor certainly can magnify the image further. But, it will still be just as blurry or sharp as the resolution. Fortunately, in general higher magnification lenses also have better resolution. In our lab a 10x objective has a resolution of 0.7 microns and a 100x objective has a resolution of 0.2 microns. One other tradeoff to consider: higher magnification lenses look at smaller fields of view, in proportion to their magnification. A 100x objective that sees a field of view of 100 x 100 micron^2 can be contrasted with a 10x objective looking at a 1000 x 1000 micron^2 field of view. So, when worrying about how good a microscope is, the most important question is what the resolution is. And in some science applications (such as my work) you care a lot about how well you can locate the centers of objects, and hopefully you can beat the resolution. Magnification is a much less useful specification (in my opinion). Technical note More technically, a microscope objective's resolution is quantified by the Numerical Aperture. This webpage has great description of this as well as a more in-depth discussion of resolution. I'll note here that the wavelength of light you use makes a difference; shorter wavelengths improve the resolution. Links How a confocal microscope works This explanation was written by Eric Weeks Send me email: weeks@physics.emory.edu. Let me know if you have further questions, or if there are parts of this explanation that are confusing.

It is easy to see how this vertical translation moves the reference parabola up or down. Let us do a simple calculation that will get us from the reference parabola to the transformed, vertically translated parabola. We will work at x = 3 and make a vertical translation upward of 4 units. First, what would be the y-coordinate on the reference parabola at x = 3? Here's the calculation: