Optics: 3) Spherical mirror

hell yeah! Here comes another post about optics. This time I will be writing about spherical mirror, its properties and how light behaves when it hits its surface.

On the left you can see the spherical mirror. Black line indicates the edge of the mirror while the other line is the axis of the mirror. Pic1, Optics 3

C is the center of the mirror. F is the focus and v is the top, or vortex.

Below you can see the light rays colored.

There are three main types of light rays hitting this kind of mirror.

There is the blue one which is called parallel ray. It is parallel to the axis and when it hits the surface of the mirror it is reflected toward focus. Actually when it is far away from the axis it may hit some different point (but that is not important right now).

Green one is called the vortex ray. It is coming to vortex where it is reflected in the same angle. On vortex, light behaves the same as when it is hitting normal flat mirror. The last and red one is focus ray, which comes through focus and is reflected in parallel with axis. The last one is actually reverted parallel ray.

Using these three rays you can get the place of the reflection. Of course in normal light there is tremendous amount of rays which than create an image.

Pic2, Optics3

On the next picture you can see the image of candle/flame reflected using these three light rays.
This kind of picture is turned upside down and when the candle is so far away it will be smaller. If it gets closer at one point, both sides will be the same and then at on point when the candle will be between focus and vortex, another thing happens.Pic3, Optics 3

So here you have the candle between focus and vortex. This means that there is no red ray/focus ray because it is not reflected  by the mirror. But your mind things that blue and green ,which should never meet, are going in straight line behind into the mirror which than creates bigger apparent picture.

So this is for the reflection basics now lets see the math behind.
This one on the left is rendering equation.
“a” is the distance of the object from the mirror.
“a'” is the distance of the reflection from the mirror.

“f” is the distance of vortex and focus. You can see that if “f” is bigger the fraction is smaller which means that the other side has to be smaller too so “a” and “a'” will get bigger too since the distances are bound to each other (and they are in denominator). If “a” is bigger than the reflection is closer to vortex which means that “a'” is smaller.

The last one is called something like transversal enlargement. From Z which is dimensionless you can know if the reflection is bigger or not because “y'” is the height of the reflection from the axis and “y” is the height of the real object. If Z is 1 then both real object and reflection are the same size. If Z is bigger than one than the reflection is bigger which hints for the last picture of reflection. If Z is smaller than 1 than the reflection is smaller.

Z can be also calculated using the distance from vortex:
This works similarly. If Z is smaller than 0 then the picture is reverted. And everything else works as well because it is actually the exact same equation.

Also you can get the place where the focus is when you are measuring something which is far away. This is because when it is far away almost all of the light rays are going parallel to the axis. When it is about 10 meters there is problem with your accuracy.


The pictures which were drawn by hand are made by me so I have all the right for them (actually feel free to use them using Creative Commons License)!

1st picture
Rendering equation
Transversal enlargement