Reflection property (1 Viewer)

[Drsoccerball]

Can someone tell me what it is and how you use it? Its in ellipses.

 

[glittergal96]

Can someone tell me what it is and how you use it? Its in ellipses.

If P is a point on an ellipse, then the normal to the ellipse at P bisects SPS', where S and S' are the focii.
So if you and I were standing at the focii of a room with an ellipsoidal roof, then you would heard things I said really loudly.

Its the elliptical analogue of a parabola reflecting straight lines parallel to its axis to straight lines passing through the focus.

You should try to prove this .


The uses of this property are abundant, you should definitely understand it.

 

[Ekman]

If P is a point on an ellipse, then the normal to the ellipse at P bisects SPS', where S and S' are the focii.
So if you and I were standing at the focii of a room with an ellipsoidal roof, then you would heard things I said really loudly.

Its the elliptical analogue of a parabola reflecting straight lines parallel to its axis to straight lines passing through the focus.

You should try to prove this .


The uses of this property are abundant, you should definitely understand it.

I believe there are questions that require you to prove ratios of lengths initially and using the sine rule, to prove the reflection property

 

[Drsoccerball]

If P is a point on an ellipse, then the normal to the ellipse at P bisects SPS', where S and S' are the focii.
So if you and I were standing at the focii of a room with an ellipsoidal roof, then you would heard things I said really loudly.

Its the elliptical analogue of a parabola reflecting straight lines parallel to its axis to straight lines passing through the focus.

You should try to prove this .


The uses of this property are abundant, you should definitely understand it.

So when can you determine when to use it because from the definition it doesn't sound too useful... What information can it give you?

 

[glittergal96]

So when can you determine when to use it because from the definition it doesn't sound too useful... What information can it give you?

How does it not sound useful? Reflection properties are literally the most practically applicable property of conics.

Keep it in mind whenever you are talking about conics and angles at the same time. Or even conics and lengths, due to trig stuff.

 

[leehuan]

At the very least the reflection property in the parabola, I feel is one of the biggest discoveries ever.

Entertainment use of reflection property - Elliptical pool table: https://www.youtube.com/watch?v=4KHCuXN2F3I

 

[porcupinetree]

Make sure you are aware of this property (as you obviously are) - it popped up in the 2011 HSC without an explanation for what it actually is. The BOS were allowed to do this because it is explicitly mentioned in the syllabus.

It can be proved myriad ways:
1. Using sine rule and then ratios of adjacent sides of triangles (the syllabus explicitly states that this style of proof is valid for HSC 4U)
2. Using perp distance formula from focus to tangent and using focus-directrix definition. This proof can be found in Terry Lee's book (as can the above proof)
3. Using the angle between two lines formula (tan theta = blah blah blah) - this is tedious.

My preferred method is the 2nd.

 

[braintic]

Make sure you are aware of this property (as you obviously are) - it popped up in the 2011 HSC without an explanation for what it actually is. The BOS were allowed to do this because it is explicitly mentioned in the syllabus.

It can be proved myriad ways:
1. Using sine rule and then ratios of adjacent sides of triangles (the syllabus explicitly states that this style of proof is valid for HSC 4U)
2. Using perp distance formula from focus to tangent and using focus-directrix definition. This proof can be found in Terry Lee's book (as can the above proof)
3. Using the angle between two lines formula (tan theta = blah blah blah) - this is tedious.

My preferred method is the 2nd.

You can also prove it using the triangle inequality - ala 2000 HSC Q7b

 

[InteGrand]

Basically, if you are standing in an elliptical room and are standing at one focus and someone else is at the other focus, if they whisper something, you can hear them like they were right next to you (assuming there isn't too much attenuation of the sound or absorption of it by the walls etc.).

I've also heard that if you had an elliptical room with walls made of mirrors, if you made a fire at one focus, and stood at the other focus, you would feel very hot indeed.

Another cool thing is that if you have an elliptical pool table with a hole in one focus and you put the ball at the other focus, no matter which direction you strike in, you will sink it (assuming there isn't too much friction or spin etc. etc... in a perfectly mathematical world it'd always work though! There's a Numberphile video on this:

)

Edit: lol, realised most of these were mentioned in this thread.

 

[Drsoccerball]

Dont you think the fire is kind of bs ??

 

[Drsoccerball]

Also how does this apply to hyperbola's ?

 

[InteGrand]

Also how does this apply to hyperbola's ?

https://www.math.hmc.edu/funfacts/ffiles/10001.2-3.shtml

Look at the right-hand picture at the top of this page for a pictorial representation: http://mathworld.wolfram.com/Hyperbola.html

 

[braintic]

Dont you think the fire is kind of bs ??

It depends. Firstly, I'm not sure how much heat from a fire is transmitted by radiation and how much is transmitted by convection. My suspicion is that at any reasonable distance from the fire, most heat comes from convection. Only radiated heat reflects (it is infrared radiation) - convected heat doesn't care too much for reflection laws.

The second issue is the properties of the mirror. Most mirrors are designed only to reflect visible light, and any reflection of longer or shorter wavelengths would be only incidental.

Interestingly, I found a reference to what are called "hot mirrors" and "cold mirrors".
The cold mirrors I saw are designed to reflect 90% of visible light (so a normal mirror in that respect), while transmitting 80% of infrared radiation.
But the hot mirrors are the exact opposite. They reflect 90% of infrared and transmit 80% of visible light. So visibly, it doesn't act like a mirror, but it will reflect any heat you radiate on it.

 

[Drsoccerball]

It depends. Firstly, I'm not sure how much heat from a fire is transmitted by radiation and how much is transmitted by convection. My suspicion is that at any reasonable distance from the fire, most heat comes from convection. Only radiated heat reflects (it is infrared radiation) - convected heat doesn't care too much for reflection laws.

The second issue is the properties of the mirror. Most mirrors are designed only to reflect visible light, and any reflection of longer or shorter wavelengths would be only incidental.

Interestingly, I found a reference to what are called "hot mirrors" and "cold mirrors".
The cold mirrors I saw are designed to reflect 90% of visible light (so a normal mirror in that respect), while transmitting 80% of infrared radiation.
But the hot mirrors are the exact opposite. They reflect 90% of infrared and transmit 80% of visible light. So visibly, it doesn't act like a mirror, but it will reflect any heat you radiate on it.

But wouldn't it be illogical to say that standing at focus would be hotter than standing closer than the distance from the fire to the focus? So if i stand right next to the fire vs standing at the focus

 

[Paradoxica]

But wouldn't it be illogical to say that standing at focus would be hotter than standing closer than the distance from the fire to the focus? So if i stand right next to the fire vs standing at the focus

If we assume the air cannot convect to you (an absurd assumption, but play along for now), then the only heat from the fire is radiation. Assuming we use perfectly reflective infrared mirrors (again, absurd), then (assuming we are in a mathematical universe), the radiation will perfectly converge onto the focus, heating the air to about the same temperature as the fire itself is (in terms of infrared heat). Convection wise, the fire is still hotter than the unoccupied focus.

Now, would anyone like to work in a frictionless vacuum? *puts fingertips together dr. doom style*
Or how about the infinite cartesian plane of uniform density? *MUAHAHAHAHAHA*

 

[braintic]

Restating that:

If you are in an ellipsoid (not just an elliptical prism), if no radiated heat is lost to convection or other forms of dissipation, if the fire is located at a focus, and you are located at the other focus, then every ray of infrared light from the fire would hit you. The only way you could achieve that by standing 'close' to the fire would be if the fire was inside you.

Further, if a short pulse of light was radiated from the focus, every reflected ray would reach you at the same time, because they all travel the same distance.