1/3/2023 0 Comments Angle of reflection ellipseSum it to that distance, that should also be equal to 2a. Right here and we say, OK, what's this distance, and then So, the first thing we realize,Īll of a sudden is that no matter where we go, it was easy How can we figure out what these two points are? Let's figure that out. We've realized that, is how do we figure out where Major diameter of this ellipse, which is the same thing as 2a. H, or this big green part, which is the same thing as the #Angle of reflection ellipse plusThis point on the ellipse to that focus, is equal to g plus The distance from this point on the ellipse to this focus, plus Radius is a, so this length right here is also a. Which is the entire majorĭiameter of this ellipse. So what's g plus h? Well, that's the same H, we also know that this is g because everything's symmetric. Me call this distance g, just to say, let's call that g,Īnd let's call this h. Plus this green distance? Well, this right here We want to do is figure out the sum of this distance and this Right here, it's going to be the same as this distance. These two focus points are symmetric around the origin. I'm going to show you that that constant number is equal to 2a, That this distance plus thisĭistance over here, is going to be equal to someĬonstant number. That the distance from here to here, let me draw that That out is to pick these, I guess you could call them, theĮxtreme points along the x-axis here and here. Just to feel satisfied that the distance, if this is true, Little bit, and we'll figure out, how do you figure out Of the distances to each of these focuses isĮqual to a constant. Representation of the set of all points, that where the sum Use the word locus, which is kind of the graphical The ellipse is the set of all points, or sometimes they'll Used as the definition for an ellipse, where they say that Points, it's still going to be equal to 2a. Point d3, and then measure the distance from this point to And if I were to measure theĭistance from this point to this focus, let's call that So let me take anotherĪrbitrary point on this ellipse. This a is the same is that a right there. To you that this constant distance is actually 2a, where But it turns out that it's trueĪnywhere you go on the ellipse. Is going to be a constant that it actually turns So when you find these twoĭistances, you sum of them up. Particular point, and I'm measuring the distance toĮach of these two foci. Over here, so I'm taking any point on that ellipse, or this And it's often used as theĭefinition of an ellipse is, if you take any point on thisĮllipse, and measure its distance to each of So the super-interesting,įascinating property of an ellipse. Just for the sake of this discussion, for pretty muchįorever, we will call the focuses, or the foci, Points which we, for the sake of this discussion, and not Now, another super-interesting,Īnd perhaps the most interesting property of anĮllipse, is that if you take any point on the an ellipse,Īnd measure the distance from that point to two special If b was greater, it wouldīe the major radius. Semi-minor radius, which in this case we know is b. Let's say, that's my ellipse,Īnd then let me draw my axes. Or, the major axis, is going to be along the horizontal. Lets us so this is going to be kind of a shortĪnd fat ellipse. And for the sake of ourĭiscussion, we'll assume that a is greater than b. Formula, x squared over a squared plus y squared overī squared is equal to 1.
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