High school geometry
Sal introduces the four conic sections and shows how they are derived by intersecting planes with cones in certain ways. Created by Sal Khan.
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- Will two same parabolas, placed side to side ( like this <>) , make a perfect ellipse ?(29 votes)
- Nope. They will have a sharp angle in the joint. If you see, for them to fit, a parabola must be vertical at some point, which is actually not true, it gets always steeper but don't get vertival until reaching infinity.
However, If you considered possible to reach infinity, then placing them infinitely separated could make them actually fit, but I doubt that shape would form an ellipse. (By the way, all shapes are either ellipses or aren't ellipses at all. There is no such a thing as a perfect ellipse : )(55 votes)
- So these conic sections are 2D shapes that intersect a 3D cone? Are there 3 dimensional shapes that intersect 4 dimensional shapes?(31 votes)
- Camerxn, you're right that time is a temporal dimension, not a spatial one, although ANY method of distinguishment (color, musical note, shape, etc.) could be a dimension, however '4D' is considered to be 4 spatial dimensions (x axis, y axis, z axis, w axis).
The more dimensions (particularly spatial ones) the more fun you can have. You can define the foci for an ellipse by using a cone, two spheres and zero math. In four dimensions, I'm willing to bet that there are ways of defining 3-dimensional shapes in a similar manner (I can't think of any off the top of my head). For the cone, make a sphere just big enough to touch the desired ellipse at one point inside the cone, and the other sphere just small enogh to touch the same ellipse in a second point, nestled on top of the cone (think of an Ice cream cone), those two points are the foci.
Three dimensional knots wouldn't but four dimensional knots would (think of a bow knot with 3 bows instead of two, something like that I think).(8 votes)
- Can't you also get a triangle by going perpendicular to the base of the cone and going through the tip?(16 votes)
- Good catch! It's not quite a triangle, because the theoretical cones in this example have infinite height, but it is two intersecting straight lines. Other "weird" examples are a single point, a single straight line, and two parallel straight lines. (See if you can figure out how to get those!) All of these examples are called degenerate conic sections, and we will never speak of them again. ^_^(23 votes)
- Dose the Hyperbola ever touch the asmpotote?(8 votes)
- No, hyperbolas never reach the asymptotes, which is why they are called asymptotes. As the hyperbola gets further and further away from the center, the hyperbola approaches the asymptotes, but is unable to touch it.(14 votes)
- So I understand how the circle, ellipse, parabola, and the hyperbola are related, but I still don't understand what the meaning of conic sections is.(4 votes)
- To 'section' something is to cut it... like in biology class when you 'dissect' a frog.
Imagine a cone... an ice cream cone (with no ice cream) or one of those orange cones they put around utility vehicles in the street.
If you hold such a cone so its central axis is vertical, then cut it with a horizontal plane, the cut edge will be a circle. If you tilt the cutting plane a bit, but not so much that it is parallel to the outer edge of the cone and section the cone the cut edge will be an ellipse. If you section the cone with a plane that is parallel to the outer surface of the cone the cut edge will be a parabola and if you tilt the cutting plane past that point and on to vertical you will get a hyperbola.
So the 'conic sections' are literally the shapes you get when you section a cone.
See http://en.wikipedia.org/wiki/Conic_section for a lot more detail.(13 votes)
- At5:19, Sal draws a "cone", however it appears to be two cones on top of each other (tip to tip). Why is this?(4 votes)
- The correct term for the solid is "double-napped cone". Essentially two congruent cones with the same axis and a common vertex. Sal just didn't use the formal name for the solid.(7 votes)
- Can I differentiate or integrate conic section (circle, parabola, ellipse, hyperbola) equations?(5 votes)
- Yes, absolutely, if you want to learn partial differentiation early! Since there are two variables in these equations, you can take two separate derivatives. When you take the partial derivative of the function for the X variable, consider Y a constant (and vice versa). For example, given
x^2 + y^2then the partial derivative with respect to X would be just 2x + 0. The result is the change in the X direction when Y is not changing at all. This concept is built upon in Multivariable Calculus and is really fun :)(5 votes)
- I have a question about how Sal got the parabola form the cone: If both the plane and the cone go on infinitely, shouldn't the plane intercept the cone again and therefore close the parabola (and form an ellipse instead)? From my understanding, The only way the parabola could remain open is if the plane was parallel to the cone but translated just so it only meets half of the cone.(5 votes)
- > The only way the parabola could remain open is if the plane was parallel to the cone but translated just so it only meets half of the cone.
That's exactly what he does. At8:13:
> I'm drawing it in such a way that it only intersects this bottom cone as the surface of the plane is parallel to the side of this top cone.
If the plane were angled less than parallel, it would intersect the bottom cone twice and become an ellipse. If the plane were angled more than parallel, it would also intersect the top cone and become a hyperbola.(2 votes)
- Why only circle,ellipse,parabola and hyperbola be conic sections ,others can't?(5 votes)
Let's see if we can learn a thing or two about conic sections. So first of all, what are they and why are they called conic sections? Actually, you probably recognize a few of them already, and I'll write them out. They're the circle, the ellipse, the parabola, and the hyperbola. That's a p. Hyperbola. And you know what these are already. When I first learned conic sections, I was like, oh, I know what a circle is. I know what a parabola is. And I even know a little bit about ellipses and hyperbolas. Why on earth are they called conic sections? So to put things simply because they're the intersection of a plane and a cone. And I draw you that in a second. But just before I do that it probably makes sense to just draw them by themselves. And I'll switch colors. Circle, we all know what that is. Actually let me see if I can pick a thicker line for my circles. so a circle looks something like that. It's all the points that are equidistant from some center, and that distance that they all are that's the radius. So if this is r, and this is the center, the circle is all the points that are exactly r away from this center. We learned that early in our education what a circle is; it makes the world go round, literally. Ellipse in layman's terms is kind of a squished circle. It could look something like this. Let me do an ellipse in another color. So an ellipse could be like that. Could be like that. It's harder to draw using the tool I'm drawing, but it could also be tilted and rotated around. But this is a general sense. And actually, circles are a special case of an ellipse. It's an ellipse where it's not stretched in one dimension more than the other. It's kind of perfectly symmetric in every way. Parabola. You've learned that if you've taken algebra two and you probably have if you care about conic sections. But a parabola-- let me draw a line here to separate things. A parabola looks something like this, kind of a U shape and you know, the classic parabola. I won't go into the equations right now. Well, I will because you're probably familiar with it. y is equal to x squared. And then, you could shift it around and then you can even have a parabola that goes like this. That would be x is equal to y squared. You could rotate these things around, but I think you know the general shape of a parabola. We'll talk more about how do you graph it or how do you know what the interesting points on a parabola actually are. And then the last one, you might have seen this before, is a hyperbola. It almost looks like two parabolas, but not quite, because the curves look a little less U-ish and a little more open. But I'll explain what I mean by that. So a hyperbola usually looks something like this. So if these are the axes, then if I were to draw-- let me draw some asymptotes. I want to go right through the-- that's pretty good. These are asymptotes. Those aren't the actual hyperbola. But a hyperbola would look something like this. They get to be right here and they get really close to the asymptote. They get closer and closer to those blue lines like that and it happened on this side too. The graphs show up here and then they pop over and they show up there. This magenta could be one hyperbola; I haven't done true justice to it. Or another hyperbola could be on, you could kind of call it a vertical hyperbola. That's not the exact word, but it would look something like that where it's below the asymptote here. It's above the asymptote there. So this blue one would be one hyperbola and then the magenta one would be a different hyperbola. So those are the different graphs. So the one thing that I'm sure you're asking is why are they called conic sections? Why are they not called bolas or variations of circles or whatever? And in fact, wasn't even the relationship. It's pretty clear that circles and ellipses are somehow related. That an ellipse is just a squished circle. And maybe it even seems that parabolas and hyperbolas are somewhat related. This is a P once again. They both have bola in their name and they both kind of look like open U's. Although a hyperbola has two of these going and kind of opening in different directions, but they look related. But what is the connection behind all these? And that's frankly where the word conic comes from. So let me see if I can draw a three-dimensional cone. So this is a cone. That's the top. I could've used an ellipse for the top. Looks like that. Actually, it has no top. It would actually keep going on forever in that direction. I'm just kind of slicing it so you see that it's a cone. This could be the bottom part of it. So let's take different intersections of a plane with this cone and see if we can at least generate the different shapes that we talked about just now. So if we have a plane that goes directly-- I guess if you call this the axis of this three-dimensional cone, so this is the axis. So if we have a plane that's exactly perpendicular to that axis-- let's see if I can draw it in three dimensions. The plane would look something like this. So it would have a line. This is the front line that's closer to you and then they would have another line back here. That's close enough. And of course, you know these are infinite planes, so it goes off in every direction. If this plane is directly perpendicular to the axis of these and this is where the plane goes behind it. The intersection of this plane and this cone is going to look like this. We're looking at it from an angle, but if you were looking straight down, if you were listening here and you look at this plane-- if you were looking at it right above. If I were to just flip this over like this, so we're looking straight down on this plane, that intersection would be a circle. Now, if we take the plane and we tilt it down a little bit, so if instead of that we have a situation like this. Let me see if I can do it justice. We have a situation where it's-- whoops. Let me undo that. Edit. Undo. Where it's like this and has another side like this, and I connect them. So that's the plane. Now the intersection of this plane, which is now not orthogonal or it's not perpendicular to the axis of this three-dimensional cone. If you take the intersection of that plane and that cone-- and in future videos, and you don't do this in your algebra two class. But eventually we'll kind of do the three-dimensional intersection and prove that this is definitely the case. You definitely do get the equations, which I'll show you in the not too far future. This intersection would look something like this. I think you can visualize it right now. It would look something like this. And if you were to look straight down on this plane, if you were to look right above the plane, this would look something-- this figure I just drew in purple-- would look something like this. Well, I didn't draw it that well. It'd be an ellipse. You know what an ellipse looks like. And if I tilted it the other way, the ellipse would squeeze the other way. But that just gives you a general sense of why both of these are conic sections. Now something very interesting. If we keep tilting this plane, so if we tilt the plane so it's-- so let's say we're pivoting around that point. So now my plane-- let me see if I can do this. It's a good exercise in three-dimensional drawing. Let's say it looks something like this. I want to go through that point. So this is my three-dimensional plane. I'm drawing it in such a way that it only intersects this bottom cone and the surface of the plane is parallel to the side of this top cone. In this case the intersection of the plane and the cone is going to intersect right at that point. You can almost view that I'm pivoting around this point, at the intersection of this point and the plane and the cone. Well this now, the intersection, would look something like this. It would look like that. And it would keep going down. So if I were to draw it, it would look like this. If I was right above the plane, if I were to just draw the plane. And there you get your parabola. So that's interesting. If you keep kind of tilting-- if you start with a circle, tilt a little bit, you get an ellipse. You get kind of a more and more skewed ellipse. And at some point, the ellipse keeps getting more and more skewed like that. It kind of pops right when you become exactly parallel to the side of this top cone. And I'm doing it all very inexact right now, but I think I want to give you the intuition. It pops and it turns into a parabola. So you can kind of view a parabola-- there is this relationship. Parabola is what happens when one side of an ellipse pops open and you get this parabola. And then, if you keep tilting this plane, and I'll do it another color-- so it intersects both sides of the cone. Let me see if I can draw that. So if this is my new plane-- whoops. That's good enough. So if my plane looks like this-- I know it's very hard to read now-- and you wanted the intersection of this plane, this green plane and the cone-- I should probably redraw it all, but hopefully you're not getting overwhelmingly confused-- the intersection would look like this. It would intersect the bottom cone there and it would intersect the top cone over there. And then you would have something like this. This would be intersection of the plane and the bottom cone. And then up here would be the intersection of the plane and the top one. Remember, this plane goes off in every direction infinitely. So that's just a general sense of what the conic sections are and why frankly they're called conic sections. And let me know if this got confusing because maybe I'll do another video while I redraw it a little bit cleaner. Maybe I can find some kind of neat 3D application that can do it better than I can do it. This is kind of just the reason why they all are conic sections, and why they really are related to each other. And will do that a little more in depth mathematically in a few videos. But in the next video, now that you know what they are and why they're all called conic sections, I'll actually talk about the formulas about these and how do you recognize the formulas. And given a formula, how do you actually plot the graphs of these conic sections? See you in the next video.